TS1 Sharma.vp

EFFECTS OF OHMIC HEATING AND VISCOUS DISSIPATION ONSTEADY MHD FLOW NEAR A STAGNATION POINT ON AN

ISOTHERMAL STRETCHING SHEET

by

Pushkar Raj SHARMA and Gurminder SINGH

Orig i nal sci en tific pa perUDC: 532.543.5:536.628:66.011

BIBLID: 3354-9836, 13 (2009), 1, 5-12DOI: 10.2298/TSCI0901005S

Aim of the pa per is to in ves ti gate ef fects of ohmic heat ing and vis cous dis si pa tion on steady flow of a vis cous in com press ible elec tri cally con duct ing fluid in the pres ence of uni form trans verse mag netic field and vari able free stream near a stag na tionpoint on a stretch ing non-con duct ing iso ther mal sheet. The gov ern ing equa tions ofcon ti nu ity, mo men tum, and en ergy are trans formed into or di nary dif fer en tial equa -tions and solved nu mer i cally us ing Runge-Kutta fourth or der with shoot ing tech -nique. The ve loc ity and tem per a ture dis tri bu tions are dis cussed nu mer i cally andpre sented through graphs. Skin-fric tion co ef fi cient and the Nusselt num ber at thesheet are de rived, dis cussed nu mer i cally, and their nu mer i cal val ues for var i ousval ues of phys i cal pa ram e ters are com pared with ear lier re sults and pre sentedthrough ta bles.

Key words: steady, MHD, stagnation point, stretching sheet, viscous dissipation,ohmic heating, skin-friction coefficient, Nusselt number

In tro duc tion

Flow and heat trans fer of an in com press ible vis cous fluid over stretch ing sheet find ap -pli ca tions in man u fac tur ing pro cesses such as the cool ing of the me tal lic plate, nu clear re ac tor,ex tru sion of poly mers, etc. Flow in the neigh bour hood of a stag na tion point in a plane was ini ti -ated by Hiemenz [1]. Crane [2] pre sented the flow over a stretch ing sheet and ob tained sim i lar -ity so lu tion in closed an a lyt i cal form. Fluid flow and heat trans fer char ac ter is tics on stretch ingsheet with vari able tem per a ture con di tion have been in ves ti gated by Gurbka et al. [3].Watanabe [4, 5] dis cussed sta bil ity of bound ary layer and ef fect of suc tion/in jec tion in MHDflow un der pres sure gra di ent. Noor [6] stud ied the char ac ter is tics of heat trans fer on stretch ingsheet. Chiam [7] dis cussed the heat trans fer in fluid flow on stretch ing sheet at stag na tion pointin pres ence of in ter nal dis si pa tion, heat source/sink and Ohmic heat ing. Chamka et al. [8] con -sid ered Hiemenz flow in the pres ence of mag netic field through po rous me dia. Sharma et al. [9]in ves ti gated steady MHD flow through hor i zon tal chan nel: lower be ing a stretch ing sheet andup per be ing a per me able plate bounded by po rous me dium. Mahapatra et al. [10] in ves ti gatedthe magnetohydrodynamic stag na tion-point flow to wards iso ther mal stretch ing sheet and re -ported that ve loc ity de creases/in creases with the in crease in mag netic field in ten sity when freestream ve loc ity is smaller/greater than the stretch ing ve loc ity. Mahapatra et al. [11] stud ied heattrans fer in stag na tion-point flow on stretch ing sheet with vis cous dis si pa tion ef fect. Attia [12]

THERMAL SCIENCE: Vol. 13 (2009), No. 1, pp. 5-12 5

ana lysed the hydromagnetic stag na tion point flow on po rous stretch ing sheet with suc tion andin jec tion. Pop et al. [13] dis cussed the flow over a stretch ing sheet near a stag na tion point tak ing ra di a tion ef fect.

