Research ArticleSimilarity Solutions for Flow and Heat Transfer ofNon-Newtonian Fluid over a Stretching Surface

Atta Sojoudi1 Ali Mazloomi2 Suvash C Saha3 and Y T Gu3

1 Department of Mechanical Engineering College of Engineering University of Tehran PO Box 11155-4563 Tehran Iran2 School of Mechanical Engineering Sharif University of Technology PO Box 11155-8639 Tehran Iran3 School of Chemistry Physics and Mechanical Engineering Queensland University of Technology 2 George StreetGPO Box 2434 Brisbane QLD 4001 Australia

Correspondence should be addressed to Suvash C Saha s c sahayahoocom

Received 19 January 2014 Revised 24 April 2014 Accepted 24 April 2014 Published 11 May 2014

Academic Editor Subhas Abel

Copyright copy 2014 Atta Sojoudi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Similarity solutions are carried out for flow of power law non-Newtonian fluid film on unsteady stretching surface subjected toconstant heat flux Free convection heat transfer induces thermal boundary layerwithin a semi-infinite layer of Boussinesq fluidThenonlinear coupled partial differential equations (PDE) governing the flow and the boundary conditions are converted to a systemof ordinary differential equations (ODE) using two-parameter groupsThis technique reduces the number of independent variablesby two and finally the obtained ordinary differential equations are solved numerically for the temperature and velocity using theshootingmethodThe thermal and velocity boundary layers are studied by themeans of Prandtl number and non-Newtonian powerindex plotted in curves

1 Introduction

Non-Newtonian fluids such as foodstuffs pulps glues inkpolymers molten plastics or slurries are increasingly usedin various manufacturing industrial and engineering appli-cations particularly in the chemical engineering processesMany simultaneous transport processes exist in industrial orengineering applications For instance mechanical formingprocesses include extrusion and melt spinning where theextruded material issues through a die The ambient fluidstatus is stagnant but a flow is induced adjacent to thematerial being extruded due to the moving surface Notingthat the fluids employed in material processing or protectivecoatings are in general non-Newtonian so the study of non-Newtonian liquid films is important

The problem of heat and mass transfer from vertical platehas been the subject of great interest for several researchersin past decades As an earlier study Wang [1] considered

the flow problem within a finite liquid film of Newtonianfluid over an unsteady stretching sheet Later Anderssonet al [2] investigated the heat transfer characteristics ofthe hydrodynamic problem solved by Wang [1] Usha andSridharan [3] regarded a similar problem of axisymmetricflow in a liquid film The effect of thermocapillarity on theflow and heat transfer in a thin liquid film was studied byDandapat et al [4] The free-surface flow of non-Newtonianliquids in thin films is a widely occurring phenomenon invarious industrial applications for instance in polymer andplastic fabrication food processing and coating equipmentThere are limited papers on gravity-driven power-law filmflows [5ndash7] and the studies of non-Newtonian film flows onan unsteady stretching surface remain littleThe heat transferaspect of such problem has also been considered by Chen [8]

The most distinctive works of Vajravelu and Hadjinico-laou [9] and Mehmood and Ali [10] are available in theliterature describing the heat transfer characteristics in the

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 718319 8 pageshttpdxdoiorg1011552014718319

2 Journal of Applied Mathematics

laminar boundary layer of a viscous fluid over a linearlystretching continuous surface with variable wall tempera-ture and the incompressible generalized three-dimensionalviscous flow with heat transfer analysis in the presence ofviscous dissipation generated due to uniform stretching of theplane wall respectively Unsteadiness of the similar problemsis stated by many researchers For instance Tsai et al [11]studied nonuniform heat sourcesink effect on the flow andheat transfer fromanunsteady stretching sheet through a qui-escent fluidmediumextending to infinityTheboundary layerequations are converted by using similarity analysis to be aset of ordinary differential equations including unsteadinessparameter

Recently Sahoo and Do [12] investigated the entrainedflow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partialslip The constitutive equation of the non-Newtonian fluidis modeled by that for a third grade fluid They found outthat slip decreases the momentum boundary layer thicknessand increases the thermal boundary layer thickness whereasthe third grade fluid parameter has an opposite effect on thethermal and velocity boundary layers

