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UNIVERSITATEA "BABES-BOLYAI" CLUJ-NAPOCA
Facultatea de Matematică şi Informatică
Catedra de Mecanică şi Astronomie
Sinteza finala a lucrarilor (2006-2008) Grant de excelenta pentru tineri cercetatori: cod 90
Titlul grantului: FENOMENE DE TRANSFER IN MEDII POROASE SI FLUIDE
VASCOASE CU PROPRIETATI FIZICE VARIABILE
Director de grant: Lect. dr. Teodor Groşan
S-au elaborat şi publicat, sau sunt în curs de publicare, următoarele lucrări ştinţiifice:
1. A.J. Chamkha, C. Bercea (casatorita Revnic) and I. Pop, Free convection flow over a truncated cone embedded in a porous medium saturated with pure or saline water at low temperatures, Mech. Res. Comm., Vol. 33, 433-440, 2006. (ISI)
2. T. Grosan, I. Pop, A note on the effect of radiation on free convection over a vertical flat
plate embedded in a non-Newtonian fluid saturated porous media, Int. J. App. Mech. Engng., vol 11, pp. 715-722, 2006.
3. M. Kumari, T. Grosan, I. Pop, Rotating flow of power-law fluids over a stretching
surface, Technische Mechanik, Vol. 26, pp.11-19, 2006. 4. T. Grosan, I. Pop, T. Mahmood, Thermal radiation effect on fully developed free
convection in a vertical rectangular duct, Studia Univ. Babes-Bolyai, Mathematica, vol. 51, pp. 117-128, 2006.
5. T. Grosan, I. Pop, Non-linear density variation effects on the fully developed mixed
convection flow in a vertical channel, Bulletin of Transilvania Universiy of Brasov, vol. 13, pp. 31-38, 2006.
6. C.Bercea (casatorita Revnic) and I. Pop: Forced convection boundary layer flow over a
flat plate with variable thermal conductivity embedded in a porous medium, Bulletin of the Transilvania Univerity of BRAŞOV, Romania, ser. B1, vol 13, pp 31-38, 2006.
7. M. Kumari, C. Bercea (casatorita Revnic), I. Pop, Mixed convection along a vertical
wavy surface with a discontinuous temperature profile in a porous medium, Proceedings of 3ICAPM 3rd International Conference on Applications of Porous Media, May 29–June 3, 2006, Marrakesch, Morocco, Paper number 4.
2
8. M.Kumari, C.Bercea (casatorita Revnic), I.Pop: Mixed convection flow along a thin vertical cylinder with localized cooling or heating in a porous medium, Int Jour Flud Mech Res, vol 34, 66-78, 2007.
9. T. Grosan, I. Pop, Thermal radiation effect on fully developed mixed convection flow in
a vertical channel, Technische Mechanik, Vol. 27, pp. 37-47, 2007.
10. M. Kumari, C. Bercea (casatorita Revnic), I. Pop: Effect of non-uniform suction or injection on mixed convection boundary layer flow over a vertical permeable cylinder embedded in a porous medium, Malayasian J. Mathematical Sci., 1(2): 31-42 (2007)
11. M. Kumari, T. Grosan, I. Pop, Boundary layers growth on a moving surface due to an
impulsive motion and sudden increase of wall heat flux, Int. J. Appl. Mech. Engng., Vol. 13, pp.203-215, 2008.
12. C. Bercea (casatorita Revnic), T. Grosan, I. Pop,. Heat Transfer in Axisymmetric
Stagnation Flow on a Thin Cylinder, Studia Univ. Babes-Bolyai, Mathematica, vol. 53, pp. 119-132, 2008
13. Ş. M. Şoltuz, Teodor Grosan, Data Dependence for Ishikawa Iteration When Dealing with Contractive-Like Operators, Fixed Point Theory and Applications, Vol. 2008, Article ID 242916, 7 pages.( doi:10.1155/2008/242916).
14. T. Grosan, I. Pop and S.R. Pop, Radiation and variable viscosity effects in forced
convection from a horizontal plate embedded in a porous medium, Stuida Univ. Babes-Bolyai, Mathematica, (in curs de publicare).
15. C. Revnic, T. Grosan, J. Merkin and I. Pop, Mixed convection near an axisymmetric
stagnation point on a vertical cylinder, Journal of Engineering Mathematics (acceptata) 16. A. Postelnicu, T. Grosan and I. Pop, Brinkman Flow of a Viscous Fluid Through a
Spherical Porous Medium Embedded in Another Porous Medium, Transport in Porous Media (sent for publication)
17. T. Grosan, C. Revnic, I. Pop, and D.B. Ingham, Magnetohydrodynamics oblique
stagnation-point flow, Meccanica (sent for publication) 18. C.Revnic, T.Grosan and I.Pop: Unsteady boundary layer flow and heat transfer over a
stretching sheet, International Conference of Numerical Analysis and Applied Mathematics (ICNAAM), Corfu, Grecia, 16-20 September 2007. In: American Institut of Physics (AIP) – Conferences Proceedings, November 2007(in press)
19. T. Grosan, C. Revnic, I. Pop, D.B. Ingham, Magnetic field and internal heat generation
effects on the free convection in a rectangular cavity filled with a porous medium, Int. J. Heat Mass Transfer, (trimisa spre publicare)
Ca rezultat al grantului s-au publicat sau au fost trimise spre publicare un număr de 17 lucrari :
3
Juranale ISI : 2 lucrari publicată (1, 13)
2 lucrări acceptate (15,18)
3 lucrări trimise spre publicare (16, 17, 19)
Jurnale din străinătate:
6 lucrari publicate (2,3,7,8,9,10,11)
Jurnale din ţară:
4 lucrări publicate (4,5,6,12)
o lucrare acceptată (14)
De asemenea membrii echipei de cercetare au participat la conferinţe în ţară (4 conferinţe) şi
străinătate (o conferinţă) şi la stagii de cercetare în străinătate (2 stagii).
S-au elaborat program numerice în Matlab şi Mathematica pentru rezolvarea sistemelor de
ecuaţii diferenţiale şi ecuaţii cu derivate parţiale.
Lucrările ISI au fost adăugate în extenso în anexă, pentru celelalte lucrari fiind adăugat
abstractul.
Cluj-Napoca, 30 Iulie 2008
Director de proiect,
Lect. dr. Teodor Groşan
Anexa I
Articole gi preprinturi publicate sau trimise spre publicare tnjurnale indexate ISI
ELSEVIER
Avai lable onl ine at www,sciencedirect ,com
a c r E N c = f , l o r R E c r '\/
Mechanics Research Communicat ions 33 (2006) 433-440
MECHANICSR E S E A R C H C O M M U N I C A T I O N S
www.elsevier .com/locate/nrechrescom
Free convection flow over a truncated cone embeddedin a porous medium saturated with pure
or saline water at low temperatures
A.J. Chamkha u, C. Bercea b, I . Pop b'*
. 'Manu[uct t t r i t lgEngtneer ingDepur|mer l l 'ThePu| l l i t .ALt thor i t1, ' / t l rAppl iedEdttcat ih Fctcultl, oJ Muthentcttits, IJniuersitt' o/ Cluj, R-3400 Cluj, CP 253. Rotttttrtier
Ava i iab le on l ine l7 November 2005
Abstract
Steady free convection boundary layer about a truncated cone enbedded in a porous medium satLlrated with pure or
sal ine water at iow temperatnres has been studied in this paper. The goveming cottpled part ial dif ferentizr l eqr' tztt ions are
solved numerical ly usir.rg a very eff icient f ini te-dif fet 'ence method. Several new paramelet 's arise and the reslt l ts are given for
some specif ic values of t l .rese parameters. The obtained results lor a Boussinesq f luid are conrpared with known results from
Ihc open l i te la tu re and i1 i s shown tha t the agreenrent be tween these resu l ts i s very good.
e l ( )a5 E. lsev ie r L td . A l l r i sh ts lese lved.
Kertrord.s; Tntncated conel Bortndary layerl Po: 'otts nredir in; Cold or sal ine water
l lntroduction
Convective flow in porolts media has been a subject of great interest for the last several decades due to its
nLlmerous thermal engineer ing appl icat ions in var ior- rs d isc ip l ines, such as geophysical thermal insulat ion,
n-rodeling of packed sphere beds, cooling of electronic systems, groundwater hydrology, petroleum teservorrs,
coal combustors, ground watef pol lu t ion, ceramic processes, to name just a few of these appl icat ions. Sonre of
rhe most important ar- ra ly t ica l , numer ical and exper imenta l s tudies wi th such appl icat ions, which present the
current s tate-of - the,ar t in the area of convect ive heat t ransfer in porous media, have been gathered in the
r l r onog raphs by N ie ld and Be jan (1999 ) . I ngham and Pop (1998 .2002 ) , Va fa i ( 2000 ) . Pop and Ing l t an t
(2001 ) . and Be jan and K raus (2003 ) .
Str - rd ies of convect ive heat t rar . rs ler in porous media have been cal r ied out in the past us ing the Bor. rss iuescl
approximat ion, namely the f lLr id densi ty p var ies l inear ly wi th temperatut 'e . However. th is is inappropr iate for
$ 'ater at low temperatures because of the extrern l lm at about 4 'C in p l l re $ 'ater t r t I a tm. Sucl i condi t ions
occLl r commor-r ly in porous nedium, such as permeable soi ls f looded by cold lake o l sea watef . wutet ' - ice
- C*ponAing authot ' . Tel . : *40 264 5943 l5; lax; *40 164 591906E-tnui I uddra. t . t . popi l i lmath.ubbclu j . ro ( l . Pop).
0093-6,113/Si - see f ront mat ler O 2005 Elsevier Ltd. Al l r ights reserved.
c l o i : 10 . I 0 l 6 / 1 .nech rescom.2005 . I 0 . 001
434 A.J. Chantkhq et ul. I Methanics Reseqrch Contntuni.cutions 33 (2006)4-13-440
slurries, etc. A l imited nunber of studies have been devoted in the past to the problem of convective boundary
layer adjacent to heated or cooled bodies immersed in a polous medium saturated with cold water wherein a
density extremum may arise. It sl-rould be mentioned that the buoyancy flow with an extremum may become
very complicated, with local f low reversals and convective inversions. Density differences may then not be
expressed as a l inear funct ion of the temperature. Rami l ison and Gebhart (1980) examined the possib le s im-
ilarity solutions for vertical, buoyancy induced flow in a porolrs medium saturated with cold water. Lin and
Gebhart (1986) have considered the corresponding case of a hor izonta l sur face in a porous medium saturated
wi th cold or sa l ine water . Gebhart et a l . (1983) obta ined mul t ip le s teady state solut ions for the problem con-
s idered by L in and Gebhart (1986) us ing two numer ical codes. A rev iew of the convect ive f low in the v ic in i ty
of the maximum-densi ty condi t ion in wate l at low temperatures, a long wi th re levant c i ta t ions. is avai lable i r l
the survey by Kukula et a l . (1987).
Tl-re present paper concerns the steady free convection boundary layer adjacent to a heated trttucated cone
embedded in an extensive porous medium satr-rrated with either pllre or saline water under the conditions in
which a densi ty ext reml ln might occur . The densi ty s tate equat ion used here is that proposed by Gebhart anci
Mol le ldor f (1977), which has been shown to be very accl r rate for both pure and sal ine water to a pressr . r re level
of 1000 bars up to 20 "C, and to 40 '% sal in i ty . To the best of our knowledge, th is problem has not been con-
sidered before. However, Yih (1999) made an analysis for free convection boundary layer about a truncated
cone in a porous medium saturated with a Boussinesq fluid subjected to the coupled effects of thermal and
mass d i f fus ion.
2. Basic equations
Consider the steady free convection over a trl lncated cone (with half angle 7) embedded in a saturated por-
ous medium f i l led wi th pure or sa l ine water . I t is assumed that the sur face of the t runcated cone is mainta ined
at the constant temperature I* , whi le the temperature of the ambient f fu id is Z- , where I*> ' f - . F ig l
shows the f low model and physical coordinate system. The governi r . rg boundary la tyer eqLrat io t ts are g ive l t
by, see Chamkha et a l . (2004),
a a^ 1r ' r i ) - t ^ ( r r ' ) : Qcx crv
t t - Pn' !K,,4 - , , , , ,u - l r - - rn, l" ] cos i ,p '
aT aT a2Tl t - L n -- *nl ^ .
0x ay oY'
subject to the boundary conditions
( t )
( 2 )
( 3 )
-l (
Fig. l . Physrcal model and coordl l la te syslenl
A.J. Chant l ihu et u l . I Mer. l tunrc.s l leseurcl t ContntuniLcr t iou.s 33 (2006t1-13 410
r ) : 0 , T : T * o n , y : O , 0 . ,
T : T- as -v - cc
where r -,xsini, ir and.), are the streamwise and transverse Cartesian coordinates, respectively, u and u are thevelocity components in the r and ,tr directiol-rs. respectively. I is the fluid temperature, K is the permeabil ity oft hepo rousmed ium,g i s themagn i t l l deo l t l - r eg rav i t a t i ona l acce le ra t i on , p , ! andqn ra re thedens i t y , v i scos i t yand effective thermal diffusivity of the porous medium. The new density equation, which applies to both pureand saline water is given by
p : p , . ( s , p ) [ l - B , , , ( s . p ) l I - I , ( r . p ) l u ]
where p is the pressure, s is the sal ini ty and p,, , and I . denote the maximum density and temperature. respec-t ively. for given pressure and sal ini ty levels. The forms and values of c1, pn, p, , , and f . , are given in the paper 'by Gebhz i r t and Mol lendor f (1977) .
We now introduce the fol lowing new variables:
( : x * l xa , x - : ( j r - xa) lxo , a : na t l lQ lx -1
,1 , : t ^ rRat l t . f ( ( , , t ) , 0 ( ( ,n ) : ( r - r * ) l (T* - I * ) lo l
where / is the stream funct ion which is def ined in the usual way as u:( | l r )Al . / lOy and u: -( l l r )dr l t ldx,respect ively and Ra".:9, , ,9K0,, ,)Tn, - I - l "x 'cos 7/trran, is the modif ied local Rayleigh number. Subst i tut ing(6) in to Eqs . (1 ) - (3 ) we ge t
/ ' ' : 0 - R l q - 1 P ; r
/ 1 : \ / A 0 a / \( / ' ' + ( ;+,= ) r o ' : <( t '#- r /+ )\ l | - ; / \ d i c c t
sr,rbject to the bor"rndary conditions (4), which become
/ ( ( , 0 ) : 0 , 0 ( i , 0 ) : 1 , 0 ( i , c c ) : Q
u'here primes denote partial differentiation with respect to 4 and the parameter R is defined as
p -T - T
T _ T
It is worth mentioning that the parameter R places the prescribed tenrperatr.rre I* and 7- with respect toI,.(s,p). It also indicates the local direction of the buoyancy force across the thermal region and thus, alsothe d i rect ion of f low (see Rami l ison ancl Gebhart , 1980).
In terms of the new variables, the velocity components in r- and .y-directions are given by
y : ( a , .Ra , . f x . ) J
, : _ ( z - ,Ra t , . 1 " , ) [ ( ! * : \ , _o ! _ ! r , , ]' \ -n ' " - ' ' " )L \ t - l - t / ' - *e i - t " ' )
We are a lso in terested in the local Nr"rssel t number, which is g iven by
Nu , . I Ra t i l : - 0 ' ( i , 0 )
( i l )
( 1 2 )
I t i s wo r th men t i on ing to t h i s end tha t Eqs . (7 ) and (8 ) become s im i l a r f o r ( : 0 and ( : cc and theydescr ibe the l ree convect ion over a ver t ica l f la t p late and, respect ive ly . over a f r . r l l cone embedded in a porousmedir-rm saturated with cold water.
3. Results and discussion
Eqs. (7) ancl 18) subject to the boundary condi t ions (9) have been solved numer ical ly for some val r " res of thetemperat l l re paraneter R in the range between -10 and 0.194 at some Llpstream coordinates i : 0 .0- l ,0 us ingthe f inr te-d i f fe1slss 56 ' l renre , leveln^pr l h . , B lot tner . (1970). The values of 17 gsed are q: l , 1 .121 141 and
4 3 5
( 5 )
( 7 )
( 8 )
(e)
( l 0 )
436 A.J. Chuntkhu et u l . I Meclrunics Rescurt l t Contntunical . ion.s 33 (2006)433-440
Table IConrpar ison of the valLres oi -0 ' ( ( ,0)
C h e n g e t . r l . ( 1 9 8 5 ) Y i h ( i 9 9 9 ) PIesen(
0 . 51 . 02 . 06 , 01 0 . 020.010.0
0.44370 . 5 4 1 20.59910.65720.72 t90 . 7 3 9 10 . 7 5 3 20.7 6070.768 5
0.44390 . 5 2 8 50 ,58070.63 7 30 7 1 2 30 , 7 3 3 00,75000.15920.768 6
0.44440.52940 . 5 8 1 20.63990 . 7 r 3 00.7 3 360.15070.75960.7 690
1.894816 (cold water approximat ion) a lso considered by Rami l ison and Gebhart (1980). I t shor-r ld be men-t ioned that c lose to R:0.194 the convefgence of the numer ical soh-r t ion is very s low. This is expected dueto the occurrence of the flow reversal across the convective layer. A conpal' ison of the present resll l ts forthe local Nussel t number, -0 ' ( i ,0) , wi th those repor ted by Cheng et a l . (1985) and Yih ( 1999) is g iven i r r TableI for R:0 and 4: I (c lass ical Boussinesq approximat ion) , and some values of the st reamwise parameter i . I tcan be seen fron-) Table I and Fig. 8 that the present results are in excellent agreement with those of Chenget a l .11985) and Yih (1999), and we are, therefore, conf ident that the present numer ical resul ts are veryaccurate.