Aim of the pres ent pa per is to in ves ti gate ef fects of Ohmic heat ing, vis cous dis si pa -tion, and vari able free stream on flow of a vis cous in com press ible elec tri cally con duct ing fluidand heat trans fer near a stag na tion point on an iso ther mal non-con duct ing stretch ing sheet.

For mu la tion of the prob lem

Con sider steady two-di men sional flow of a vis cous in com press ible elec tri cally con -duct ing fluid in the vi cin ity of a stag na tion point on a stretch ing sheet in the pres ence of trans -verse mag netic field of con stant in ten sity Bo. The stretch ing sheet has con stant tem per a ture Tw,lin ear ve loc ity uw(x) and free stream ve loc ity is U(x). It is as sumed that ex ter nal field is zero, the

elec tric field ow ing to po lar -iza tion of charges and Hallef fect are ne glected. Stretch -ing sheet is placed in theplane y = 0 and x-axis is taken along the sheet as shown infig. 1. The fluid oc cu pies thehalf plane (y > 0).

The gov ern ing equa tionsof con ti nu ity, mo men tum,and en ergy (Pai [14], Bansal[15], Schlichting et al.[16],etc.) un der the in flu ence ofuni form trans verse mag neticfield (Jeffery [17], Bansal[18]) with Ohmic dis si pa tionare:

¶

¶

¶

¶

u

x

v

y+ = 0 (1)

uu

xv

u

yU

U

x

u

y

Bu U x

¶

¶

¶

¶

¶

¶+ = + - -

d

dn

s

r

2

2

02

[ ( )] (2)

r k s mC uT

xv

T

y

T

yB u U x

u

yp

¶

¶

¶

¶

¶

¶

¶

¶+

æ

èçç

ö

ø÷÷ = + - +

2

2 02 2[ ( )]

æ

èçç

ö

ø÷÷

2

(3)

The bound ary con di tions are:y u u x cx v T T

y u U x bx T T

w w= = = = =

® = = =

0 0: ( ) , ,

: ( ) ,4 4

(4)

Method of so lu tion

In tro duc ing the stream func tion y (x, y) as de fined by:

uy

vx

= = -¶

¶

¶

¶

y yand (5)

6 Sharma, P. R., Singh, G.: Effects of Ohmic Heating and Viscous Dissipation on ...

Fig ure 1. Phys i cal model

and the sim i lar ity vari able

h y n h= =yc

vx y c xfand ( , ) ( ) (6)

into the eqs. (2) and (3), we get:

¢¢¢ + ¢¢ - ¢ - ¢ - + =f ff f f2 2 2 0M ( )l l (7)

¢¢ + ¢ + ¢ - + ¢¢ =q q lPr Pr ( ) Prf O f fh EcM Ec2 2 2 0 (8)

where Oh = 0 or 1 is the Ohmic heat ing pa ram e ter. It is ob served that eq. (1) is iden ti cally sat is fied. The cor re spond ing bound ary con di tions are re duced to:

f f f( ) , ( ) , ( ) , ( ) , ( )0 0 0 1 0 1 0= ¢ = = ¢ = =q l q4 4 (9)

The gov ern ing eqs. (7) and (8) with the bound ary con di tions (9) are solved us ingRunge-Kutta fourth or der tech nique (Jain et al. [19], Krishnamurthy et al. [20], Jain [21], etc.)along with shoot ing tech nique (Conte et al. [22]). First of all, higher or der non-lin ear dif fer en tial eqs. (7) and (8) are con verted into si mul ta neous lin ear dif fer en tial equa tions of or der first andthey are fur ther trans formed into ini tial value prob lem ap ply ing the shoot ing tech nique. Oncethe prob lem is re duced to ini tial value prob lem, then it is solved us ing Runge-Kutta fourth or dertech nique.

Skin-fric tion

Skin-fric tion co ef fi cient at the stretch ing sheet is given by:

Cc cv

xfwf = = ¢¢

t

r( )0 (10)

where tw = m(¶ u/¶y + ¶v/¶x)y=0 is the shear stress at the stretch ing sheet.