The non-dimensional velocity/((,4) and non-dimensional temperatLrre 0((,4) profl les are shown in Figs. 2-8. Also, the var iat ion of the localNr-rssel t nnmber g iven by Eq. (12) is p lot ted in F igs. 9 and 10. I t is seen f ronFigs. 2 and 3 that for a fixed value of q and i. the velocity profi les increase, while the temperatl lre profi lesdecrease as the parameter R decreases from zero, The same happens for these profi les when q iucreases fora f ixed value of R and i as can be seen l rom Figs. 4 and 5. However, F igs. 6 and 7 show that both the veloc i tyand temperature prof i les decrease zrs the st reamwise coordinate ( increase f ion ( :0 to q : cc ( fu l l cone). Thepresent resr . r l ts are a lso compared in F ig. 8 wi th those of Rami l ison anci Geblrar t (1980) lor the correspondingproblem of a ver t ica l f fa t p late ( i :0) embedded in a porous medium saturated wi th cold water . An excel lentagreement between these resr.rlts can be again noticed. We notice tl-rai a small f low reversal occLlrs for the val-r les of R in the range 0.1 < R < 0.194 that conf i rms the f ind ings of Rami l ison and Cebhart (1980t . Fr" r r ther .F igs. 9 ar- rd l0 show that the values of the local Nussel t numbel increases wi th a declease f rom zelo of thetemperatl lre parameter R and with the increase of the parameter 17. The valiation of the heat transfer is verylarge over the whole range of R lor the value o l rT considered. However, the var iat ion of the local Nussel t num-
ber is a lmost l inear wi th .
0 . 5 1 . 0 1 . 5
I
Fig. 2. Ef fects o l -R on the tarrgent ia l veloci ty prof i les
R=0 . - 1 . - 1 . - 5 . - l t )
0 . 0
Chunt l ;ho et u l , I Met l tani t . t Re:eurth (ontntuni tcr t ion. : 33 t )006 I 43-1.440
rl
Fig. 3. Effecrs of R on t lre tenrperatule pr-ohles
t 3tl
F ig, 4. Ef fects of 4 on the tangent ia l veloci ty prof i les
rl
Fig. 5. Efiects of q on the tempefl l ture prof i les
q = 1 . 0 . 1 . 7 2 7 l 4 l , 1 . 8 9 . 1 8 1 6
q= | .0 , L717 1 ;17 . L i 9 . 8 I6
4 1 8 A.J Chttnt l ihu ct t r l I Mcchtrn i r ' : Rasatrr t l t Contnuniut iot t . \ l3 t2006 ) 1-13"414
6
5
'1
?
I
0
0.5 1 .0 l . -5 2.0 2.5
rl
F ig. 6. Developrr lent o l the tangent ia l veloci ty p lof i les
3 .0
0 .75
o 0 .50
0.25
0 0 00 . 5 1 . 0 1 . 5 2 . 0 ? . 5
rl
F ig. 7. Deve)opnrent of the ter l lpef t l ture prof i les.
0 .9
0 .8
0 .1
0 .6
0 .5
! t i +
0 .1
0 .1
0 . I
0 . 0
-0. I0
F i g . 8
. 1 3 l l l o
rl
Reserved f lou' corrditrons l 'ot ' variolts values ot ' R
q = 1 . 8 9 1 8 1 6
R = 1 . 0
i =0 .0 .5 .1 .0 . - 5 .0 . 10 . oo
q = l 8 9 ' 1 8 1 6
R=- l 0
0 . 0 .5 .1 . 0 , 5 . 0 .10 . oo
q = 1 . 8 9 4 8 1 6
- Pfe\ent wofk. i=t)o Ihnr i l i son and Cebhan l6 l
R = ( J . 1 . ( r 1 5 . 0 L 9 - 1
9 = 1 . 8 9 ' 1 8 1 6
A.J. C' l r rnt l r l rcr ct u l . I McrhuniLs Re.surr th Crt t t t r t t t r r t i r t t t ior t . \ 3J 12006)433-140
i
F r , ' 9 F " I e c r . o l R c r r t l r c l , ' e i r l \ r r . s e l t r t u n t b e t
C
t r i o l r I l : e , r . o f , r ' r r r h r . l o r ' : r l \ L r t s e l t I i L l l t t b e f
4. Conclusions
The f ree convect io l . r boundary layer f low over a heated ver t ic i l l t r l lncated cone embedded in a porous med-
ium str tur"ated wi th cold watef wherein ?t densi ty ext remul l may ar ise is invest igzr ted. Numel ica l invest igat io t ls
sLrpported by an exact analys is wi th the f in i te-d i f ference are made tbr an isothermal s l l r face and ovef a wide
fange of rhe te l rperat l l re parameter R and three values of the exponent q in der- rs i ty Eq. (5) ' Two parameters
R, and q arise ancl they derelnrine the fi indamental l.ratlrre of t l-re density field and the effects of t l-re pressr:re
and sal in i ty levels , r 'espect ive ly . The convent ional f iee convect ion approxi luat ion is a lso inc luded in the pfesel l t
fornr- r le t t ior . r by choosing 4: I i r - r those l lows for which R:0 (c lerss ical Boussinesq approxin lat ion) . I t is
shoq,r . r that the ef f 'ect o1 'q on heat t rar . rs l 'er is great when lR is h igh. and i t increases w' i t l . t an incfease of iR .
References
Be j rLn . A . . K l aus . A .D . (Eds . ) . 1001 . Hca t T rans le l Handboo l ' Wr [ ' \ . Ne \ \ \ ' ( ) r l (
B l o t t n e r ' . F C . . 1 9 7 0 A I A A J . 8 . 1 9 3 .
C h a n r k h a . A . J . . B e l c e a . C . . P o p . 1 . . 1 0 0 4 , I n t . . l A p p l . M a t h E n g . 9 . 2 7 3 .
Cheng . P . Le , T .T . . Pop . 1 . , 1985 . I t i t . Co t r r t l r ' t n . Hea l Mass T rans ie r l l . 717 .
Gebha l t . B , , N o l l endo l f ' . J .C . . 1977 . Deep Sea Res . 14 . 831
Gebha r t . B . . Hassa rd . B . . Has t rngs , S .P . . Kaza l i no f f , N . . l 9 t t 3 . NL tue t . F Iea t T l ans lb r 6 . J l 7
I ngha : . 1 . D .8 . . Pop . l . (Eds . ) . 1998 . T ranspo l t Phcnon rena r r Po rous Mec l i a . Pe t ' gamon , Ox ib l d . vo l . I I . 2O0 l .
1 .21 .0Llt1 .61..11 .2
r ^ 1 . 8< " 1 .6
1 . 00 .80 .60.,10 .2
r.tt
1 . 6
t : 1
1 . 0
0,u
U . b
0.:l
=Z
q = l . 7 2 7 l a l
440 A.J. C' l turnl i l ta at u l . I NI t thunic. : Rc.veurt l t CotnutLrni tu l ior t .v 3- ] (2006 ) 133"140
Kr . r k r , r l a . D . J . , Gebha r t . B . . Mo i l endo r l ' , J .C . . 1987 Adv . Hea t T l ans fe r 18 .325
L i n . D . - S . , C e b h a r t , B . , 1 9 8 6 . l n t . J . H e a t M a s s T r a n s l e r - 2 9 , 6 l l
N ie l d . D .A . , Be jan . A . . 1999 . Convec t i on i n Po rous Med ia . l nd ed . Sp r t nge r . Neu 'Yo l l < .
P o p . l ' . [ n g l r a n r . D ' B ' . 2 0 0 i . C o n v e c 1 i v e H e a t T l a r l s | e t . : C o r - r r p r t t a t i o r r a l a t l d M a t l r e t r r l r tMed ia . Pe rgamon , Ox fo rd .
R a m r l i s o n . J . M , . G e b h a l t . B . . 1 9 8 0 . I n t . J . H e a t M a s s T r a n s l ' e t l 3 , l 5 l l .
Va la i . l ( . (Ed , ) , 1000 . Handbook o1 'Po :o r . r s Med ia . Ma rce l De l t ke r ' . Ne r l 'Yo r l < .
Y i h . K . A . . 1 9 9 9 . A c t a M e c l r . 1 i 7 . 8 3 .
Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2008, Article ID 242916, 7 pagesdoi:10.1155/2008/242916
Research ArticleData Dependence for Ishikawa Iteration WhenDealing with Contractive-Like Operators
S. M. Soltuz1, 2 and Teodor Grosan3
1Departamento de Matematicas, Universidad de los Andes, Carrera 1 No. 18A-10, Bogota, Colombia2 The Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania3Department of Applied Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Correspondence should be addressed to Teodor Grosan, [email protected]
Received 13 February 2008; Accepted 27 May 2008
Recommended by Hichem Ben-El-Mechaiekh
We prove a convergence result and a data dependence for Ishikawa iteration when applied tocontraction-like operators. An example is given, in which instead of computing the fixed point ofan operator, we approximate the operator with a contractive-like one. For which it is possible tocompute the fixed point, and therefore to approximate the fixed point of the initial operator.
Copyright q 2008 S. M. Soltuz and T. Grosan. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.
1. Introduction
Let X be a real Banach space; let B ⊂ X be a nonempty convex closed and bounded set. LetT, S : B → B be two maps. For a given x0, u0 ∈ B, we consider the Ishikawa iteration (see [1])for T and S:
xn+1 =(1 − αn
)xn + αnTyn, yn =
(1 − βn
)xn + βnTxn, (1.1)
un+1 =(1 − αn
)un + αnSvn, vn =
(1 − βn
)un + βnSun, (1.2)
where αn ⊂ (0, 1), βn ⊂ 0, 1), and
limn→∞
αn = limn→∞
βn = 0,∞∑
n=1
αn = ∞. (1.3)
Set βn = 0, ∀n ∈ N, to obtain the Mann iteration, see [2].The map T is called Kannan mappings, see [3], if there exists b ∈ (0, 1/2) such that for
all x, y ∈ B,
‖Tx − Ty‖ ≤ b(‖x − Tx‖ + ‖y − Ty‖). (1.4)
2 Fixed Point Theory and Applications
Similar mappings are Chatterjea mappings, see [4], for which there exists c ∈ (0, 1/2)such that for all x, y ∈ B,
‖Tx − Ty‖ ≤ c(‖x − Ty‖ + ‖y − Tx‖). (1.5)
Zamfirescu collected these classes. He introduced the following definition, see [5].
Definition 1.1 (see [5, 6]). The operator T : X→X satisfies condition Z (Zamfirescu condition)if and only if there exist the real numbers a, b, c satisfying 0 < a < 1, 0 < b, c < 1/2 such thatfor each pair x, y in X, at least one condition is true:
(i) (z1) ‖Tx − Ty‖ ≤ a ‖x − y‖,(ii) (z2) ‖Tx − Ty‖ ≤ b (‖x − Tx‖ + ‖y − Ty‖),(iii) (z3) ‖Tx − Ty‖ ≤ c (‖x − Ty‖ + ‖y − Tx‖).
It is known, see Rhoades [7], that (z1), (z2), and (z3) are independent conditions.Consider x, y ∈ B. Since T satisfies condition Z, at least one of the conditions from (z1), (z2),and (z3) is satisfied. If (z2) holds, then
‖Tx − Ty‖ ≤ b(‖x − Tx‖ + ‖y − Ty‖) ≤ b
(‖x − Tx‖ + (‖y − x‖ + ‖x − Tx‖ + ‖Tx − Ty‖)).(1.6)
Thus
(1 − b)‖Tx − Ty‖ ≤ b‖x − y‖ + 2b‖x − Tx‖. (1.7)
From 0 ≤ b < 1 one obtains,
‖Tx − Ty‖ ≤ b
1 − b‖x − y‖ + 2b
1 − b‖x − Tx‖. (1.8)
If (z3) holds, then one gets
‖Tx − Ty‖ ≤ c(‖x − Ty‖ + ‖y − Tx‖) ≤ c
(‖x − Tx‖ + ‖Tx − Ty‖ + ‖x − y‖ + ‖x − Tx‖) (1.9)
Hence,
(1 − c)‖Tx − Ty‖ ≤ c‖x − y‖ + 2c‖x − Tx‖, (1.10)
that is,
‖Tx − Ty‖ ≤ c
1 − c‖x − y‖ + 2c
1 − c‖x − Tx‖. (1.11)
Denote
δ := maxa,
b
1 − b,
c
1 − c
, (1.12)
S. M. Soltuz and T. Grosan 3
to obtain
0 ≤ δ < 1. (1.13)
Finally, we get
‖Tx − Ty‖ ≤ δ‖x − y‖ + 2δ‖x − Tx‖, ∀x, y ∈ B. (1.14)
Formula (1.14)was obtained as in [8].Osilike and Udomene introduced in [9] a more general definition of a quasicontractive
operator; they considered the operator for which there exists L ≥ 0 and q ∈ (0, 1) such that
‖Tx − Ty‖ ≤ q‖x − y‖ + L‖x − Tx‖, ∀x, y ∈ B. (1.15)
Imoru and Olatinwo considered in [10], the following general definition. Because theyfailed to name them, we will call them here contractive-like operators.
Definition 1.2. One calls contractive-like the operator T if there exist a constant q ∈ 0, 1) and astrictly increasing and continuous function φ : [0,∞)→[0,∞) with φ(0) = 0 such that for eachx, y ∈ X,
‖Tx − Ty‖ ≤ q‖x − y‖ + φ(‖x − Tx‖). (1.16)
In both papers [9, 10], the T -stability of Picard and Mann iterations was studied.
2. Preliminaries
The data dependence abounds in literature of fixed point theory when dealing with Picard-Banach iteration, but is quasi-inexistent when dealing with Mann-Ishikawa iteration. As faras we know, the only data-dependence result concerning Mann-Ishikawa iteration is in [11].There, the data dependence of Ishikawa iteration was proven when applied to contractions.In this note, we will prove data-dependence results for Ishikawa iteration when appliedto the above contractive-like operators. Usually, Ishikawa iteration is more complicated butnevertheless more stable as Mann iteration. There is a classic example, see [12], in whichMann iteration does not converge while Ishikawa iteration does. This is the main reason forconsidering Ishikawa iteration in Theorem 3.2.
The following remark is obvious by using the inequality (1 − x) ≤ exp(x), ∀x ≥ 0.
Remark 2.1. Let θn be a nonnegative sequence such that θn ∈ (0, 1], ∀n ∈ N. If∑∞
n=1θn = ∞, then∏∞n=1(1 − θn) = 0.
The following is similar to lemma from [13]. (Note that another proof for this lemma[13] can be found in [11].)
Lemma 2.2. Let an be a nonnegative sequence for which one supposes there exists n0 ∈ N, such thatfor all n ≥ n0 one has satisfied the following inequality:
an+1 ≤(1 − λn
)an + λnσn, (2.1)
where λn ∈ (0, 1), ∀n ∈ N,∑∞
n=1λn = ∞, and σn ≥ 0 ∀n ∈ N. Then,
0 ≤ limn→∞
supan ≤ limn→∞
supσn. (2.2)
4 Fixed Point Theory and Applications
Proof. There exists n1 ∈ N such that σn ≤ lim supσn, ∀n ≥ n1. Set n2 = maxn0, n1 such thatthe following inequality holds, for all n ≥ n2:
an+1 ≤(1 − λn
)(1 − λn−1
) · · · (1 − λn1
)an1 + lim
n→∞supσn. (2.3)
Using the above Remark 2.1 with θn = λn, we get the conclusion. In order to prove (2.3),consider (2.1) and the induction step:
an+2 ≤(1 − λn+1
)an+1 + λn+1σn+1 ≤
(1 − λn+1
)(1 − λn
)(1 − λn−1
) · · · (1 − λn1
)an1
+(1 − λn+1
)limn→∞
supσn + λn+1σn+1
=(1 − λn+1
)(1 − λn
)(1 − λn−1
) · · · (1 − λn1
)an1 + lim
n→∞supσn.
(2.4)
3. Main results
Theorem 3.1. Let X be a real Banach space, B ⊂ X a nonempty convex and closed set, and T : B→B acontractive-like map with x∗ being the fixed point. Then for all x0 ∈ B, the iteration (1.1) converges tothe unique fixed point of T.
Proof. The uniqueness comes from (1.16); supposing we have two fixed points x∗ and y∗, weget
‖x∗ − y∗‖ = ‖Tx∗ − Ty∗‖ ≤ q‖x∗ − y∗‖ + φ(‖x∗ − Tx∗‖) = q‖x∗ − y∗‖, (3.1)
that is, (1 − q)‖x∗ − y∗‖ = 0. From (1.1) and (1.16) we obtain
∥∥xn+1 − x∗∥∥ ≤ (1 − αn
)∥∥xn − x∗∥∥ + αn
∥∥Tyn − Tx∗∥∥
≤ (1 − αn
)∥∥xn − x∗∥∥ + αnq∥∥yn − x∗∥∥
≤ (1 − αn
)∥∥xn − x∗∥∥ + αnq(1 − βn
)∥∥xn − x∗∥∥ + qαnβn∥∥Txn − Tx∗∥∥
≤ (1 − αn
(1 − q
))(1 − (1 − q)βn
))∥∥xn − x∗∥∥
≤ (1 − αn
(1 − q
))∥∥xn − x∗∥∥ ≤ · · · ≤( n∏
k=1
(1 − αkq
))∥∥x0 − x∗∥∥.
(3.2)
Use Remark 2.1 with θk = αkq to obtain the conclusion.
This result allows us to formulate the following data dependence theorem.
Theorem 3.2. Let X be a real Banach space, let B ⊂ X be a nonempty convex and closed set, and letε > 0 be a fixed number. If T : B→B is a contractive-like operator with the fixed point x∗ and S : B→Bis an operator with a fixed point u∗, (supposed nearest to x∗) , and if the following relation is satisfied:
‖Tz − Sz‖ ≤ ε, ∀z ∈ B, (3.3)
S. M. Soltuz and T. Grosan 5
then
∥∥x∗ − u∗∥∥ ≤ ε
1 − q. (3.4)
Proof. From (1.1) and (1.2), we have
xn+1 − un+1 =(1 − αn
)(xn − un
)+ αn
(Tyn − Svn
). (3.5)
Thus
∥∥xn+1 − un+1
∥∥ =
∥∥(1 − αn
)(xn − un
)+ αn
(Svn − Tyn
)∥∥
≤(1 − αn
)∥∥xn − un
∥∥ + αn
∥∥Svn − Tvn + Tvn − Tyn
∥∥
≤(1 − αn
)∥∥xn − un
∥∥ + αn
∥∥Tvn − Svn
∥∥ + αn
∥∥Tvn − Tyn
∥∥
≤(1 − αn
)∥∥xn − un
∥∥ + αnε + qαn
∥∥yn − vn
∥∥ + αnφ(∥∥yn − Tyn
∥∥)
≤(1−αn
)∥∥xn−un
∥∥+αnε+qαn
(1 − βn
)∥∥xn−un
∥∥+qαnβn∥∥Txn−Sun
∥∥+αnφ(∥∥yn−Tyn
∥∥)
≤(1 − αn
)∥∥xn − un
∥∥ + αnε + qαn
(1 − βn
)∥∥xn − un
∥∥
+ αnβnq(∥∥Txn − Tun
∥∥ +∥∥Tun − Sun
∥∥) + αnφ(∥∥yn − Tyn
∥∥)
≤(1 − αn
)∥∥xn − un
∥∥ + αnε + qαn
(1 − βn
)∥∥xn − un
∥∥
+ q2αnβn‖xn − un‖ + qαnβnφ(∥∥xn − Txn
∥∥) + qαnβnε + αnφ
(∥∥yn − Tyn
∥∥)
=(1 − αn
(1 − q
(1 − βn
) − βnq2))∥∥xn − un
∥∥ + αnε + qαnβnε
+ qαnβnφ(∥∥xn − Txn
∥∥) + αnφ(∥∥yn − Tyn
∥∥)
=(1−αn(1−q)
(1+qβn
))∥∥xn−un
∥∥+αn
(qβnφ
(∥∥xn−Txn
∥∥)+φ(∥∥yn−Tyn
∥∥)+qβnε+ε)
≤(1−αn(1−q))∥∥xn−un
∥∥+(αn(1 − q)
)qβnφ(∥∥xn−Txn
∥∥)+φ(∥∥yn−Tyn
∥∥)+qβnε+ε1 − q
.
(3.6)
Note that limn→∞φ(‖xn − Txn‖) = limn→∞φ(‖yn − Tyn‖) = 0 because φ is a continuous map andboth xn, yn converge to the fixed point of T. Set
λn := αn(1 − q),
σn :=qβnφ
(∥∥xn − Txn
∥∥) + φ(∥∥yn − Tyn
∥∥) + qβnε + ε
1 − q,
(3.7)
6 Fixed Point Theory and Applications
and use Lemma 2.2 to obtain the conclusion∥∥x∗ − u∗∥∥ ≤ ε
1 − q. (3.8)
Remark 3.3. (i) Set βn = 0, ∀n ∈ N, to obtain the data dependence for Mann iteration.
(ii) The Zamfirescu operators and implicitly (Chatterjea and Kannan) are contractive-likeoperators, therefore our Theorem 3.2 remains true for these classes.
4. Numerical example
The following example follows the example from [8].
Example 4.1. Let T : R→R be given by
Tx = 0, if x ∈ (−∞, 2]
= −0.5, if x ∈ (2,+∞).(4.1)
Then T is contractive-like operator with q = 0.2 and φ = identity.
Note the unique fixed point is 0. Consider now the map S : R→R,
Sx = 1, if x ∈ (−∞, 2]
= −1.5, if x ∈ (2,+∞)(4.2)
with the unique fixed point 1. Take ε to be the distance between the two maps as follows:
‖Sx − Tx‖ ≤ 1, ∀x ∈ R. (4.3)
Set u0 = x0 = 0, αn = βn = 1/(n + 1). Independently of above theory, the Ishikawa iterationapplied to S, leads to
Iteration step Ishikawa iteration1 0.510 0.9100 0.99
(4.4)
Note that for n = 1,
0.5 =1
n + 10 +
1n + 1
S
(12
), (4.5)
since y1 = (1/(n + 1))0 + (1/n + 1)1 = 1/2. (The above computations can be obtained alsoby using a Matlab program.) This leads us to “conclude” that Ishikawa iteration applied to Sconverges to fixed point, (x∗ = 1). Eventually, one can see that the distance between the twofixed points is one. Actually, without knowing the fixed point of S (and without computing it),via Theorem 3.2, we can do the following estimate for it:
∥∥x∗ − u∗∥∥ ≤ 11 − q
=1
1 − 0.2=108
= 1.2. (4.6)
As a conclusion, instead of computing fixed points of S, choose T more closely to S and thedistance between the fixed points will shrink too.
S. M. Soltuz and T. Grosan 7
Acknowledgments
The authors are indebted to referee for carefully reading the paper and for making usefulsuggestions. This work was supported by CEEX ET 90/2006-2008.
References
[1] S. Ishikawa, “Fixed points by a new iterationmethod,” Proceedings of the AmericanMathematical Society,vol. 44, no. 1, pp. 147–150, 1974.
[2] W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol.4, no. 3, pp. 506–510, 1953.
[3] R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp.71–76, 1968.
[4] S. K. Chatterjea, “Fixed-point theorems,” Comptes Rendus de l’Academie Bulgare des Sciences, vol. 25, pp.727–730, 1972.
[5] T. Zamfirescu, “Fix point theorems in metric spaces,” Archiv der Mathematik, vol. 23, no. 1, pp. 292–298,1972.
[6] B. E. Rhoades, “Fixed point iterations using infinite matrices,” Transactions of the AmericanMathematicalSociety, vol. 196, pp. 161–176, 1974.
[7] B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of theAmerican Mathematical Society, vol. 226, pp. 257–290, 1977.
[8] V. Berinde, “On the convergence of the Ishikawa iteration in the class of quasi contractive operators,”Acta Mathematica Universitatis Comenianae, vol. 73, no. 1, pp. 119–126, 2004.
[9] M. O. Osilike and A. Udomene, “Short proofs of stability results for fixed point iteration proceduresfor a class of contractive-type mappings,” Indian Journal of Pure and Applied Mathematics, vol. 30, no.12, pp. 1229–1234, 1999.
[10] C. O. Imoru andM. O. Olatinwo, “On the stability of Picard andMann iteration processes,” CarpathianJournal of Mathematics, vol. 19, no. 2, pp. 155–160, 2003.
[11] S. M. Soltuz, “Data dependence for Ishikawa iteration,” Lecturas Matematicas, vol. 25, no. 2, pp. 149–155, 2004.
[12] C. E. Chidume and S. A. Mutangadura, “An example of the Mann iteration method for Lipschitzpseudocontractions,” Proceedings of the American Mathematical Society, vol. 129, no. 8, pp. 2359–2363,2001.
[13] J. A. Park, “Mann-iteration process for the fixed point of strictly pseudocontractive mapping in someBanach spaces,” Journal of the Korean Mathematical Society, vol. 31, no. 3, pp. 333–337, 1994.
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Brinkman Flow of a Viscous Fluid through a Spherical
Porous Medium Embedded in another Porous Medium A. Postelnicu1,, T. Grosan2 and I. Pop2 1Department of Thermal Engineering and Fluid Mechanics, Transilvania University,
500036, Brasov, Romania 2Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania
Abstract A mathematical model for the two-dimensional steady viscous and incompressible flow past a permeable sphere embedded in another porous medium is presented under the assumption of the Darcy-Brinkman equation model and a uniform shear flow away from the sphere. Closed form analytical solutions are presented for streamlines inside and outside the sphere and shearing stress at the surface of the sphere. The streamlines and the shearing stress at any point on the surface of the sphere are shown in several graphs for different values of the governing parameters. It is shown that the dimensionless shearing stress on the sphere is periodic in nature and its absolute value increases with an increase in both porous parameters. Keywords: permeable sphere · porous media · Darcy-Brinkman model · analytical solution
Nomenclature
a radius of the sphere, m
iK permeability of the porous medium, m 2
ip pressure, Pa
ip non-dimensional pressure
r radial coordinate, m
r non-dimensional radial coordinate
ii vu , radial and transversal coordinates, m.s -1
ii vu , non-dimensional radial and transverse components of velocity
∞U constant velocity away from the sphere, m.s -1
1
Greek symbols
iφ porosity of the porous medium
φ ratio of the porosities
θ angular coordinate
iσ parameter of the porous medium, iKa /
)()( , irrir ττ θ non-dimensional skin friction at the surface of the sphere in the radial and transversal directions
iψ non-dimensional stream function
Subscript
2,1=i
1. Introduction
Flow in porous media has been an area of intensive investigation for the last several
decades. The growing emphasis on effective granular and fibrous insulation systems for
the successful containment of the transport of radio-nuclide from deposits of nuclear
waste materials has stimulated various studies in fluid saturated porous media and many
results were obtained for the forced and convective flow in the fundamental geometries
of internal (cavities, annulus, etc.) and external (flow over surfaces) flows. In
comprehensive reviews of the heat transfer mechanisms in geothermal systems, Cheng
(1987,1985) and Bejan (1987) presented the work in this field with emphasis on its
applications in geothermal and energy systems research. Since then a very large number
of practical applications, both industrial and environmental, have caused a rapid
extension of the research, and a substantial number of papers, which relate the boundary-
layer flow past surfaces of various configurations, have been published. Nield and Bejan
(2006), Ingham and Pop (1998,2002), Pop and Ingham (2001), Vafai (2000,2005)
Ingham et al. (2004) and Bejan et al. (2004) gathered many applications which highlight
the directions where further theoretical and experimental developments are required.
Earlier studies of a uniform shear flow past a body embedded in a porous medium
in the Stokes flow were mainly concerned with the use of the Darcy model because the
2
dimensionless particle diameter (i.e., the ratio of particle diameter to the characteristic
length) is usually small. Flows in rocks, soil sand and other composite media encountered
in hydrology and geothermal problems are usually studied using the Darcy flow model.
Rudraiah (1984) and Rudraiah et al. (2003) have shown that in many practical
applications of flow in porous media which involve porous materials, such as foam
metals and fibrous media of high porosity, where dimensionless particle diameter is not
small, the Darcy law is not adequate and the non-Darcy equation is more accurate to
describe the flow. The non-Darcy equation which incorporates both boundary and inertia
effects in addition to the Darcy resistance may alter the characteristics of flow past a
body. Padmavathi et al. (1993), Masliyah et al. (1987) and Berman (1996) have studied
the Stokes flow past a sphere embedded in a porous medium using Stokes and Brinkmann
flow equations with specifying a uniform velocity far away from the sphere. Also,
Srivastava and Srivastava (2005) have recently discussed the steady flow of an
incompressible viscous fluid streaming past a porous sphere at small Reynolds number
with a uniform velocity by dividing the flow in three regions. The region –I is the region
inside the porous sphere in which the flow is governed by Brinkman equation with the
effective viscosity different from that of the clear (non-porous) fluid. In regions II- and
III- clear fluid flows and Stokes and Ossen solutions are respectively valid. In all the
three regions Stokes stream function is expressed in powers of Reynolds number. Further,
in a very recent paper, Rudraiah and Chandrashekhar (2005) have studied the two-
dimensional steady incompressible flow past an impermeable sphere embedded in a
porous medium using the Brinkman model with a uniform shear instead of a uniform
velocity away from the sphere. This problem occurs in a wide variety of technological
applications like removing impurities in the integrated circuits used in computers,
lubrication process in porous bearings, etc.. However, results of this paper are,
unfortunately, wrong because the authors have wrongly considered the expression for the
stream function of the flow outside the porous sphere. The existence and uniqueness of
the solution for the two-dimensional flow with porous inclusions based on Brinkman
equation has been studied by Kohr and Raja Sekar (2006).
The aim of this paper is to study the flow of a viscous fluid past a permeable
sphere embedded in another porous medium using the Brinkman equation model for the
3
flow inside and outside the sphere. A rigorous theoretical justification of Brinkman’s
equation was given by Tam (1969) and later by Lundgren (1972). It is important that we
succeed to present a closed form solutions of the non-dimensional governing equations
and the evolution of the flow field is demonstrated by plotting the streamlines, velocity
and shearing stress for the flow inside and outside the sphere. These quantities are
numerically computed and depicted graphically for various values of the porous medium
parameters 1σ and 2σ . By way of correspondence to a real physical phenomenon of such
a situation, one may well consider the application to pollutants flow at the interface
between two porous media or oil drilling in a porous medium having insertions with
others porous media.
2. Basic equations Consider the steady forced convection flow of a viscous and incompressible fluid past a
permeable sphere of radius with the constant velocity away from the sphere. The
sphere is filled with a porous medium of permeability placed in a fluid-saturated
porous medium of permeability . The problem is discussed by dividing the flow area
in two zones. Zone I is the region inside the porous sphere and zone II is the region
outside it. Let the index i in the subscript of any entity
a ∞U
1K
2K
iΧ , 2,1=i indicate the zone in
which the entity is represented.
The continuity and Brinkman equations for this problem with inertial terms
omitted are of the form, see Nield and Bejan (2006):
0=⋅∇ iv (1)
ii vv 2~ ∇+−=∇ µµ
ii K
p (2)
where ip is the pressure, iv is the superficial velocity vector, µ is the dynamic
viscosity of the fluid, µ~ is an effective or Brinkman viscosity. It is common practice for
µ~ to be taken equal with µ for high porosity medium, i.e. 1/~ =µµ (see Nield and
Bejan, 2006). Further, we use a spherical coordinate system ),,( ϕθr with the origin at
4
the centre of the sphere and the axis 0=θ along the direction of the undisturbed flow
as it is shown in Fig. 1. Due to the symmetry of the problem we have ∞U 0/ =∂∂ ϕ . It is
convenient to non-dimensionalise all variable by writing
)/(,/,/,/ ∞∞∞ ==== UpapUvvUuuarr iiiiii µ (3)
where iu and iv are the radial and transverse components of velocity.
Using the new non-dimensional variables (3), Eqs. (1) and (3) can be written as
0)sin(sin
)( 2 =∂∂
+∂∂ θ
θθ ii vrurr
(4)
⎭⎬⎫
⎩⎨⎧
−∂∂
−−∂∂
+∂∂
+∂∂
+∂∂
−=∂
∂− 22222
2
22
22 cot222cot12
rvv
rruu
ru
rru
rru
urp iiiiiii
iii θ
θθθ
θσ (5)
⎭⎬⎫
⎩⎨⎧
−∂∂
+∂∂
+∂∂
+∂∂
+∂∂
−=∂∂
− 2
2
222
2
22
22 cosec2cot121
rvu
rv
rv
rrv
rrv
vp
riiiiii
iii θ
θθθ
θσ
θ (6)
where ii Ka /=σ are the parameters of the porous media. Permeabilities of the
porous media are given by Carman-Kozeny relationship, see Nield (2002),
iK
( )2
32
1180 i
iii
dK
φφ−
= (7)
where iφ are the porosities and are the mean particles diameters for the two porous
media. Thus the parameters of the porous media are given by:
id
2/3
)1(180
i
iii φ
φγσ
−= (8)
where ii da /=γ . In Eqs. (4) – (6) variables r and θ vary in zone I in the ranges
and in zone II in the ranges . oo 180180,10 ≤≤−≤≤ θr oo 180180,1 ≤≤−∞<≤ θr
The matching conditions at the surface of the sphere can be written as (see,
Merikh and Mohamad, 2002)
5
1at
1at1at1at
)2()1(
)2()1(
21
21
==
======
r
rrvvruu
rrrr
rr
ττ
ττ θθ (9)
where )(ir θτ and )(irrτ are the non-dimensional shear stresses on the surface of the sphere.