Nusselt num ber

The rate of heat trans fer in terms of the Nusselt num ber at the stretch ing sheet is given by:

Nu w

w

=-

= - ¢v

c

q

T Tkq

( )( )

4

0 (11)

where qw = –k(¶T/¶y)y=0

Par tic u lar cases

(1) In the absence of magnetic field i. e. M = 0, the results of the present paper are reduced tothose obtained by Pop et al. [13] and Mahapatra et al. [11].

(2) In the absence of magnetic field and viscous dissipation i. e. M = 0 and Ec = 0, the results ofthe present paper are reduced to those obtained by Mahapatra et al. [11].

(3) At l = 1, there is no formation of boundary layer as the surface velocity is equal to fluidvelocity (Pai [14]).

Re sults and dis cus sion

Equa tions (7) and (8) are solved us ing Runge-Kutta fourth or der tech nique for dif fer -ent val ues of M, l, Ec, and Oh when Pr = 1.0 tak ing step size 0.005.


It is ob served from tab. 1 that the nu mer i -cal val ues of ¢¢f ( )0 of the pres ent pa per whenM = 0 are in good agree ment with those ob -tained by Pop et al. [13] and Mahapatra et al.[11].

It is ob served from tab. 2 that shear stress at the sheet de creases due to in crease in the mag -netic field in ten sity when l < 1, while it in -creases with the in crease in the mag netic fieldin ten sity when l > 1. It also in creases due toin crease in l when the mag netic field in ten sityis fixed.

It is seen from tab. 3 that the nu mer i cal re -sults of –q'(0) of the pres ent pa per when M == 0, Oh = 0, and Pr = 1.0 are in full agree mentwith the re sults ob tained by Mahapatra et al.[11].

It is seen from tab. 4 that the Nusselt num ber de creases with the in crease in the mag netic field in ten sity when l = 0.1. For l = 2.0, due to in -crease in the mag netic field in ten sity, theNusselt num ber slightly in creases in the ab -sence of vis cous dis si pa tion; while it de creaseswhen Ec = 1.0 and 2.0. Hence re ver sal ef fect inheat trans fer rate is ob served in pres ence ofOhmic heat ing.

Ta ble 5 re veals that the Ohmic heat ing(when Oh = 1.0) de creases the Nusselt num ber when only vis cous dis si pa tion is con sid eredand ef fect is more pro nounced as mag neticfield in ten sity in creases.

Fig ure 2 shows that the bound ary layerthick ness de creases con sid er ably as l in -creases in the ab sence of mag netic field in ten -sity, which is shown by dot ted ver ti cal lines at the points where f ' reaches at bound ary con di -tion. Hence for l > 1, the bound ary layer isthin. Fig ure 3 de picts that for l = 0.1 with thein crease in the mag netic field in ten sity, thefluid ve loc ity de creases which is fully agreewith phys i cal phe nom ena. How ever, re versephe nom ena is ob served for l = 2.0, which isdue to the fact that in verted bound ary layer isformed for l >1 i. e. when free stream ve loc -ity is greater than stretch ing sheet pa ram e ter(Mahapatra et al. [10, 11]).

It is ob served from fig. 4 that at l = 0.1with the in crease in mag netic field in ten sity,


Ta ble 1. Val ues of ¢¢f ( )0 for dif fer ent val ues of l are com pared with the re sults ob tained by Pop et al. [13] and Mahapatra et al. [11]

l

¢¢f ( )0

Pop et al.[13]

Mahapatraet al. [11]

Pres ent pa per re sults

0.1 –0.9694 –0.9694 –0.969386

0.2 –0.9181 –0.9181 –0.9181069

0.5 –0.6673 –0.6673 –0.667263

2.0 2.0174 2.0175 2.01749079

3.0 4.7290 4.7293 4.72922695

Ta ble 2. Val ues of ¢¢f (0) for dif fer ent val ues of l and M

l¢¢f ( )0

M = 0.0 M = 0.5 M = 1.0

0.1 –0.969386 –1.0678983 –1.321111

0.5 –0.667263 –0.7118914 –0.8321261

2.0 2.017502 2.0777247 2.249103

Ta ble 3. Val ues of –q'(0) for dif fer ent val ues of l are com pared with the re sults ob tained by Mahapatra et al. [11] when Pr = 1.0

l- ¢q ( )0

Mahapatra et al.[11]

Pres ent pa per re sults

0.1 0.603 0.602157

0.5 0.692 0.692445

2.0 0.974 0.978726

Ta ble 4. Val ues of –q'(0) for dif fer ent val ues of l, M, and Ec when Pr = 1.0 and Oh = 1.0

l = 0.1 - ¢q ( )0

M Ec = 0.0 Ec = 1.0 Ec = 2.0

0.0 0.602157 0.228906 –0.144344

1.0 0.550806 –0.217832 –0.986471

2.0 0.474701 –1.037764 –2.550229

l = 2.0 - ¢q ( )0

0.0 0.978726 0.060048 –0.858629

1.0 0.985968 –0.205953 –1.397875

2.0 1.001650 –0.858963 –2.719576

fluid tem per a ture in creases and is even higher thanthe sheet tem per a ture. In the ab sence of vis cous dis -si pa tion and l = 2.0, the re sults in tab. 4 show that

the fluid tem per a ture de creases with the in crease in the mag netic field in ten sity be cause in verted ther mal bound ary layer for ma tion but when Ec ¹ 0.0 with the in crease in mag netic field in ten -sity, fluid tem per a ture increases near the sheet and is ap prox i mately same as the dis tance from


Ta ble 5. Val ues of –q'(0) for dif fer ent val ues of l, M andOh when Pr = 1.0 and Ec = 2.0

- ¢q ( )0

M l = 0.1 l = 2.0

Oh = 0.0 Oh = 1.0 Oh = 0.0 Oh = 1.0

1.0 –0.507073 –0.986471 –1.054078 –1.397875

2.0 –1.235249 –2.550229 –1.568088 –2.719576

Fig ure 2. Ve loc ity dis tri bu tion vs. h whenM = 0.0

Fig ure 3. Ve loc ity dis tri bu tion vs. h for dif fer ent val ues of l

Figure 4. Temperature distribution vs. h for different values of l when Ec = 2.0 and Oh = 1.0

sheet in creases. Hence a re ver sal char ac ter is tic in fluid tem per a ture is ob served in the pres enceof Ohmic heat ing. The ther mal bound ary layer thick ness in creases with the in crease in mag neticfield in ten sity for l = 0.1 and change is neg li gi ble for l = 2.0.

It is seen from fig. 5 that with the in crease in the vis cous dis si pa tion pa ram e ter, fluidtem per a ture in creases and ef fect is more pro nounced with the in crease in mag netic field in ten sity,be cause of the ef fect of Ohmic heat ing when l = 0.1. The change in the ther mal bound ary layerthick ness is neg li gi ble. Fig ure 6 shows the ef fects of Ec and M on fluid tem per a ture when l = 2.0.The tem per a ture pro files are same as in the case of l = 0.1 ex cept that steep in crease in fluid tem -per a ture near the sheet is ob served when l = 0.1 in com par i son to the case when l = 2.0.

Fig ure 7 rep re sents that the fluid tem per a ture in creases in the pres ence of vis cous dis -si pa tion and Ohmic heat ing. In crease of fluid tem per a ture near the sheet is ob served in the pres -ence of Ohmic heat ing, when l = 0.1 in com par i son to the case when l = 2.0.

Con clu sions

Fluid ve loc ity in creases and bound ary layer thick ness de creases with the in crease in li. e. when the free stream pa ram e ter is dom i nat ing. Fluid ve loc ity de creases due to in crease inmag netic field in ten sity when l = 0.1, while it in creases due in crease in the mag netic field in ten -sity when l = 2.0.