We introduce now the stream function iψ defined in the usual way as
rr
vr
u ii
ii ∂
∂−=
∂∂
=ψ
θθψ
θ sin1,
sin1
2 (10)
Outsideb the porous sphere the stream function 2ψ is given by
1allforsin121),( 22
2 >⎟⎠⎞
⎜⎝⎛ −= r
rrr θθψ (11)
Using (10) and (11), the boundary conditions for the velocity components in zone II are
∞→− rvu assin~,cos~ 22 θθ (12)
The expression of )(ir θτ and )(irrτ are given by
ru
p
rv
rvu
r
iiirr
iiiir
∂∂
+−=
−∂∂
+∂∂
=
2
1
)(
)(
τ
θτ θ
(13)
3. Mathematical analysis
We introduce now the stream function iψ defined in the usual way
rr
vr
u ii
ii ∂
∂−=
∂∂
=ψ
θθψ
θ sin1,
sin1
2 (14)
and eliminate the pressure between Eqs. (5) and (6). Then, using (14), we get ip
( ) 0222 =−ℑℑ ii ψσ (15)
where the operator is defined as 2ℑ
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+∂∂
=ℑθθθ
θsin
1sin22
22
rr (16)
6
The boundary conditions far away from the sphere (12) in terms of 2ψ become
∞→rrr assin2
~),( 22
2 θθψ (17)
The boundary condition (17) suggests the following similarity solution to Eq. (15)
θθψ 2sin)(),( rfr ii = (18)
Substituting (18) into Eq. (15), we obtain the following ordinary differential equation
02''88''42
2432 =⎟
⎠⎞
⎜⎝⎛ −−−+− iiiii
vii f
rf
rf
rf
rf σ (19)
where primes denote differentiation with respect to r . If we make the transformation
)(2'' 2 rgrfr
f iii =− (20)
the fourth order equation (19) reduces to the second order equation
( ) 023''' 2
22 =
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛−+ iii grgrgr σ (21)
This is the modified Bessel differential equation and hence his general solution is given
by
)()()( 2/32/3 rKBrIArg iiiii σσ += (22)
where )(2/3 rI iσ and )(2/3 rK iσ are the modified Bessel functions of first and second
kind of order 3/2, respectively, and and are arbitrary constants of integration. Thus
on using (22), Eq. (20) becomes
iA iB
)()(2'' 3/23/22 rKBrIAfr
f iiiiii σσ +=− (23)
and has the general solution
)()()( 2/322/322 rKrBrIrArD
rC
rf ii
iii
iii
i σσ
σσ
+++= (24)
where and are also arbitrary constants of integration. iC iD
7
Using (18) and (24) we can describe the flow in the both zones I and II with a
proper choice of the constants . For the flow in zone I where iiii DCBA and,, 1<r and
origin occurs, the constants 011 == BC and the expression for can be written as )(1 rf
)()( 12/321
12
11 rIrArDrf σσ
+= (25)
On the other hand, for the flow in zone II where 1>r and ∞→→ ras21)( 2
2 rrf , we
get and the expression for is given by 0and2/1 22 == AD )(2 rf
)(21)( 22/32
22
222 rKrBr
rCrf σ
σ++= (26)
To determine the constants we have to use the matching and
boundary conditions (9). To do it, we have to determine the expression for
1221 and,, DCBA
)(ir θτ and
)(irrτ given by (13) and also the pressure . Substituting (14) and (18) into Eqs. (5) and
(6), we get after some algebra
ip
θσ cos4'282 32422 ⎟
⎠⎞
⎜⎝⎛ +++−= ∫∫ ii
iiii f
rf
rdr
rf
drrf
p (27)
where the constant of integration has been taken to be zero. Thus, relations (13) become
θστ
θτ θ
cos'28122
sin''12'2
24322
)(
32)(
⎟⎠⎞
⎜⎝⎛ +−−=
⎟⎠⎞
⎜⎝⎛ −−=
∫ ∫ ii
ii
iirr
iiiir
fr
drrf
fr
drrf
fr
fr
fr
(28)
Now, the four matching conditions (9) in terms of can be written as )(rfi
141
21
11
42
22
2
12
12
12
44
)1('')1('')1(')1('
)1()1(
==
⎟⎠⎞
⎜⎝⎛ −=⎟
⎠⎞
⎜⎝⎛ −
==
=
∫∫∫ ∫rr
drrfdr
rf
rfdr
rf
ffffff
σσ
(29)
8
These relations can be used to determine the values of the constants
for some values of the parameters
1221 and,, DCBA
21 and σσ .
Thus, we have the stream functions ),( θψ ri in the both zones I and II given by
θθψθθψ 222
211 sin)(),(,sin)(),( rfrrfr == (30)
From these equations we obtain the normal and tangential components of velocities in
zones I and II, using (14) as
θθ
θθ
sin)('
,cos)(
2
sin)('
,cos)(
2
222
22
112
11
rrf
vr
rfu
rrf
vr
rfu
−==
−== (31)
The case when the sphere is impermeable corresponds to and 02 =u 02 =v at
the surface of the sphere. Thus, the values of and are given by and
from (29). On using (24) and the properties of the Bessel functions
(Abramowitz and Stegun, 1972) calculation gives the following expressions for and
2B 2C 0)1(2 =f
0)1('2 =f
2B
2C
)(2
)(321,
)(23
22/12
22/32
22/1
22 σσ
σσ
σK
KC
KB −−== (32)
Thus, is given by )(2 rf
)()(
231
)()(3
121
21)(
22/1
22/3
222/12
22/322 σ
σσσσ
σK
rKrrK
Krrf +⎟⎟
⎠
⎞⎜⎜⎝
⎛+−= (33)
This expression is exactly the same with that reported by Pop and Ingham (1996) for the
problem of flow past an impermeable sphere embedded in a porous medium using the
Brinkman model if we change 2σ by 2/1 σ . Thus, using (28) and (33) we get that the
dimensionless shearing stress at any point on the surface of the sphere (i.e., 1=r ) is
given by
θστ θ sin)1(23
2)2( +−=r (34)
Also this expression is identical with that found by Pop and Ingham (1996) if we change
2σ by 2/1 σ . Therefore, we can conclude that the present analysis is correct.
9
4. Results and discussion
Based on the governing equations of motion it is seen that the parameters affecting the
flow in the present problem are the porous media parameters 1σ and 2σ given by Eq. (8).
Values of the porosities for different porous materials can be found in the book by Nield
and Bejan (2006). For a high porosity porous media a typical value is 8.0=iφ so that Eq.
(8) reduces to:
ii γσ 5.3= (35)
In most physical problems of interest iγ is large and hence iσ is large. In this paper
calculation have been carried out using the symbolic calculus software Mathematica and
it was found that for large values of the porous media parameters, except when 21 σσ = ,
in all other cases, basically diminishes and takes a very large value. In an attempt
to understand, from the numerical point of view, this fact, we have to examine Eq. (23).
Thus, we remark from this equation that as
1A 2B
r increases towards 1=r and 1σ takes large
values, the weight of the second term becomes important, due to the behaviour of the
Bessel function )( 12/3 rI σ , which grows very fast to infinity. On the other hand, the
modified Bessel function )( 22/3 rK σ in Eq. (24) decreases to zero as r increases. For
example, = 1.9793e-005. Therefore, the effect of the coefficient, which has
a very large value, is damped, due to the above mentioned behavior of
)10(2/3K 2B
)( 22/3 rK σ . In
order to have confidence in this analysis, we have determined both numerically and
analytically the values of the coefficients A1, B2, C2 and D1, and it was found a very good
agreement for large values of the parameters 1σ and 2σ . However, in order to save space
we will not present here this comparison.
Figures 2 to 4 illustrate the streamlines computed from Eq. (30) are drawn for
some values of the parameters of the porous media 1σ and 2σ , namely the cases of
2121 , σσσσ >< and 21 σσ = have been considered. Figure 2 shows the streamlines for
11 =σ and 100,102 =σ , Fig. 3 illustrates the streamlines for 101 =σ and 12 =σ , 10 ,
, and Fig. 4 show the streamlines for100 1001 =σ and 12 =σ , 10 . It is clearly seen from
10
these figures that there is a substantial difference between the cases considered when
2121 and σσσσ >< , respectively. Thus, it is noticed that the meandering of the
streamlines near the surface of the sphere and approaching a constant value away from
the sphere for both 2121 and σσσσ >< . In addition, in the case 21 σσ > a boundary
layer near the surface of the sphere can be observed, see Figs.3a, 4a and 4b. Further we
can notice from Figs. 2a, 2b and 3c that the streamlines are distorted towards the center of
the porous sphere in the case 21 σσ < . This is in accordance with the properties of the
porous media. In the special case of 21 σσ = , the flow is not perturbed by the presence of
the porous sphere. This is again in accordance with the properties of the porous media
that is because the two porous media have the same properties.
The variation of the dimensionless shearing stress )2(θτ r at any point on the
surface of the sphere (i.e., 1=r ) with θ is illustrated in Figs. 5 to 7 for several values of
the porous parameters 1σ and 2σ . We can see that the absolute value of the shearing
increases with an increase in both 1σ and 2σ . It is also seen that at the front and rear
points of the sphere, and , the shearing stress vanishes, whereas it attains
the maximum value at . It should also be pointed out that there is no flow
separation occurring for the flow past a sphere which is embedded in a constant porosity
medium based on the Brinkman equation model. The positive or negative value of the
shearing stress
o0=θ o180=θo90±=θ
)2(θτ r depend on the velocity direction at the interface between the
interface of two porous media.
5. Conclusion
The flow of a viscous fluid past a permeable sphere embedded in another porous medium
has been investigated using the Brinkman equation model. An exact analytical solution of
the governing equations for the flow inside and outside the sphere has been calculated. It
has been found that for a particular porous medium outside the sphere, the dimensionless
shearing stress at any point on the surface of the sphere )2(θτ r increases with the increase
11
of the parameter 2σ . It is shown that there is a substantial difference between the cases
when 212121 and, σσσσσσ >=< .
References
Bejan, A.: 1987, Convective heat transfer in porous media. In: Handbook of Single-Phase
Convective Heat Transfer (Kakaç, S., Shah, R.K. and Aung, W.,eds.), Chapter 16.
Wiley, New York.
Bejan, A., Dincer, I., Lorente, S., Miguel, A.F. and Reis, A.H., 2004, Porous and
Complex Flow Structures in Modern Technologies, Springer, New York.
Berman, B.: 1996, Flow of a Newtonian fluid past an impervious sphere embedded in a
porous medium, Indian J. Pure Appl. Math. 27, 1249-1256.
Cheng, P.: Heat transfer in geothermal systems, in: Adv. Heat Transfer 14, 1-105.
Cheng, P.: Natural convection in a porous medium: external flows, in: Natural
Convection: Fundamentals and Applications (Kakç, S., Aung, W. and Viskanta, R.
eds.), Hemisphere, Washington, DC, 1985.
Ingham, D.B. and Pop, I. (eds.): 1998, Transport Phenomena in Porous Media,
Pergamon, Oxford.
Ingham, D.B. and Pop, I. (eds.): 2002, Transport Phenomena in Porous Media,
Pergamon, Oxford, Vol. II.
Ingham, D.B. and Pop, I. (eds.): 2005, Transport Phenomena in Porous Media. Elsevier,
Oxford, Vol. III.
Kohr, M. and Raja Sekar, G.P.: 2007, Existence and uniqueness result for two-
dimensional porous media flows with porous inclusions based on Brinkman equation,
Engineering Analysis with Boundary Elements (in press).
Lundgren, T.S.: 1972, Slow flow through stationary random beds and suspensions of
spheres, J. Fluid Mech. 51, 273-299.
Masliyah, J.H., Neale, G., Malysa, M. and Van De Ven, T.G.M.: 1987, Creeping flow
over a composite sphere: Solid core with porous shell, Chem. Engng. Sci. 4, 245-253.
12
Merrikh, A.A., Mohamad, A.A., 2002, Non-Darcy effects in buoyancy driven flows in an
enclosure filled with vertically layered porous media, Int. J. Heat Mass Transfer 45,
4305-4313.
Nield, D.A. and Bejan, A.: 2006, Convection in Porous Media (3nd edition), Springer,
New York.
Niled, D.A., 2002, Modeling fluid flow in saturated porous media and at interfaces. In:
Transport Phenomena in Porous Media (Ingham, D.B. and Pop, I., eds.), Pergamon,
Oxford, 1-19.
Padmavathi, B.S., Amaranath, T. and Nigam, S.D.: 1993, Stoke’s flow past a porous
sphere using Brinkman model, J. Appl. Math. Phys. (ZAMP) 44, 929-939.
Pop, I. and Ingham, D.B.: 1996, Flow past a sphere embedded in a porous medium based
on the Brinkman model, Int. Comm. Heat Mass Transfer 23, 865-874.
Pop, I. and Ingham, D.B.: 2001, Convective Heat Transfer: Mathematical and
Computational Modelling of Viscous Fluids and Porous Media, Pergamon, Oxford.
Rudraiah, N.: 1984, Non-linear convection in porous medium with convective
acceleration and viscous force, Arab. J. Sci. and Engng. 9, 153-167.
Rudraiah, N., Pradeep, G.S. and Masuoka, T.: 2003, Non-linear convection in porous
media: a review, J. Porous Media 6, 1-32.
Rudraiah, N. and Chandrashekhar, D.V.: 2005, Flow past an impermeable sphere
embedded in a porous medium with Brinkman model, Int. J. Appl. Mech. Engng. 10,
145-158.
Srivastava, A.C. and Srivastava, N.: 2005, Flow past a porous sphere at small Reynolds
Number, J. Appl. Math. Phys. (ZAMP) 56, 821-835.
Tam, C.K.W.: 1969, The drag on a cloud of spherical particles in low Reynolds number
flow, J. Fluid Mech. 38, 537-546.
Vafai, K. (ed.): 2000, Handbook of Porous Media, Marcel Dekker, New York.
Vafai, K. (ed.): 2005, Handbook of Porous Media, Taylor & Francis, New York.
13
θ
r
v u
a
11 ,vu zone I 22 ,vu
zone II
∞U
Fig. 1 Physical model and coordinate system.
14
a)
b)
Fig. 2. Streamlines for 11 =σ : a) 102 =σ ; b) 1002 =σ .
a)
15
b)
c)
Fig. 3. Streamlines for 101 =σ : a) 12 =σ ; b) 102 =σ ; c) 1002 =σ .
16
a)
b)
Fig. 4. Streamlines for 1001 =σ : a) 12 =σ ; b) 102 =σ .
17
Fig. 5. Variation of tangential shear stress along the surface of the sphere
for 11 =σ and 100,50,12 =σ .
Fig. 6. Variation of tangential shear stress along the surface of the sphere
18
for 101 =σ and 100,50,12 =σ .
Fig. 7. Variation of tangential shear stress along the surface of the sphere
for 1001 =σ and 50,10,12 =σ .
19
Magnetohydrodynamics oblique stagnation-point flow
T. Grosan ⋅ C. Revnic ⋅ I. Pop ⋅ D.B. Ingham
Abstract Laminar two-dimensional stagnation flow of an incompressible viscous electrically conducting fluid obliquely impinging on a flat plate is formulated as a similarity solution of the Navier-Stokes equations. The relative importance of this flow is measured by the dimensionless strain rate γ and magnetohydrodynamic M parameters. The viscous problem is reduced to a coupled pair of ordinary differential equations governed by γ and M . It is found that the parameter M causes a shift in the position of the point of zero skin friction along the plate. Keywords Magnetohydrodynamics; Oblique stagnation-point flow; Similarity solution; Numerical methods 1 Introduction The steady two-dimensional stagnation-point flow of an incompressible viscous fluid impinging obliquely on a plane rigid wall has been studied by many researchers. Stuart [1], Tamada [2], Dorrepaal [3,4], Labropulu et al. [5], Liu [6] and Tittleyand Weidman [7] have considered the case when the plane is fixed while Reza and Gupta [8], Lok et al. [9] and Mohapatra et al. [10] have considered the two-dimensional oblique T. Grosan ⋅ I. Pop ( ) Faculty of Mathematics and Computer Science, Babes-Bolyai University of Cluj, R-3400 Cluj, Romania email: [email protected] C. Revnic Institute of Mathematics of the Romanian Academy of Sciences, Cluj-Napoca, Romania D.B. Ingham Centre for Computational Fluid Dynamics, University of Leeds, Leeds LS2 9JT, UK
stagnation-point flow towards a surface which is stretched with a velocity proportional to the distance stagnation-point flow towards a surface which is stretched with a velocity proportional to the distance from a fixed point. The case of axisymmetric flow stagnating obliquely on a circular cylinder has been considered by Weidman and Putkaradze [11]. Dorrepaal [3] and Labropolu et al.[5] have shown that for oblique flow impinging on a flat rigid wall that the slope of the dividing streamline at the wall divided by its slope at infinity is independent of the angle of incidence.
Several aspects of steady and unsteady two-dimensional stagnation-point flow of an electrically conducting fluid in the presence of a uniform applied magnetic field towards a fixed rigid plane wall or a stretching surface has been discussed in recent years by several authors, see for example Pavlov [12], Chakrabarti and Gupta [13], Andersson [14], Chiam [15], Mohapatra and Gupta [16], Ariel [17] and Xu et al.[18]. The study of magnetohydrodynamic (MHD) flow of an electrically conducting fluid caused by the deformation of the wall of a vessel containing a fluid is of considerable interest in modern metallurgical and metal-working processes.
The aim of the present paper is to discuss the steady two-dimensional oblique flow of a viscous and electrically conducting fluid impinging on a flat rigid wall in the presence of a constant applied magnetic field. It is found that the presence of the magnetic field causes a shift in the stagnation point and that this shift depends upon the magnetic parameter. To the best of our knowledge this problem has not been studied before. 2 Basic equations Consider the steady two-dimensional MHD flow of an electrically conducting fluid near a stagnation point when an incompressible viscous fluid impinges obliquely on a rigid wall coinciding with the plane 0=y in the presence of a uniform transverse magnetic field, see Fig. 1, where x and y are the Cartesian coordinates along the wall and normal to it, respectively. It is assumed that the strength of the magnetic field is and that the magnetic Reynolds number is small. It is also assumed that the
0B
induced magnetic field is negligible. Under these assumptions the basic equations are given by
Fig. 1. Physical model and coordinate system.