Figure 5. Temperature distribution vs. h for different values of M and Ec when l = 0.1

Figure 6. Temperature distribution vs. h for different values of M and Ec when l = 2.0

The skin-fric tion co ef fi cient at the sheet de creases due to in crease in the mag netic field in ten sity when l = 0.1, while re verse be hav iour is ob served when l = 2.0.

Fluid tem per a ture in creases due to in crease in mag netic field in ten sity when l = 0.1.Fluid tem per a ture de creases with the in crease in the mag netic field in ten sity in the ab sence ofvis cous dis si pa tion, but with the in crease in mag netic field in ten sity fluid tem per a ture in creasesnear the sheet and is ap prox i mately same as the dis tance from sheet in creases in the pres ence ofvis cous dis si pa tion when l = 2.0. Steep in crease in fluid tem per a ture near the sheet is ob servedwhen l = 0.1 in com par i son to the case when l = 2.0.

The ther mal bound ary layer thick ness in creases with the in crease in mag netic field in -ten sity when l = 0.1 and neg li gi ble change is ob served when l = 2.0. Fluid tem per a ture in -creases due to in crease in the Eckert num ber and ef fect is more pro nounced at higher Hartmannnum ber, ir re spec tive val ues of l. The ef fect of in crease in the Eckert num ber on ther mal bound -ary layer thick ness is neg li gi ble.

Ohmic heat ing pa ram e ter can not be ne glected for large val ues of the Eckert num berand Hartmann num ber.


No men cla ture

Bo – magnetic field intensity, [T]b – free stream velocity parameter, [s–1]Cf – skin-friction coefficient, [–]Cp – specific heat at constant pressure, [Jkg–1K–1]c – stretching sheet parameter, [s–1]Ec – Eckert number [= c2x2/Cp(Tw – T4)], [–]f – dimensionless stream function, [–]M – Hartmann number [= (sBo

2/rc)1/2], [–]Nu – Nusselt number, [–]Oh – Ohmic heating parameter, [–]Pr – Prandtl number (= mCp/k), [–]qw – rate of heat transfer, [Wm–2]T – fluid temperature, [K]Tw – temperature of stretching sheet, [K]T4 – free stream temperature, [K]U(x) – free stream velocity, [ms–1]u, v – velocity components along x- and

– y-axes, respectively, [ms–1]uw – velocity of stretching sheet, [ms–1]

x, y – Cartesian coordinates along x- and– y-axes, respectively, [m]

Greek let ters

q – dimensionless temperature – [(T – T4)/(Tw – T4)], [–]

k – thermal conductivity, [Wm–1K–1]l – ratio of free stream velocity parameter

– and stretching sheet parameter, [–]m – coefficient of viscosity, [kgms–1]n – kinematic viscosity (= m/r), [m2s–1]h – similarity variable [= (c/v)1/2y]r – density of fluid, [kgm–3]s – electrical conductivity , [Wm–2K–4]tw – shear stress, [Pa]

Su per script

' – differentiation with respect to h

Figure 7. Temperature distribution vs. h for different values of Ec and l when M = 2.0

Ref er ences

[1] Hiemenz, K., The Bound ary Layer on a Cir cu lar Cyl in der in Uni form Flow (in Ger man), Dingl. Polytec. J, 326 (1911), pp. 321-328

[2] Crane, L. J., Flow Past a Stretch ing Plate, ZAMP, 21 (1970), 4, pp. 645-647[3] Grubka, L. J., Bobba, K. M., Heat Trans fer Char ac ter is tics of a Con tin u ously Stretch ing Sur face with