0=∂∂
+∂∂
yv
xu (1)
uB
uxp
yuv
xuu
ρσ
νρ
2021
−∇+∂∂
−=∂∂
+∂∂ (2)
vyp
yvv
xvu
21∇+
∂∂
−=∂∂
+∂∂ ν
ρ (3)
where u and v are the velocity components along the – x and – y directions, respectively, p is the pressure, ρ is the density, ν is the kinematic
viscosity, σ is the electrical conductivity and 2
∇ is the two-dimensional Laplacian. We consider an outer (inviscid) flow consisting of a linear superposition of an irrotational stagnation-point flow of strain rate a ( > 0) and a uniform shear flow parallel to the wall of strain b. Thus, we assume that the boundary conditions appropriate to Eqs. (1) – (3) are given as
∞→===
===
∞ yppvvuu
yvu
ee as,,
0at0 (4)
We introduce now the stream function ψ defined as
xv
yu
∂∂
−=∂∂
=ψψ , (5)
Substituting (5) into Eqs. (2) and (3), and eliminating the pressure p from the resulting equations, we obtain
+⎟⎠⎞⎜
⎝⎛∇
∂
∂
∂
∂−⎟
⎠⎞⎜
⎝⎛∇
∂
∂
∂
∂ ψψψψ 22
xyyx
02
2204
=∂
∂−∇+
y
B ψρ
σψν (6)
and the boundary conditions (4) become
∞→+=
==∂
∂=
yybyxa
yy
as
0at0,0
2ψ
ψψ (7)
Further we define the following non-dimensional variables
( )2,, allyylxx ψψ === (8)
where ( ) 2/1/ al ν= is a characteristic length. Substituting (8) into (6), we obtain the following equations, in non-dimensional form,
( ) ( )+∇∂∂
∂∂
−∇∂∂
∂∂ ψψψψ 22
xyyx
02
24 =
∂∂
−∇+y
M ψψν (9)
and the boundary conditions (7) become
∞
0
→+=
==∂∂
=
yyxy
yy
as
at0,0
2γψ
ψψ (10)
Where γ is the shear parameter and M is the magnetic parameter which are defined as
aB
Mab
ρσ
γ 0, ==2
(11)
A physical quantity of interest is the skin friction, or the shear stress wτ at the wall, which is defined as
0=2
2
2
2
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂+
∂
∂=
y
wyx
ψψµτ (12)
or in non-dimensional form
0=2
2
2
2
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂
∂=
yw yx
ψψµτ (13)
where ( )µττ aww = .
3..Solution procedure
The boundary conditions (10) suggest that ( )yx,ψ has the form
( ) )()(, yGyxFyx +=ψ (14) Substituting (14) into Eq. (9), we obtain the following ordinary differential equations
0''''''' =−+− MFFFFFF IV (15)
0'''''''' =−+− MGGFGFG IV (16)
which have to be solved subject to the boundary conditions
γ2)('',1)('0)0(')0(,0)0(')0(
=∞=∞====
GFGGFF
(17)
where primes denote differentiation with respect to y. Integrating Eqs. (15) and (16) once, we obtain
12 ''''''' CMFFFFF =−−+ (18)
2'''''''' CMGFGFGG =−−+ (19)
where C1 is a constant and C2 = C2(y). Using the boundary conditions (17) for F(y), we obtain
so that Eq. (18) becomes )1(1 M - C +=
0)'1('1''''' 2 =−−−++ FMFFFF (20) This equation describes the classical MHD boundary layer flow near a two-dimensional orthogonal stagnation-point, see Ariel [17]. Equation (20) gives
∞→+≈ yAyyF as)( (21)
where A is a constant which depends on M. Further, using (21) we obtain from Eq. (19) that
yMAC γγ 222 −= and Eq.(19) becomes
( ) γγ AGyMFGFGG 2'2''''''' =−+−+ (22)
Finally, we have to solve Eqs.(20) and (22) subject to the boundary conditions
γ=∞=∞====
)('',1)('0)0(')0(,0)0(')0(
GFGGFF
(23)
Introducing the new variable
)y(2)(' HyG γ= (24)
then Eq. (22) becomes
( ) γAHyMHFFHH =−+−+ '''' (25) which has to be solved subject to the boundary conditions
1)(,0)0( =∞= HH (26) Further, from (24) we have
∫=y
sHyG0
)(2)( γ ds
C
(27)
Integrating numerically Eq. (25) with the boundary conditions (26), we obtain
H =)0(' (28)
where C is a constant which depends on M. The non-dimensional skin friction (13) can be
written as
)0('2)0('' HxFw γτ += (29) The dimensionless velocity components ( )vu, are
defined as )/(aluu = and )/(alvv which give, from (14)
=
)y
(),(')(' FvyGyxFu −=+= (30) The streamline 0=ψ meets the wall at the point
0xx = where 0=wτ (point of zero skin friction), which from (29) is given by
)0(''('2
0 FHx γ
−=)0 (31)
On the other hand, following Labropulu et al. [5], we expand F(y) and G(y) in Taylor series about 0≈y (near the wall) and Eq. (29) gives for 0=wτ
[ ] ++ yGxF )0('')0(''21
[ ] h.o.t)0(''')0('''61 2 =++ yGxF 0
B C
(32)
where , F =)0('' G γ2)0('' = ,
)1()0(''' MF +−= and γAG 2)0(''' = . Dividing
Figure 2 shows the variation of the velocity profiles with y at a fixed value of x = 1.0 for several
values of M and),( yxu
γ . It is seen that at a given location (x = 1.0) the velocity (which is simply taken to represent a typical result) increases with increasing M.
(31) by and letting 0)0('' ≠F BCxX /2γ+= , we obtain
0h.o.t)1(
32
2=
⎭⎬⎫
⎩⎨⎧
+⎥⎦
⎤⎢⎣
⎡ +++ y
BMC
BA
XB γγ (33)
Figures 3 to 7 show the streamline pattern for orthogonal ( 0=γ ) and oblique ( 0≠γ ) flows for several values of M. The location of the point of zero skin friction on the axis, , is also shown in these figures. It can be seen that for a fixed value of
From (33) we can see that the dividing streamline meets the wall at or 0=y 0=X BCx /2γ−= and its slope near the wall is given by 0m 0x
γ that these locations are at a larger distance from the stagnation point as the value of M increases. Further, it is clear from these figures that for a fixed value of γ that the streamlines become more and more closer to the wall with increasing M.
[ ])1(23 2
0 MCABBm
++−=
γ (34)
On the other hand, the dividing streamline which comes into the wall from infinity is defined by
0),( =yxψ and its slope at infinity is given by ∞mFurther, for a given value of M the streamlines are
more and more oblique towards the left of the stagnation- point with increasing γ . On the other hand, for a fixed value of M the streamlines are more and more oblique towards the right of the stagnation-point with increasing γ when 0<γ . This is consistent with the fact that increasing the shear in the γ results in an increase in the shearing motion which in turn leads to increased obliquity of the flow towards the surface. However, it should be noticed that the streamlines for negative values of γ are not shown in Figs. 5 to 7 because they are almost mirror images in the plane y = 0 to the streamlines for positive values of γ .
γ1
−=∞m (35)
The ratio ∞= mmR 0 is found to be
[ ])1(23 2
MCABBR
++= (36)
which is independent of the parameter γ .
5 Conclusions 4 Results and discussion
A numerical solution of the steady oblique flow of a viscous and electrically conducting fluid impinging on a flat plate has been investigated. The governing partial differential equations are reduced to a set of two ordinary differentials equation which are solved numerically for different values of the governing parameters M and γ . This solution provides useful information about the MHD stagnation-point flow and, in limited cases that have already been investigated when the magnetic field is absent, there is quantitatively good agreement.
Equations (20) and (22) subject to the boundary conditions (23) have been solved numerically using the Runge-Kutta method for several values of the parameters γ and M. Values of A, B and C are given in Table 1 for 0=γ (orthogonal stagnation-point flow) and different values of M. The corresponding values reported by Labropulu et al. [5] and Ariel [17] have also been included in this table. It is seen that the present results are in very good agreement with those determined by Labropulu et al. [5] and Ariel [17]. We are therefore confident that the present results are accurate. It is seen, as expected, that
increases steadily as M increased from zero and this appears to be more plausible physically. Also some values of R given by (36) for several values of M are given in Table 1. The value of R = 3.748513 for M = 0 reported by Dorrepaal [3] and Labropulu et al. [5] has been also included in this table.
)0(''F
Acknowledgements The authors I. Pop and D.B. Ingham wish to express
their thanks to the Royal Society for partial financial support. The work of the authors T. Grosan and C. Revnic was supported form the grant CEEX-ET90/2006-2008.
)0(''F )0('H A R M
present Ariel [17] present Labropulu et al. [5]
present Labropulu et al. [5]
Present Labropulu et al. [5]
0.00 1.232588 1.232588 1.406616 1.406544 -0.647900 -0.647900 3.748069 3.748513
0.16 1.295368 1.295368 1.373280 -0.626277 3.219676
0.64 1.467976 1.467976 1.300013 -0.572893 2.503764
1.00 1.585331 1.585331 1.261685 -0.541007 2.263267
4.00 2.346663 2.346663 1.127294 -0.393590 1.752707
25.00 5.147965 5.147965 1.027863 -0.190725 1.544224
100.00 10.074741 10.074741 1.007358 -0.098774 1.481927
Table 1. Values of , , A and R for several values of M. )0(''F )0('H
(a)
(b)
(c)
Fig. 2. Velocity profile for x = 1 and several values of M :
(a) 0=γ (orthogonal stagnation-point flow); (b) 5.0=γ (c) 1=γ .
Fig.3. Streamline pattern for M = 0, 0=γ (orthogonal
stagnation point).
(a)
(b)
Fig. 4.Streamlines for M = 0; (a) 5.0=γ (b) 5.0−=γ
Fig. 5. Streamlines for M = 0 and 1=γ .
(a)
(b)
(c)
Fig. 6. Streamline pattern for M = 1; (a) 0=γ (b) 5.0=γ (c) 1=γ .
(a)
(b)
(c)
Fig. 7. Streamline pattern for M = 25; (a) 0=γ (b)
5.0=γ (c) 1=γ . References
1. J.T. Stuart (1959) The viscous flow near a stagnation point when the external flow has uniform vorticity. J. Aerospace Sci. 26:124-125
2. K.J. Tamada (1979) Two-dimensional stagnation point flow impinging obliquely on a plane wall. J. Phys. Soc. Jpn. 46: 310-311
3. J.M. Dorrepaal (1986) An exact solution of the Navier-Stokes equation which describes non-orthogonal stagnation-point flow in two dimensions. J. Fluid Mech. 163:141-147
4. J.M. Dorrepaal (2000) Is two-dimensional oblique stagnation point flow unique?. Canadian Appl. Math. Quart. 8:61-66
5. F. Labropulu, J.M. Dorrepaal, O.P. Chandna (1996) Oblique flow impinging on a wall with suction or blowing. Acta Mech. 115:15-25
6. T. Liu (1992) Nonorthogonal stagnation flow on the surface of a quiescent fluid - an exact solution of the Navier-Stokes equation. Quart. Appl. Math. 50:39-47
7. B.S. Tittley, P.D. Weidman (1998) Oblique two-fluid stagnation-point flow. Eur. J. Mech. B/Fluids 17:205-217
8. M. Reza, A.S. Gupta (2005) Steady two-dimensional oblique stagnation-point flow towards a stretching surface. Fluid Dyn. Res. 27:334-340
9. Y.Y. Lok, N. Amin, I. Pop (2007) Comments on: “Steady two-dimensional oblique
stagnation-point flow towards a stretching surface”: M. Reza and A.S. Gupta. Fluid Dynamics Research 37 (2005) 334–340, Fluid Dyn. Res. 39:505-510
10. P.D. Weidman, V. Putkaradze (2003) Axisymmetric stagnation flow obliquely impinging on a circular cylinder. Eur. J. Mech. B/Fluids 22:123-131
11. K.B. Pavlov (1974) Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a plane surface. Magnitnaya Gidrodinamika 4:146-147
12. Chakrabarti, A.S. Gupta (1979) Hydromagnetic flow and heat transfer over a stretching sheet. Quart. Appl. Math. 37:73-78
13. H.I. Andersson (1992) MHD flow of a viscoelastic fluid past a stretching surface. Acta Mech. 95:227-230
14. T.C. Chiam (1994) Stagnation-point flow towards a stretching plate. J. Phys. Soc. Japan 63: 2443-2444
15. T.R. Mahapatra, A.S. Gupta (2001) Magnetohydrodynamic stagnation-point flow towards a stretching sheet. Acta Mech. 152:191-196
16. T. Ray Mahatra, S. Dholey, A.S. Gupta (2007) Heat transfer in oblique stagnation point flow of an incompressible viscous fluid towards a stretching surface. Heat Mass Transfer 43:767-773
17. P.D. Ariel (1994) Hiemenz flow in hydromagnetics. Acta Mechanica 103:31-43
18. H. Xu, S.-J. Liao, I. Pop (2007) Series solutions of unsteady three dimensional MHD flow and heat transfer in a boundary layer over an impulsively streatching plate. Eur. J. Mech. B/Fluids 26:15-27
Unsteady Boundary Layer Flow and Heat Transfer Over aStretching Sheet
Cornelia Revnic∗, Teodor Grosan† and Ioan Pop1†
∗“Tiberiu Popoviciu" Institute of Numerical Analysis, P.O.Box. 68-1, 400110, Cluj-Napoca, Romania†Babes-Bolyai University, Applied Mathematics, R-3400 Cluj, CP 253, Romania
Abstract. Unsteady two-dimensional boundary layer flow and heat transfer over a stretching flat plate in a viscous andincompressible fluid of uniform ambient temperature is investigated in this paper. It is assumed that the plate is isothermal andis stretched in its own plane. Using appropriate similarity variables, the basic partial differential equations are transformedinto a set of two ordinary differential equations. These equations are solved numerically for some values of the governingparameters, using Rungge-Kutta method of fourth order. Flow and heat transfer characteristics are determined and representedin some tables and figures. It is found that the structure of the boundarylayer depends on the ratio of the velocity of thepotential flow near the stagnation point to that of the velocity of the stretching surface. In addition, it is shown that the heattransfer from the plate increases when the Prandtl number increases.Our results are shown to include the steady situation asa special case considered by other authors. Comparison with known results is very good.
Keywords: heat transfer, stretching surface, the external inviscid flow, stagnation-point flow, boundary layer.PACS: 02.60Lj, 44.20+b, 44.27+g, 47,10Ad, 47.85-g
INTRODUCTION
The unsteady boundary layers are important in several physical problems in aero - nautics, missile dynamics, acousticsetc. The work in this area was initiated by Moore [1], Lighthill [2] and Lin [3]. Critical reviews of unsteady boundarylayers were presented by Stuart [4], Riley [5], Telionis [6], [7] and Pop [8]. In recent years certain aspects of theunsteady flows were investigated by Ma and Hui [9] and Ludlow et al. [10] using the classical method of Lie-group.The essence of the Lie-group method is that each of the variables in the initial equation is subjected to an infinitesimaltransformation and the demand that the equation is invariant under these transformations leads to the determinationof the possible symmetries (see Ludlow et al. [10]). The fundamental governing equations for fluid mechanics are theNavier-Stokes equations. This nonlinear set of partial differential equations have no general solutions, and only a smallnumber of exact solutions have been found (see Wang [11]). Exact solutions are important for the following reasons:(i) the solutions represent fundamental fluid-dynamic flows. Also, owing to the uniform validity of exact solutions, thebasic phenomena described by the Navier-Stokes equations can be more closely studied. (ii) the exact solutions serveas standards for checking the accuracies of the many approximate methods, whether they are numerical, asymptotic,or empirical.Flow of a viscous fluid over a stretching sheet has an important bearing on several technological processes. Inparticular in the extrusion of a polymer in a melt-spinning process, the extruded from the die is generally drawnand simultaneously stretched into a sheet which is then solidified through quenching or gradual cooling by directcontact with water. Further, glass blowing, continuous casting of metals and spinning of fibres involve the flow due toa stretching surface, see Lakshmisha et al. [12]. In all these cases, a study of the flow field and heat transfer can be ofsignificant importance since the quality of the final productdepends to a large extent on the skin friction coefficientand the surface heat transfer rate. Crane [13] presented a simple closed form exponential solution of the steady two-dimensional flow caused solely by a linearly stretching sheet in an otherwise quiescent incompressible fluid. Thesimplicity of the geometry and the possibility of obtainingfurther exact solutions through simple generalizationshave generated a lot of interest in extending it to more general situations. Such extensions include consideration ofmore general stretching velocity, application to non-Newtonian fluids, and inclusion of other physical effects suchas suction or blowing, magnetic fields, etc. Unsteady two-dimensional boundary layer flow over a stretching surface
1 Corresponding author: Tel.:-40-264594315; fax:+40-264591906; E-mail adress: [email protected] (Pop Ioan)
has been studied by Na and Pop [14], Wang et al. [15], Elbashbeshy and Badiz [16], Sharidan et al. [17] and Aliand Magyari [18], while Lakshmisha et al. [12], Devi et al. [19] and Takhar et al. [20] have considered the unsteadythree-dimensional-flow due to the impulsive motion of a stretching surface. The aim of the present analysis is tostudy the unsteady flow and heat transfer in the stagnation-point flow on a heated stretched surface in a viscousand incompressible fluid when both velocities of the stretching sheet and of the external flow (inviscid flow) areproportional to the distance from the stagnation-point andinversely to time. The geometry is similar to that proposedby Mahapatra and Gupta [21] for the steady two-dimensional stagnation-point flow towards a stretching sheet. Theparabolic partial differential equations governing the flow and heat transfer have been reduced to a system of twoordinary differential equations which are solved using an implicit finite-difference scheme in combination with theshooting method.
PROBLEM FORMULATION
We consider the unsteady two-dimensional forced convection flow and heat transfer of a viscous and incompressiblefluid near a stagnation point on a surface coinciding with theplain y = 0, the flow being confined toy > 0. Two equaland opposite forces are applied along thex - axis at the initial timet = 0, so that the surface is stretched keeping theorigin fixed as shown in Fig.1. It is assumed that the uniform temperature of the plane isTw , while the temperatureof the ambient fluid isT∞, whereTw > T∞ (heated plate). It is also assumed that the viscous dissipation effects areneglected. Under these assumptions, the system of boundarylayer equations are given by
∂u∂x
+∂v∂y
= 0 (1)
∂u∂ t
+u∂u∂x
+v∂u∂y
=∂ue
∂ t+ue
∂ue
∂x+v
∂ 2u∂y2 (2)
∂T∂ t
+u∂T∂x
+v∂T∂y
= α∂ 2T∂y2 (3)
subject to the initial and boundary conditions are of the form:
t < 0 : u = 0, v = 0, T = T∞ f or any y> 0
t > 0 : u = uw(t,x), v = 0, T = Tw f or y = 0 (4)
t = 0 : u = uws(x), v = 0, T = Tw
u→ ue(t,x), T → T∞ as y→ ∞
50 100 150 200 250 300
20
40
60
80
100
120
140
160
180
200
220
FIGURE 1. Physical model and coordinate system.
whereu andv are the velocity components along thex− andy− axis,T is the fluid temperature,ν is the kinematicviscosity,uws = cx (c is a positive constant) andα is the thermal diffusivity. Following Surma Devi [19] et al., weassumed thatuw(t,x) andue(t,x) are given by
uw(t,x) =cx
(1− γ t), ue(t,x) =
ax(1− γ t)
(5)
wherea is a positive constant. The momentum and energy equations can be transformed to the corresponding ordinarydifferential equations by the following substitutions:
ψ = (cν/(1− γ t))1/2x f(η),
θ(η) = (T −T∞)/(Tw−T∞), (6)
η = (c/ν(1− γ t))1/2y
whereψ is the stream function which is defined in the usual way asu = ∂ψ/∂y andv = −∂ψ/∂x.Substituting (6) into Eqs. (2) and (3), we obtain the following two ordinary differential equations:
f ′′′ + f f ′′ +a2
c2 − f ′2 +γc
(ac−
η2
f ′′− f ′)
= 0 (7)
1Pr
θ ′′ + f θ ′−γ2c
ηθ ′ = 0 (8)
subject to the boundary conditions (4) which become
f (0) = 0, f ′(0) = 1,θ(0) = 1 (9)
f ′(∞) =ac,θ(∞) = 0 (10)
wherePr is the Prandtl number and primes denote differentiation with respect toη .The physical quantities of interest are the skin friction coefficientCf and the local Nusselt numberNux, which aredefined as
Cf =τw
ρu2ws
, Nux =xqw
k(Tw−T∞), (11)
whereτw is the skin friction andqw is the heat transfer from the plate which are given by
τw = µ(
∂u∂y
)
y=0, qw = −k
(
∂T∂y
)
y=0(12)
with µ andk being the dynamic viscosity and thermal conductivity, respectively. Using (6), we get
(1− γ t)3/2Re1/2x Cf = f ′′(0), (13)
(1− γ t)1/2Re−1/2x Nux = −θ ′(0)
WhereRe= (cx)x/ν is the low Reynolds number. It is important to notice that forthe steady-state case,Eqs. (7) and(8) reduced to
f ′′′ + f f ′′− f ′2 +a2
c2 = 0 (14)
1Pr
θ ′′ + f θ ′ = 0 (15)
with the boundary conditions (9)-(10). Equations (14) and (15) with the boundary conditions (9)-(10) where estab-lished by Mahapatra and Gupta [21].
TABLE 1. Values of f ′′(0) for some values ofa/c when theflow is steady. ( ) values reported by Mahapatra and Gupta [21].
a/c 0.10 0.20 0.50 2.00
f ′′(0) -0.9696 -0.9182 -0.6673 2.0175(-0.9694) (-0.9181) (-0.6673) (2.0175)
TABLE 2. Values ofθ ′(0) for some values ofa/c andPrwhen the flow is steady. ( ) values reported by Mahapatra andGupta [21].
a/c / Pr 0.05 0.5 1 1.5
0.1 -0.081 -0.381 -0.603 -0.777(-0.081) (-0.383) (-0.603) (-0.777)
0.5 -0.137 -0.472 -0.691 -0.863(-0.136) (-0.473) (-0.692) (-0.863)
2 -0.248 -0.711 -0.978 -1.171(-0.241) (-0.709) (-0.974) (-1.171)
SOLUTION
The systems of ordinary differential equations (7)-(8) and(14)-(15) subject to the boundary condition (9)-(10) havebeen solved numerically for some values of the parametersa/c, t andPr using Rungge-Kutta method of fourth ordercombined with the shooting technique. For the physical considearation we takeγ = −1. Some values off ′′(0) andθ ′(0) are given in Tables 1 and 2 for the case of the steady flow.
We can see from these tables that there is a very good agreement between our results and those obtained byMahapatra and Gupta [21]. Therefor, we are confident that theresults obtained using the present method are accurate.
Figures 2 - 5 show the velocities profilesf and f ′ along with the corresponding streamlines patterns for the case ofunsteady flow, Eqs. (7) and (8). The values of the parameters are a = 0.1, c = 1 andt = 0,1,2,3. It is interesting tonotice that the solution of Eq. (7) is not unique. Thus, thereare two solutions, one Fig. 2 representing an attached flowand the other one Fig. 4 the reversed flow. These is in agreement with the results obtained by Ma and Hui [9] for theunsteady two-dimensional boundary layer flow near a stagnation point of a fixed plat plate.
0 1 2 3 40
0.2
0.4
0.6
0.8
1
1.2
1.4
η
f and
f’
ff ’
0.1
FIGURE 2. The first solution off (η) and f ′(η) for a/c = 0.1.
−20 −10 0 10 200
2
4
6
8
10
−20 −10 0 10 200
5
10
−20 −10 0 10 200
5
10
15
−20 −10 0 10 200
5
10
15
20
FIGURE 3. The streamlines function for:t = 0,1,2 and 3 corresponding to the first solution fora/c = 0.1.
0 2 4 6 8 10−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
η
f and
f ’
ff ’
FIGURE 4. The second solution off (η) and f ′(η) for a/c = 0.1.
Fig. 6 illustrates the dimensionless temperature profilesθ(η) for some values ofPr whena/c = 2. We notice thattemperature profile increase whenPr decreases. Further, Fig. 7 shows the variation of the heat transfer from the wall− θ ′(0) with a/c and different values ofPr. It is evident from Fig. 7 that an increase inPr result in a decrease in thethermal boundary layer thickness and as a consequence the heat transfer from the wall increases withPr.
−20 −10 0 10 200
2
4
6
8
10
−20 −10 0 10 200
5
10
−20 −10 0 10 200
5
10
15
−20 −10 0 10 200
5
10
15
20
FIGURE 5. The streamlines function for:t = 0,1,2 and 3 corresponding to the second solution fora/c = 0.1.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
η
θ Pr = 0.05, 0.5, 0.72, 1, 1.5
FIGURE 6. Temperature profiles ofθ(η) for several values ofPr anda/c = 2 in respect withη .
CONCLUSION
The unsteady two-dimensional stagnation-point flow and heat transfer of a viscous and incompressible fluid over anisothermal stretching flat plate in its own plane has been numerically analyzed in detailed. Following Surma Devi etal. [19] similarity variables where used to reduced the governing partial differential equations to ordinary differentialequations. Solving numerically these equations, we have been able to determine the velocity and temperature profiles,skin friction and heat transfer from the plate. For the case of steady-state flow, we have compared our present resultswith those of Mahapatra and Gupta [21]. The agreement between the results is excelent. Effects ofa/c andPr onthe flow and heat transfer characteristic have been examenedand discussed in detail. It is shown that for small valuesof a/c the solution of the ordinary differential equation is not unique. One solution represents an attached flow andthe other one a reversed flow. It shoud be noticed that we have determined solutions of the problem for more valuesof the governing parameters but in order to save space, the reported results are limited only to some values of theseparameters.
0 0.5 1 1.5 2 2.5 30.5
1
1.5
2
2.5
3
a/c
− θ
’ (0
)
Pr=[0.05,0.5,0.72,1,1.5]
FIGURE 7. Variation of the heat transfer witha/c for several values ofPr.
REFERENCES
1. F. K. Moore,Unsteady laminar boundary layer flow, NACA TN, 1951, 3471.2. M. J. Lightill, Proc. Roy. Soc.224A, 1–23 (1954).3. C. C. Lin, Proc. 9th Int. Cong. Appl. Mech4, 155–168 (1956).4. J. T. Stuart1, 1–40 (1971).5. N. Riley, SIAM Review 17, 274–297 (1975).6. D. P. Telionis, J. Fluids Engng.101, 29–43 (1979).7. D.P. Telionis,Unsteady Viscous Flows,Springer, Berlin, 1981.8. I. Pop,Theory of Unsteady Boundary Layers(in Romanian), Ed. Stiintifica si Enciclopedica, Bucuresti–Romania, 1983.9. P.K.H. Ma, and W.H. Hui, J. Fluid Mech216, 537–559 (1990).10. D. K. Ludlow, P. A. Clarkson, and A.–P. Bassom, Q. Jl. Mech. Appl. Math. 53, 175–206 (2000).11. C. Y. Wang, Annu. Rev. Fluid Mech23, 159–177 (1991).12. K. N. Lakshmisha, S. Venkateswaran, and G. Nath, ASME J. HeatTransfer 110, 590–595 (1988).13. L.J. Crane, J. Appl. Math. Phys, (ZAMP)21, 645–647 (1970).14. I. Pop, and T.Y. Na,23, 413–422 (1996).15. C. Y. Wang, Q. Du, M. Miklav ˘cic, and C.–C. Chang, SIAM J. Appl. Math.57, 1–14 (1997).16. E. M. A. Elbashbeshy, and M.A.A. Bazid, Heat Mass Transfer41, 1–4 (2004).17. S. Sharidan, T. Mahmood, and I. Pop, Int. J. Appl. Mech. Engng., 11, 647–654 (2006).18. M. E. Ali, and E. Magyari, Int. J. Heat Mass Transfer50, 188–195 (2007).19. C. D. Surma Devi, H.S. Takhar, and G. Nath, Int. J. Heat Mass Transfer 29, 1996– 1999 (1986).20. H. S. Takhar, A.J. Chamkha, and G. Nath, Acta Mechanica146, 59–71 (2001).21. T. Ray Mahapatra, and A. S. Gupta, Heat Mass Transfer38, 517–521 (2002).
Magnetic field and internal heat generation
effects on the free convection in a rectangular
cavity filled with a porous medium
T. Grosan, a C. Revnic, b I. Pop a,∗ and D.B. Ingham c
aBabes-Bolyai University, Applied Mathematics, R-3400 Cluj, CP 253, Romania
b”Tiberiu Popoviciu” Institute of Numerical Analysis, P.O.Box 68-1, 400110,
Cluj, Romania
cCentre for Computational Fluid Dynamics, University of Leeds, Leeds LS2 9JT,
UK
Abstract
A numerical investigation of the steady magnetohydrodynamics free convectionin a rectangular cavity filled with a fluid-saturated porous medium and with internalheat generation has been performed. A uniform magnetic field, inclined at an angleγ with respect to the horizontal plane, is externally imposed. The values of thegoverning parameters are the inclined angle γ = 0, π/6, π/4 and π/2, Hartmannnumber Ha = 0, 1, 5, 10 and 50, Rayleigh number Ra = 10, 103 and 105, and theaspect ratio a = 0.01, 0.2, 0.5 and 1(square cavity). It is shown that the intensityof the core convection is considerably affected by the considered parameters. It isalso found that the local Nusselt number NuY decreases on the bottom wall asγ increases (magnetic field changes its direction from the horizontal to the verticaldirection) and vice-versa for the top wall of the cavity. The reported results are ingood agreement with all the available published work in the literature.
Key words: Porous media, Natural convection, Heat generation, MHD
∗ Corresponding author. Tel:+40 264 594 315; fax: +40 264 591 906. Email address:
[email protected] (I.Pop).
Nomenclature V velocity vector, m · s−1
a aspect ratio T0 temperature of the vertical wall, K
B applied magnetic field, Wb ·m−2 x dimensional Cartesian
cp specific heat at constant pressure, coordinate along the bottom wall, m
KJ · kg−1 ·K−1 y dimensional Cartesian coordinate
g gravitational acceleration vector, m · s−2 along the left vertical wall, m
Ha Hartmann number X,Y dimensionless Cartesian coordinates
h height of the cavity, m Greek symbols
k thermal conductivity, W ·m−1 ·K−1 αm effective thermal diffusivity, m2 · s−1
K permeability of the porous medium, m2 β coefficient of thermal expansion, K−1
l width of the cavity, m γ angle of inclination to the horizontal
Nu mean Nusselt number of applied magnetic field, radians
Nuy local Nusselt number µ dynamic viscosity, kg ·m−1 · s−1
q′′′0
heat generation, W ·m−3 θ dimensionless temperature
Ra Rayleigh number ρ fluid density, kg ·m−3
u, v velocity components along the ρ0 reference density, kg ·m−3
x and y directions, respectively m · s σ electrical conductivity, Ω−1 ·m−1
1 Introduction
Natural convective heat transfer in viscous fluids and fluid-saturated porousmedia has occupied the central stage in many fundamental heat transfer analy-ses and has received considerable attention over the last few decades. Thisinterest is due to its wide range of applications in, for example, packed spherebeds, high performance insulation for buildings, chemical catalytic reactors,grain storage and such geophysical problems as frost heave. Porous media arealso of interest in relation to the underground spread of pollutants, solar powercollectors, and to geothermal energy systems. The literature concerning con-vective flow in porous media is abundant and representative studies in thismay be found in the recent books by Nield and Bejan [1], Ingham and Pop [2],Vafai [3], Bejan et al. [4], Pop and Ingham [5], de Lemos [6] and Vadasz [7].Further, a valuable reference on convective fluids in cavities filled with viscousfluids can be found in the recent book by Martynenko and Khramtsov [8].Natural convection in enclosures in which internal heat generation is present
2
is of prime importance in certain technological applications. Examples arepost-accident heat removal in nuclear reactors and geophysical problems asso-ciated with the underground storage of nuclear waste, among others (Acharyaand Goldstein [9], Ozoe and Maruo [10], Lee and Goldstein [11], Fusegi et al.[12], Venkatachalappa and Subbaraya, [13], Shim and Hyun, [14] , Hossain andWilson [15]).
The present paper investigates the effect of a magnetic field on the steadyfree convection in a rectangular cavity filled with a porous medium saturatedwith an electrically conducting fluid. This type of problem arises in geophysicswhen a fluid saturates the earth’s mantle in the presence of a geomagnetic field.Natural convection flow in the presence of a magnetic field in an enclosure filledwith a viscous and incompressible fluid has been studied by Garandet et al.[16], Alchaar et al. [17], Kanafer and Chamka [18], Chamkha and Al-Naser[19], Mahmud et al. [20], Hossain and Ress [21], Hossain et al. [22], and Eceand Buyuk [23]. However, there are very few studies on the natural convectionof a conducting fluid saturating a porous medium in the presence of a magneticfield in an enclosure. To the best of our knowledge, the first investigation ofthis problem is due to Alchar et al. [17] who considered the stability of aconducting fluid saturating a porous medium in the presence of a uniformmagnetic field using the Brinkman model. However, some comments on theMHD convection in a porous medium have been done very recently by Nield[24]. Also a very recent paper by Barletta et al. [25] has studied the mixedconvection with heated effect in a vertical porous annulus with the radiallyvarying magnetic field.
2 Mathematical model
In this paper we consider the steady natural convection flow in a rectangu-lar cavity filled with an electrically conducting fluid-saturated porous mediumwith internal heat generation. We assume that the enclosure is permeated bya uniform inclined magnetic field. The geometry and the Cartesian coordinatesystem are schematically shown in Fig. 1, where the dimensional coordinatesx and y are measured along the horizontal bottom wall and normal to it alongthe left vertical wall, respectively. The height of the cavity is denoted by hand the width by l. Further, the angle of inclination of the magnetic field B
from the horizontal plane, and measured positively in the counterclockwisedirection is denoted by γ. It is assumed that the vertical walls are maintainedat a constant temperature T0 , while the horizontal walls are adiabatic. A uni-form source of heat generation in the flow region with a constant volumetricrate of q′′′
0[W ·m−3] is also considered. Further, it is assumed that the effect of
buoyancy is included through the well-known Boussinesq approximation. The
3
viscous, radiation and Joule heating effects are neglected. The resulting con-vective flow is governed by the combined mechanism of the driven buoyancyforce, internal heat generation and the retarding effect of the magnetic field.The magnetic Reynolds number is assumed to be small so that the inducedmagnetic field can be neglected compared to the applied magnetic field.
Under the above assumptions, the conservation equations for mass, mo-mentum under the Darcy approximation, energy and electric transfer are giveby
∇.V = 0 (1)
V =K
µ(−∇p+ ρg + I × B) (2)
(V.∇)T = αm∇2T +
q′′′0
ρ0cp(3)
∇.I = 0 (4)
I = σ (−∇φ+ V × B) (5)
ρ = ρ0[1 − β(T − T0)] (6)
where V is the fluid velocity vector, T is the fluid temperature, p is the pres-sure, B is the external magnetic field, I is the electric current, φ is the electricpotential, g is the gravitational acceleration vector, K is the permeability ofthe porous medium, αm is the effective thermal diffusivity, ρ is the density,µ is the dynamic viscosity, β is the coefficient of thermal expansion, cp isthe specific heat at constant pressure, σ is the electrical conductivity, ρ0 isthe reference density and -∇φ is the associated electric field. As discussed byGarandet et al. [16], Eqs. (4) and (5) reduce to ∇2φ = 0. The unique solutionis ∇φ = 0 since there is always an electrically insulating boundary aroundthe enclosure. Thus, it follows that the electric field vanishes everywhere (see,Alchaar et al., [17]).