Vari able Tem per a ture, Int. J. Heat Mass Trans fer, 107 (1985), pp. 248-250[4] Watanabe, T., Magnetohydrodynamic Sta bil ity of Bound ary Layer along a Flat Plate with Pres sure Gra di -

ent, Acta Mechanica, 65 (1986), 1-4, pp. 41-50[5] Watanabe, T., Ef fect of Uni form Suc tion or In jec tion on a Mag neto-Hy dro dy namic Bound ary Layer Flow

along a Flat Plate with Pres sure Gra di ent, Acta Mechanica, 73 (1988), 1-4, pp. 33-44[6] Noor, A., Heat Trans fer from a Stretch ing Sheet, Int. J. Heat Mass Trans fer, 36 (1993), 4, pp. 1128-1131[7] Chiam, T. C., Magnetohydrodynamic Heat Trans fer over a Non-Iso ther mal Stretch ing Sheet, Acta

Mechanica, 122 (1997), 1-4, pp. 169-179[8] Chamka, A. J., Khaled A. R. A., Sim i lar ity So lu tion for Hydromagnetic Mixed Con vec tion and Mass

Trans fer for Hiemenz Flow though Po rous Me dia, Int. J. Num. Meth ods Heat & Fluid Flow, 10 (2000), pp. 94-115

[9] Sharma, P. R., Mishra, U., Steady MHD Flow through Hor i zon tal Chan nel: Lower Be ing a Stretch ingSheet and up per Be ing a Per me able Plate Bounded by Po rous Me dium, Bull. Pure Appl. Sci ences, In dia,20E (2001), 1, pp. 175-181

[10] Mahapatra, T. R., Gupta, A. S., Magnetohydrodynamic Stag na tion-Point Flow To wards a Stretch ingSheet, Acta Mechanica, 152 (2001), 1-4, pp. 191-196

[11] Mahapatra, T. R., Gupta, A. S., Heat Trans fer in Stag na tion-Point Flow To wards a Stretch ing Sheet, HeatMass Trans fer, 38 (2002), 6, pp. 517-523

[12] Attia, H. A., Hydromagnetic Stag na tion Point Flow with Heat Trans fer over a Per me able Sur face, Ara bian J. Sci ence & En gi neer ing, 28 (2003), 1B, pp. 107-112

[13] Pop, S. R., Grosan, T., Pop, I., Ra di a tion Ef fect on the Flow Near the Stag na tion Point of a Stretch ingSheet, Technische Mechanik, 25 (2004), 2, pp. 100-106

[14] Pai, S. I., Vis cous Flow The ory: I – Lam i nar Flow, D.Van Nostrand Co., New York, USA, 1956[15] Bansal, J. L., Vis cous Fluid Dy nam ics, Ox ford & IBH Pub. Co., New Delhi, In dia, 1977[16] Schlichting, H., Gersten, K., Bound ary Layer The ory, McGraw-Hill Book Co., New York, USA, 1999[17] Jeffery, A., Magnetohydrodynamics, Ol i ver and Boyed, New York, USA, 1966[18] Bansal, J. L., Magnetofluiddynamics of Vis cous Flu ids, Jaipur Pub. House, Jaipur, In dia, 1994[19] Jain, M. K., Iyengar, S. R., Jain, R. K., Nu mer i cal Meth ods for Sci en tific and En gi neer ing Com pu ta tion,

Wiley East ern Ltd., New Delhi, In dia, 1985[20] Krishnamurthy, E. V., Sen, S. K., Nu mer i cal Al go rithms, Af fil i ated East-West Press Pvt. Ltd., New Delhi,

In dia, 1986[21] Jain, M. K., Nu mer i cal So lu tion of Dif fer en tial Equa tions, New Age Int. Pub. Co., New Delhi, In dia, 2000[22] Conte, S. D., Boor, C., El e men tary Nu mer i cal Anal y sis, McGraw-Hill Book Co., New York, USA, 1981

Authors' affiliations:

P. R. Sharma (corresponding author)Department of Mathematics, University of Rajasthan, Jaipur – 302055, IndiaE-mail: [email protected]

G. SinghBirla Institute of Technology (Mesra, Ranchi), Jaipur Ext., India

Paper submitted: June 11, 2007Paper revised: May 2, 2008Paper accepted: December 8, 2008


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TS1 Sharma.vp