Eliminating the pressure term in Eq. (2) in the usual way, the governingequations (1) to (3) can be written as
∂u
∂x+∂v
∂y= 0 (7)
4
∂u
∂y−∂v
∂x= (8)
−gKβ
υ
∂T
∂x+σKB2
0
µ
(
−∂u
∂ysin2 γ + 2
∂v
∂ysin γ cos γ +
∂v
∂xcos2 γ
)
u∂T
∂x+ v
∂T
∂y= αm
(
∂2T
∂x2+∂2T
∂y2
)
+q′′′0
ρcp(9)
which has to be solved subject to the boundary conditions
u = 0, T = T0 at x = 0 and x = l, 0 ≤ y ≤ h (10)
v= 0,∂T
∂y= 0 at y = 0 and y = h, 0 ≤ x ≤ l
where B0 is the magnitude of B and υ is the kinematic viscosity of the fluid.Further, we introduce the following non-dimensional variables
X =x
l, Y =
y
h, U =
h
αm
u, V =l
αm
v, θ =T − T0
(q′′′0 l2/k)
(11)
where k is the thermal conductivity. Introducing the stream function ψ definedas U = ∂ψ/∂Y and V =-∂ψ/∂X, and using expressions (11) in Eqs. (7) - (9),we obtain the following partial differential equations in non-dimensional form:
∂2ψ
∂X2+ a2
∂2ψ
∂Y 2= −Ra
∂θ
∂X−Ha2
(
a2∂2ψ
∂Y 2sin2 γ + 2a
∂2ψ
∂X∂Ysin γ cos γ
+∂2ψ
∂X2cos2 γ
)
(12)
∂2θ
∂X2+ a2
∂2θ
∂Y 2+ 1 = a
(
∂ψ
∂Y
∂θ
∂X−∂ψ
∂X
∂θ
∂Y
)
(13)
which have to be solved subject to the boundary conditions
ψ = 0, θ = 0, at X = 0 and X = 1, 0 ≤ Y ≤ 1 (14)
ψ= 0,∂ψ
∂Y= 0,
∂θ
∂Y= 0 at Y = 0 and Y = 1, 0 ≤ X ≤ 1
where a = l/h is the aspect ratio of the cavity, Ra is the Rayleigh number andHa = σKB2
0/µ is the Hartmann number for the porous medium. It should
5
mentioned that γ = 0 corresponds to a horizontal magnetic field and γ = π/2corresponds to a vertical magnetic field, respectively.
Once we know the temperature we can obtain the rate of heat transferfrom each of the vertical walls, which are given in terms of the local Nusseltnumber NuY and the mean Nusselt number Nu which are defined as
NuY = −
(
∂θ
∂X
)
X=0
, Nu = −
1∫
0
(
∂θ
∂X
)
X=0
dY (15)
3 Numerical method and validation
To obtain the numerical solution of Eqs (12) and (13) a central finite-difference scheme was used and the system of discretized equations has beensolved using a Gauss-Seidel iteration technique. The unknowns θ and ψ werecalculated iteratively until the following criteria of convergence was fulfilled:
|max[χnew(i, j) − χold(i, j)]| ≤ ε (16)
where χ represents the temperature or the stream function and ε is the con-vergence criteria. In all the results presented in this paper, ε = 10−7 was foundbe sufficiently small such that any smaller value produced results which weregraphically the same. In order to choose the size of the grid, accuracy testsusing the finite different method and Richardson extrapolation [26] for meshsensitivity analysis were perform for Ra = 103, Ha = 0, and aspect ratioa = 1, using three sets of grids: 26 × 26, 51 × 51, 101 × 101 and 201 × 201 asshown in Table 1. Reasonably good agreement was found between the 51× 51and 101× 101 grids and therefore the grid used in this problem was 101× 101and these give accurate results for Ra ≤ 103. We have also found that 201×201grids give accurate results for Ra ≤ 105.
Table 1Accuracy test for Ra = 103, Ha = 0 and a = 1
Nodes ψ(0.24, 0.24) θ(0.24, 0.24)
26 × 26 2.6368 0.0389
51 × 51 2.5987 0.0384
101 × 101 2.5800 0.0382
201 × 201 2.5707 0.0381
Richardon extrapolation 2.5614 0.0380
6
Further, in order to verify the accuracy of the code we compared the ob-tained results for the case when the magnetic field is absent (Ha = 0), a = 0.5and Ra = 10 and 103, respectively, with those obtained by Haajizadeh et al.
[27]. These results are shown in Table 2. It can be concluded from this ta-ble that the results are in good agreement and we can be confident that thepresent analysis and the code used are correct.
Table 2Comparison of ψmax and θmax for Ha = 0 and a = 0.5
Ra Haajizadeh et al. [27] Present (Richardson extrapolation)
ψmax θmax ψmax θmax
10 0.078 0.130 0.079 0.127
103 4.880 0.118 4.833(4.832) 0.116(0.116)
4 Analytical solution
For small values of a(≪ 1), the solution of Eqs. (12) and (13) is given bythe leading order terms, ψ = ψ0(x) and θ = θ0(x), see Mandar et al. [28]
(1 +Ha2cos2γ)∂2ψ
0
∂X2= −Ra
∂θ0
∂X(17)
∂2θ0
∂X2+ 1 = 0 (18)
with the boundary conditions
ψ0 = θ0 = 0 at X = 0 and X = 1 (19)
On solving Eqs. (17) and (18) with the boundary conditions (19), we obtain
θ0(X) =X(1 −X)
2(20)
ψ0(X) =Ra
2(1 +Ha2 cos2 γ)
(
1
6X −
1
2X2 +
1
3X3
)
7
5 Results and discussions
In this section we present numerical results for the streamlines, isothermsand velocity profiles on the left wall, for various values of the magnetic fieldparameter Ha, inclination angle γ of the magnetic field and the Rayleighnumber Ra. In addition, the local Nusselt numbers NuY have been calculated.Figures (2) -(6) show plots of the streamlines and isotherms for an aspectration a = 1, Rayleigh number Ra = 103 and 105, magnetic field parameterHa = 0, 1, 10 and 50 and values of the inclined angle γ = 0, π/6, π/4 and π/2.It is seen from these figures that the intensity of the convection in the core isconsiderable affected by the magnetic field. A weak convective motion with abicellular structure is induced, see Figs. 2 to 5. The two cells are symmetricalwith respect to the central plane according to the value of γ. It is also observedthat these two cells rotate for γ = π/6 and π/4. It is seen from Fig. 2 thatthe pattern of the streamlines and isotherms are similar to those predicted byHaajizadeh et al. [27]. On the other hand, Fig. 3 illustrates that for relativesmall values of Ha (Ha = 1) the maximum stream function increases and themaximum temperature decreases as γ increases. However, for larger values ofHa (Ha >> 1) the maximum of both the stream function and the temperatureincrease as γ increases. Therefore, the presence of the magnetic force tendsto accelerate the fluid motion inside the cavity when the direction of themagnetic field changes from the horizontal to the vertical direction. Further,for Ra = 103 and higher values of Ha, the isotherms are almost parallel andthis implies that conduction is dominant, see Figures 4 and 5. Also, thesefigures show that when the magnetic field is horizontal (γ = 0) and Ra = 103
the core vortex is elongated vertically as the Hartmann number increase. Forthe value of Ra and γ considered, the core streamlines start to flatten atthe top and bottom of the cavity, while the isotherms are almost parallel.This indicates that conduction is dominated, see Fig. 5. For high Rayleighnumbers, the flow and heat transfer regime is characterized by a thermallystratified core region and two thin boundary layers on the vertical walls, seeFig. 6(a). Also, as the Raigleigh number and magnetic field increase (Ra = 105
and Ha = 10, 50), stronger convective motion takes place and the core vortexbreaks up into three cells. There is also a weak distortion of the isotherms,as indicated in Figs. 6(b)-(c). This occurs at smaller inclination angles of themagnetic field (γ = π/6). This is in agreement with the results reported byAl-Najem et al. [29] for the case of natural convection in a two-dimensionalsquare cavity filled with a viscous fluid with a transverse magnetic field.
Typical velocity profiles at the vertical walls, Uw for Ra = 103 and differ-ent values of γ and Ha are shown in Fig. 7. We observe that for a fixed valueof Ha that the minimum of the wall velocity is attained when the magneticfield is in the vertical direction (γ = π/2). On the other hand, for γ = π/4 thewall velocity decreases as Ha increases. We observe that for all the values of
8
Ra considered, the maximum temperature profiles do not overshoot the limitprescribed by the analytical solution (θmax = 1/8), see Fig. 8. Numerical andanalytical solutions for the streamlines on the horizontal centerline plane ofthe cavity are shown in Figs. 9 for a small value of the aspect ratio, namelya = 0.01 and for Ra = 103 when Ha = 0, 1, 5 and γ = 0 (Fig. 9a) and whenHa = 1 and γ = 0, π/4, π/2 (Fig. 9b). We observe from these figures thatthere is very good agreement between the analytical and numerical solution.
Figure 10 shows the variation of the local Nusselt number NuY with Yfor Ra = 100 and γ = 0. In order to compare the present results with thoseobtained by Haajizadeh et al. [27] when the magnetic field is absent(Ha = 0)the value of 2NuY for a = 0.2 is presented in this figure. It is observed thatthe results are in very good agreement and therefore we are confident that thepresent results are accurate. Finally, Fig. 11 presents the variations of the localNusselt number NuY with Y for Ra = 100, Ha = 1 and for several values ofγ when a = 1. It is also found that the local Nusselt number NuY decreasesat the bottom wall as γ increases (magnetic field changes its direction fromthe horizontal to the vertical direction) and vice-versa for the top wall of thecavity.
6 Conclusion
The present numerical study exhibits many interesting features concerningthe effect of the inclined magnetic fields on the free convection flow and heattransfer characteristics in a rectangular cavity filled with a porous medium.Detailed numerical results for the temperature distribution and heat transferhave been presented in graphical and tabular form. The main conclusions ofthe present analysis are as follows:
• In general, it has been found that the effect of the magnetic field is to reducethe convective heat transfer inside the cavity.
• The convection modes within the cavity were found to depend upon boththe strength and the inclination of the magnetic field. The applied magneticfield in the horizontal direction was found to be most effective in suppressingthe convection flow for a stronger magnetic field in comparison with thevertical direction.
• It is found that strong boundary layers are formed near the vertical wallsfor Ra = 105 and γ = π/6 and the intensity increases as Ha increases.The flat isotherms in the core region indicate that there is negligible lateralheat conduction and the equal spacing of the streamlines implies a uniformvertical velocity in this region, as predicted by boundary layer theory, seeFig. 6(a).
9
• The magnetic field has a negligible effect on the heat transfer mechanism forsmall values of γ and Ha >> 1. This is true since pure conduction becomesdominant when the magnetic field is applied in the horizontal direction(γ = 0). However, for Ra = 105 the the parabolic profile is distroyed.
• For Rayleigh number Ra = 103, and small Hartmann numbers, the flow andheat transfer are characterized by a parallel flow structure in the centralregion of the cavity. The conduction is the dominant mode of heat transferand vertical velocity profiles and temperatures are almost parabolic.
• It should be pointed out that the general analysis described in this workcan represent a useful starting point to treat more complex problems, suchas, for example, time-dependent flows.
Acknowledgements
The work done by T. Grosan and C. Revnic is supported by the MEDCTunder Grant Ceex 90/2006. D.B. Ingham and I. Pop wishes to express theirthanks to the Royal Society for partial financial support.
References
[1] D.A. Nield, A. Bejan, Convection in Porous Media (3th edition), Springer, NewYork, 2006.
[2] D.B. Ingham, I. Pop (eds.), Transport Phenomena in Porous Media, Elsevier,Oxford, 2005.
[3] K. Vafai (ed.), Hanbook of Porous Media, (2nd edition), Taylor&Francis, BocaRaton, 2005.
[4] A. Bejan, I. Dincer, S. Lorente, A. F. Miguel, A. H. Reis, Porous and ComplexFlows Structures in Modern Technologies, Springer, New York, 2004.
[5] I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical andComputational Modelling of Viscous Fluids and Porous Media, Pergamon, Oxford,2001.
[6] M.J.S. de Lemosm, Turbulence in Porous Media: Modeling and Applications,Elsevier, Oxford, 2006.
[7] P. Vadasz (ed.), Emerging Topics in Heat and Mass Transfer in Porous Media,Springer, New York, 2008.
[8] O. Martynenko, P. Khramtsov, Free-Convective Heat Transfer, Springer, Berlin,2005.
[9] S. Acharya, R.J. Golstein, Natural convection in an externally heated verticalor inclined square box containing internal energy sources, J. Heat Transfer 107(1985) 855-866.
10
[10] H. Ozoe, K. Okada, The effect of the direction of the external magnetic field onthe three-dimensional natural convection in cubical enclosure, Int. J. Heat MassTransfer 32(1989) 1939-1954.
[11] J.-H. Lee, R.J. Goldstein, An experimental study on natural convection heattransfer in an inclined square enclosure containing internal energy sources, J. HeatTransfer 110 (1988) 345-349.
[12] T. Fusegi, J. M. Hyun, K. Kuwahara, Natural convection in a differentiallyheated square cavity with internal heat generation, Numer. Heat Transfer, PartA 21 (1992) 215-229.
[13] M. Venkatachalappa, C.K. Subbaraya, Natural convection in a rectangularenclosure in the presence of a magnetic field with uniform heat flux from theside walls, Acta Mechanica 96 (1993) 13-26.
[14] Y.M. Shim, J.M. Hyun, Transient confined natural convection with internalheat generation, Int. J. Heat Fluid Flow 18 (1997) 328-333.
[15] M.A. Hossain, M. Wilson, Natural convection flow in a fluid-saturated porousmedium enclosed by non-isothermal walls with heat generation, Int. J. ThermalSci. 41 (2002) 447-454.
[16] J.P. Garandet, T. Albussoiere, R. Moreau, Buoyancy driven convection in arectangular enclosure with a transverse magnetic field, Int. J. Heat Mass Transfer35 (1992) 741-748.
[17] S. Alchaar, P. Vasseur, E. Bilgen, Natural convection heat transfer in arectangular enclosure with a transverse magnetic field, J. Heat Transfer 117 (1995)668-673.
[18] K. Kanafer, A.J. Chamkha, Hydromagnetic natural convection from an inclinedporous square enclosure with heat generation, Numer. Heat Transfer, Part A 33(1998) 891-910.
[19] A.J. Chamkha, H. Al-Naser, Double-diffusive convection in an inclined porousenclosure with opposing temperature and concentration gradients, Int. J. ThermalSci. 40 (2001) 227-244.
[20] S. Mahmud, S.H. Tasnim, M.A.H.Mamun,Thermodynamic analysis of mixedconvection in a channel with transverse hydromagnetic effect, Int. J. Thermal Sci.42 (2003) 731-740.
[21] Md.A. Hossain, D.A.S.Rees, Natural convection flow of water near its densitymaximum in a rectangular enclosure having isothermal walls with heat generation,Heat Mass Transfer 41 (2005) 367-374.
[22] M.A. Hossain, M.Z. Hafiz, D.A.S. Rees, Buoyancy and thermo capillary drivenconvection flow of an electrically conducting fluid in an enclosure with heatgeneration, Int. J. Thermal Sci. 44 (2005) 676-684.
[23] M.C. Ece, E. Buyuk, Natural-convection flow under a magnetic field in aninclined rectangular enclosure heated and cooled on adjacent walls, Fluid Dyn.Res. 38 (2006) 564-590.
11
[24] D.A. Nield, Impracticality of MHD Convection in a Porous Medium, TransportPorous Media 73 (2008) 379-380.
[25] A. Barletta, S. Lazzari, E. Magayri, I. Pop, Mixed convection with heating effectin a vertical porous annulus with a radially varying magnetic field, Int. J. HeatMass Transfer (online).
[26] G.D. Smith, Numerical Solution of Partial Differential Equations. FiniteDifference Method, Oxford university Press, New York, 2004.
[27] M. Haajizadeh, A. F. Ozguc,C. L. Tien, Natural convection in a vertical porousenclosure with internal heat generation, Int. J. Heat Mass Transfer 27 (1984)1893-1902.
[28] N.M. Al-Najem, K.M. Khanafer, and M.M. El-Refaee, Numerical study oflaminar natural convection in tilted enclosure with transverse magnetic field, Int.J. Numer. Methods Heat Fluid Flow 8 (1998) 651-672
[29] V.J. Mandar, U. N. Gaitonde and K.M. Sushanta, Analytical study of naturalconvection in a cavity with volumetric heat generation, J. Heat Transfer 128 (2006)176-182.
List of Figures
Figure 1. Geometry of the problem and co-ordinate system.
Figure 2. Streamlines and isotherms for Ra = 103, Ha = 0 and when themagnetic field is in horizontal direction (θ = 00).
Figure 3. Streamlines and isotherms for Ra = 103, Ha = 1 and for differentvalues of γ.
Figure 4. Streamlines and isotherms for Ra = 103, Ha = 10 and for differentvalues of γ.
Figure 5. Streamlines and isotherms for Ra = 103, Ha = 50 and for differentvalues of γ.
Figure 6. Streamlines and isotherms for Ra = 105, γ = π/6 and for differentvalues of Ha.
12
Figure 7. The velocity profile in the vicinity of the vertical wall for differentvalues of γ and Ha.
Figure 8. Temperature profiles for a = 0.01 and Ra = 1000.
Figure 9. Comparison between the streamlines obtained analytical and nu-merical:(a) for γ = 0 and different values of Ha;(b) for Ha = 1 and different values of γ.
Figure 10. Nusselt number for Ra = 100, a = 0.2, γ = 0 and different valuesof Ha.
Figure 11. Nusselt number for Ra = 100, a = 1, Ha = 1 and different valuesof γ.
13
Fig. 1. Geometry of the problem and co-ordinate system.
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 0, ψmax
= 3.5338, Tmax
= 0.097359
0 0.5 10
0.2
0.4
0.6
0.8
1
Fig. 2. Streamlines and isotherms for Ra = 103, Ha = 0 and when the magnetic
field is in horizontal direction (θ = 00).
14
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 1, γ=0, ψmax
= 2.4677, Tmax
= 0.11189
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 1, γ=π/6, ψmax
= 2.5867, Tmax
= 0.11026
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 1, γ=π/4, ψmax
= 2.7473, Tmax
= 0.10832
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 1, γ=π/2, ψmax
= 3.2759, Tmax
= 0.10343
0 0.5 10
0.2
0.4
0.6
0.8
1
Fig. 3. Streamlines and isotherms for Ra = 103, Ha = 1 and for different
values of γ.
15
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 10, γ=0, ψmax
= 0.080034, Tmax
= 0.12657
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 10, γ=π/6, ψmax
= 0.13141, Tmax
= 0.12792
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 10, γ=π/4, ψmax
= 0.19473, Tmax
= 0.12903
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1Ha = 10, γ=π/2, ψ
max= 0.46131, T
max= 0.13168
0 0.5 10
0.2
0.4
0.6
0.8
1
Fig. 4. Streamlines and isotherms for Ra = 103, Ha = 10 and for
different values of γ.
16
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 50, γ=0, ψmax
= 0.0032072, Tmax
= 0.12505
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1Ha = 50, γ=π/6, ψ
max= 0.0054676, T
max= 0.12513
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha = 50, γ=π/4, ψmax
= 0.0082989, Tmax
= 0.12518
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1Ha = 50, γ=π/2, ψ
max= 0.022856, T
max= 0.12535
0 0.5 10
0.2
0.4
0.6
0.8
1
Fig. 5. Streamlines and isotherms for Ra = 103, Ha = 50
and for different values of γ.
17
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1Ha=1, γ=pi/6, ψ
max =22.0839, T
max=0.0263
(a)
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha=10, γ=π/6, ψmax
=4.22472, Tmax
=0.08443
(b)
0 0.5 10
0.2
0.4
0.6
0.8
1
0 0.5 10
0.2
0.4
0.6
0.8
1
Ha=50, γ=π/6, ψmax
=0.53527, Tmax
=0.13394
(c)
Fig. 6. Streamlines and isotherms for Ra = 105, γ = π/6 and fordifferent values of Ha.
18
0 0.2 0.4 0.6 0.8 1−15
−10
−5
0
Y
Uw
Ra=103, Ha=10
γ=0, π/6, π/4, π/3,π/2
0 0.2 0.4 0.6 0.8 1−60
−50
−40
−30
−20
−10
0
Y
Uw
Ra=103, γ=π/4
Ha=0, 5, 10, 20, 30
Fig. 7. The velocity profile in the vicinity of the vertical wall for differentvalues of γ and Ha.
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
x
T(x,
0.5
)
ooo Analytic___ numeric
Fig. 8. Temperature profiles for a = 0.01 and Ra = 1000.
19
0 0.2 0.4 0.6 0.8 1−10
−8
−6
−4
−2
0
2
4
6
8
x
ψ(x,
0.5
)
Ha=0, 1, 5Ra=1000γ=0a=0.01
ooo analytic___ numeric
(a)
0 0.2 0.4 0.6 0.8 1−10
−8
−6
−4
−2
0
2
4
6
8
x
ψ(x,
0.5
)
ooo analytic___ numeric
γ=π/2, π/4, 0Ra=1000,Ha=1a=0.01
(b)
Fig. 9. Comparison between the streamlines obtained analytical andnumerical:
(a) for γ = 0 and different values of Ha;(b) for Ha = 1 and different values of γ.
20
0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 NuY
Y
Ha = 0, 1, 5, 10
o Haajizadeh [24] __ Present
Fig. 10. Nusselt number for Ra = 100, a = 0.2, γ = 0 and different values ofHa.
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
NuY
Y
γ=0, π/6, π/4, π/2
γ=0, π/6, π/4, π/2
Fig. 11. Nusselt number for Ra = 100, a = 1, Ha = 1 and differentvalues of γ.
21
Anexa II
Abstractul articolelor publicate in jarnak neindexate ISI
, 2446, vol.1l, No.3
)
Lettet to lhe Edilol
A NOTE ON THE EFFECT OF RADIATION ON F'REE CONVECTIONOVER A VERTICAL FLAT PLATE EMBEDDED IN A NON.NEWTONIAN
T.LUID SATURATED POROUS MEDIUM
T. GROSAN and L POP-Faculry of Mathematics and Computer Science
Babeq-Bolyai University, R-3400 Cluj-Napoca, ROMANIAe-mail: [email protected].!o
The effect ofradiatron on the ftee convectjon fiom a v€dical plale enbedded in a lower-taw flujd satumtedlorous nedia has been considered. Similarity squations have been obtaired dd solved numedcalty. rr was fomdthar therc is u incr€ase in the boundary layer thickness wirh an increase in the 'adiation parmeter / and adecrease in ihelrqwer,law ind€x, was obsew€d.
Key wordstporous media, non-Ddcy law, bolndary layer, radiarion.
TECHMSCHE MECHANIK, Bed 26, Hefr r, (2006), l l-t 9Mdskip&ingllg: 03. Novmbd 2005
Rotating Flow of Power-Law Fluids over a Stretching Surface
M. KMi, T. Grosan, I. Pop
ne stearr $ow of a non-Ne\'toniu pawer-lN ll"id due to a st/etchins suface in a rctatins tuid het beeniwestigated in this paper Afret a similaritr bantfalnation, the set of non-linear odine", difercntial equatia6ha|e been solyed ftunzrically ,sins the Ke efiox nethod far sone wlues af the pt Meter )- which is theratia ofthe ntation rute to the st/etching rute M.l the pave/-lorr index n. It is faund that bath the skinfriction!caefrcients in the x and y dircctiolts dzclease ith the ircrcase of tha palaneter ),. Havew, fo/ smallelwluesof A the skn liiction coeficients arc highet fo/ the Abtant Jhid and ffia er for the psedoplastic luid,
' )
, )
STUDTA uNrV. (BA9E5,Boryar'] MATIIEMATIOA, VoLlse Lt, NuEber 4j De.eBbe. ?006
TIIERMAL RADIATION ET'FECT ON FULTY DEVDLOPEDFR-EE CONVECTION IN A VERIICAL RECTANGULAR DUCT
T. GROgAN, T.
Dedicoftn h Praie$or Ghturshe Conen at hi,70tk annhrers',r!
Abstract. The efiect of radiatioD on th€ steady &e€ conv€ctios flo', i.€.
the case of puely buoFecy-&iven flolv, in a veriical rectaryutar dl1cr b iD-
vestigated for lalmiEar ard tully developed iegime. :rhe Ross€la4d applon-
:!atio! iE coD€id€red aed teBperatures of tha walj! are assuIned colstads,
Th€ goverdng equations ai€ 6(p!e$ed i! non-dinensional forn ud a.re
solved both aralyticaly and aueencauy. li a"s found that the govsn-
il1g pduetas have a 6igdf.ad €fect on the aelocity aod temleraiue
Piofiles.
BULLETD"I OF TIE TRANSILVANIA ,NN€RSITY OF BRA$OV
NON-LINE"AR DENSITY VARIATION EFFECTSON THE FULLY DEVELOPED MXEDCOT\TVECTION FLOW IN A !.ERTICAL
CHANNEL
| i - r ,
T. GRO$AN' r. POP',
Abstract: 1]E efect of the qla&aii. tun oJ d;Nry rtiatia yith tenryalveon the steadJ nixed coMcnon fo\.t h d ve rcat chamet is hrrestigated Joltaliw ad!/ t devlopedlow rcgine .tn the nod2 iks of ttg hzat tosfer'lEvtscotts ditsipanu ana v6 consid.red and tenpetuhrer al the ltaus ate6same.t cMafls lhe go\ammg .guatuns dre ?4rese.t n non4ireNionalJa.n ed ne rolved both nalyticd !a nhedcal). h uatlothd that thete isa denw in rcrEtrat fow \'ith @ mcrcase n he nireu tu
Rerwords: htly developedltow. ,** **,*^,,*-^ ***n* |t
Mixed convection flow along a thin vertical cylinder withlocalized heating or cooling in a porous medium
M. KumariDepartment of Mathematics, Indian lnstitute of Bangalore, Bangalore, India
C. Bercea and I. Pop*Faculty of Mathematics University of Cluj, Cluj CP 253, Romania
The effects of localized cooling/heating on the steady mixed convection boundarylayer flow over a thin vertical cylinder embedded in a fluid saturated porous mediumunder the assumption of Darcy law has been theoretically studied. The localizedcooling/heating introduces a finite discontinuity in the mathematical formulation of theproblem, which increases its complexity. ln order to overcome this difficulty, a non-uniform distibution of the wall temperature is considered at certain sections of thecylinder. The nonlinear coupled parabolic partial differential equations have beensolved numerically by using an implicit finite-difference scheme similar to thatproposed by Blo,ttner [23].
Key words: mixed convection, vertical cylinder, porou medium, boundary layer
Seria B1, Vol 13, pag 3l-38, 2006
FORCED CONVECTION BOUNDARY LAYERFLOW OVER A FLAT PLATE WITH
VARIABLE THERMAL CONDUCTIVITYEMBEDDED IN A POROUS MEDIUM
C. BERCEA- I. POP*
Absttact: The efect of variable thermal difinivity onthe stea$t forced convection boundary layer flow past a
tlat plae which is embedded in afuid-saturated porousmedium has been studied in this paper. The basicpartial dffirential equatioru of continuity, Darcy lanand the energl arc transformed into a single ordinarydiferential equation using a simple similaritytransformation. This equation is solved analytically andmrmerically.
Keywords: forced corwection, boundary layer flo*-,variable thermal conductivin. Keller - box method.
TECHMSCHEI,ECHANIK,B^|27,t+t), eoar.t 4iMdurkipteingugt 6 tdi 2m6
T. Crrosan, I. Pop
The efect of rcdiatioa an the stead! nbed coftvediotflo,,e in a \)etical channet is ikwsngapdior tanihdl andfuny ddelaped lav legiw. Th. Rasseland apprcximation is c@sidercd in the node ing of thz conau.nan-rudiatiat heat ba er md tenpelatules of the wals are ^sumed constants. me g@ening eEations areer?ressed in nan-.liw$ionolfam ard oe sotued both aMUicaU, td nuneica r It '/6 fouhd thlt there is ade.tease in lewrsal JIow trith an ircree in the ladiation ptMeter&
Thermal Radiation Effect on Fully Developed Mixed ConvectionFlow in a Vertical Channel
Effect ofnon-uniform suction or itrjection on mired convection llow over a vertical
cylinder embedded in a porous medium
M. Kumarit, C. Bercea and I. Pop"
tDepa/tnent of Mathenatics, Indian lhstitute of Bangalare, Bangalore, India
'Faatltr ofMathematics Untuenity of Cluj, R-3400 Cluj, Claj CP 253, Ronania
Abslracl The efecl of steady non-xtdform suction or injection on mixed convection boxndary
layer flow over a vertical heated or cooled permeable cylinder, wbich js embedded in a fluid-
sahEated porous medium, is studied numericalty using the Darcy law approximatiotr. Both
assisting and opposing flow cases ate co$idered. Using suitable transformations, the coupled
goveming bomdary layer equations are transformed into a folm suitable for a DuEerical
solutiotr. The €fects of the suction or injection" bansverse curvature and mixed convection
parameteis or the local Nusselt numbei and temperatue profiles are studied. The obtain€d
results are presented graphically and discxssed in details.
Ketrwords bo]utr.rdaty layer, h€at transfer, mixed convection, porous medium suction/injection,
v€rtical cylinder
T
)
BOUNDARY LAYERS GROWTH ON A MOVING SURFACE DUE TO ANIMPULSIVE MOTION AND A SUDDEN INCREASE OF WALL
HEAT FLUX
M, KUMARIDepartment of MathematicsIndian Institute of Bangalorc
Bangalore, INDIA
T. GROSAN and I. POP'Iaculty of Mathematics
University of ClujR-3400 Ciuj, CP 253, ROMANIA
e-maiL [email protected]
In this.pap€r we investigate the d€velopment of the momentum and themal bomdary layers over acontinuous moving semi-infinite flat plate when the extemal sheam starts impulsively tom resr artime r = 0 wilh a constant velocify u- . lt is assumed lhat rhe plate starts to supply heat to the fluid a!a constant rate q* at time | = 0 a,1d maintained at this rate. The problem has been formulaied ln anew system of scaled coordinaies sirch that for I + = , ir rcduces to Rry Le igh twe of aquation and fort*-)6 (large rime) it reduces to Blasius or Sakiadis rwe of equarion. A new scale ofdimensionlesstrme € has been used which reduces the region of time inregrarion tom an infinite region (O < I < .)to a finite time region (, < e < 1) which rcduces the computarionat time considerabty. The govemingpartial differeniial equations are translormed into a singular parabolic partiai differential equatjonswhich have been solved numerically for a range of valu€s of the goveming parameteB using animplicit finiie-differ€nce schemg fte resuits show that there is a smooth transition tom Ravieishsolution to Blasius or Sakiadis solution as the dimensiorless time e increases ftom zero to one.
Key words: continuously novins Dlate, uns!€ady boundary layer, heat flu rate. numerical solution.
lnt. J. afApplied Mechanics and E in8, 2043, val !3, No.l,
sruDl uNrv. !tsaBrt BoLYA
Radiation and rariable viscosity efiects in forced convection from ahorizontal plate embedded in a porous medium
Abst .acr . Rad iar :oDabd 6mp+erupdbocDdanr v i6@s. tJ F6pc .s oD fo rc"d.ob\e.||oD oouodDry lovpr 1ow der a borizolrdt pta". abocddpd in a Ruro_$fu€ lFd po- rcG med i6 ios rud i "d i ! rb is pappr . Dd.ys .aw Dodet , Rcse_rand DodF 'q radralion and
"tr qvdse proponio!.I lsw for r-mparur
'l.pebdFDt vi€-o6i-r hava be"D .oD"idded Th" LrMsformed ordinary drt_t-r"Dral "qu6LioB de.solv"o lumF-haly. ud d v-r) sood aerFb6DL b*rgPd be prspnr -esul6 ud rbos" rppo-1-d fof parLi. utd "irus, ion" rre
Proceedings of 3ICAMP3'd International Conference on Applications of Porous MediaMay 29 - June 3, 2006, Marrakesch, MOROCCOPaper number tt4tt
MIXED CONVECTION ALONG A VERTICAL WAVY SURFACEWITH A DISCONTINUOUS TEMPERATURE PROFILE IN A
POROUS MEDIUM
M. Kumari("). C.Bercea(b)" and I. Popbl(')Department of Mathematics, lndian lnstitutc of Bangalore, Bangalore, lndia(o)Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania
ABSTRACTThe steady mtxed convection boundary layer flow along a vawy vertical wall with a discontinuoustemperature profile and embedded in a fluid-saturated porous medium is considered in this paper.The overall surface is equally divided into a heated section succeeded by an unheated sectionalternately. The basic continuity equation, the Darcy law and the energy equation are transformedusing a simple coordinate transformation which transforrns the inegular surface into a flat surt'ace.These equations are solved numerically using a very efficient implicit finite-difference method. The'influence of the rnixed convection and wavy geometry parameters on the skin friction coefficient.local Nusselt number and velocity and temperature proliles have been studied in details. Thc
numerical results demonstrate that values of the local Nusselt number are positivc (heat istransi-erre tiom thc u,ail to the fluid) in the heated rcgions and it is negative theat is transf'errcdtiom the t'luid to tlie sall) in the unheated regions. respectircll '. The obtained results for the heat
transf i . r f rorrr the rral l are also compared ui th the corresponding rcsults tbr a rert ical t lat platc
embeddc.d in a porous ntediunr s ith a unitbrnt tet'nperature distribtrtion.I
STUDIA UMV. 'BABE9-BOLYAI' ' MATHEMATICA, Volqmc !IIf, Number 2, Juac 2OOB
HEAT TRANSFER INON
AXISYMMETR,IC STAGNATION .FLO'WA THIN CYI,INDER
)
CoRNELTA REvNrc, T:oD:n.GRolAN, 1c!ND l:AN
pop
Abetract. The eteady axisymetric staga.ation flow and heat tra,nsfer on
a tbi$ infnite cyli:rder qf radius a is studied in thil rpaper., Both cases
of consta.at wall temperattle and constant wall heat ux are considered.
Usiag similarity va.riables the gwerning partial diferential equations are
transfornred into ordinary difieiential equatioib. lhe''resuttini set of
two equations is eolved numerically rising Rirng+Kuttb urethod combined
with a shootitrg technique. For the special case of the S,eyaolde number
Re >> 1 (bounary layer approximation), we obtaiired an asymptotic solu-
tion wbich include the Hiemeaz solution. The p:eseut rysyltt-s are compared
in aome particular cases with exisiing resuits from the open literature and
with the asymptotic approcimatioo, and we found a very'lood agteement.
Ii iB sh# that the Nusselt number and the skin fictiod increase and the I
boundary Layer thickness decreases witb the increase oftba Reyaolde nun-
ber. Some grapbs for the velocity and tenoperature profiles are preoented.
Also, tables with values related to tbe skia fiction and Nusselt;number
a,re given, L r:.,