Many efforts were made to find analytical and numer-ical solutions imposing certain status and using variousmathematical concepts to the same problem [13ndash19] Themathematical technique employed in the current analysisis the two-parameter group transformation which leads toa similarity representation of the problem Morgan [20]presented a theorywhich led to amendments over earlier sim-ilarity methods and Michal [21] extended Morganrsquos theoryGroup methods as a class of methods which assuage to areduction of the number of independent variables were firstpresented in [22 23] Moran and Gaggioli [24 25] indicateda general systematic group formalism for similarity studywhere a given combination of partial differential equationswas reduced to a system of ordinary differential equations[25ndash34]

In the present work we provide analytical solution for theunsteady free convection non-Newtonian fluids flow over acontinuous moving vertical plate subjected to constant heatflux using a two-parameter group The specified techniqueis explained at the following parts Under the employmentof a two-parameter group the governing partial differentialequations and boundary conditions are reduced to ordinarydifferential equations with the appropriate boundary condi-tionsThe obtained differential equations are solved using theshooting method

2 Mathematical Formulation

Theunsteady laminar flow of an incompressible fluid inducedby a ceaseless moving sheet placed in a fluid at quiescent isconsideredThe vertical flat sheet illustrated in Figure 1 arisesfrom a thin slit at 119909 = 119910 = 0 and is subsequently stretchedvertically The positive 119909 and 119910coordinates are measuredalong the direction of the moving film with the slot as theorigin and the normal to the sheet respectively Constantheat flux 119902

119908 is imposed to the flat sheet giving rise to a

g

v

x

Tinfin

ulowast

ulowast

T minus Tinfin

y 120578

uw

qw

= co

nsta

nt

Figure 1 Physical model for laminar flow over stretching sheet

buoyancy force while the ambient fluid is kept at a constanttemperature 119879

infin The boundary layer equations governing

the free convection flow over the moving sheet are expressedas follows

120597119906lowast

120597119909

+

120597V120597119910

= 0

120597119906lowast

120597119905

+ 119906lowast120597119906lowast

120597119909

+ V120597119906lowast

120597119910

= 120599119899(

120597119906lowast

120597119910

)

119899minus1

1205972

119906

1205971199102+ 119892120573 (119879 minus 119879

infin)

120597119879

120597119905

+ 119906lowast120597119879

120597119909

+ V120597119879

120597119910

=

120599

Pr1205972

119879

1205971199102

(1)

where the certain boundary conditions are

119906lowast

(119909 119910 0) = 119906lowast

0(119909 119910) V (119909 119910 0) = V

0(119909 119910)

119879 (119909 119910 0) = 1198790(119909 119910) 119906

lowast

(119909 0 119905) = 119906119908(119909 119905)

V (119909 0 119905) = 0

(2a)

120597119879 (119909 0 119905)

120597119910

=

minus119902119908

119896

119906lowast

(119909 119910 119905) = 0 119879 (119909 119910 119905) = 119879infin

as 119910 997888rarr +infin

(2b)

119906lowast and V are velocity components related to the 119909 and119910 coordinates 119879 119892 and 119899 are fluid temperature gravityand power index of non-Newtonian fluid respectively Pris defined as the ratio of kinematic viscosity (120592) to thermaldiffusion (120572) 120573 is thermal expansion coefficient at constantpressure where 119906lowast

0 V0 and 119879

0are initial velocity components

and initial temperature 119906119908and 119879

119908are fluid velocity and

temperature on vertical moving sheet Dimensionless 119909-velocity and temperature are defined using the following

Journal of Applied Mathematics 3

relationships 119906 = 119906lowast

119906119908and 120579(119909 119910 119905) = ((119879(119909 119910 119905) minus

119879infin)119902119908119909)119896radicRe

119909 Now (1) is rewritten in the form

119906

120597119906119908

120597119909

+ 119906119908

120597119906

120597119909

+

120597V120597119910

= 0 (3)

119906

120597119906119908

120597119905

+ 119906119908

120597119906

120597119905

+ 1199062

119908119906

120597119906

120597119909

+ 1199062

119906119908

120597119906119908

120597119909

+ V119906119908

120597119906

120597119910

= 120599119899119906119899

119908(

120597119906

120597119910

)

119899minus1

(

1205972

119906

1205971199102) +

119892120573119902119908

119896radicRe119909

120579119909

(4)

119909

120597120579

120597119905

+ 119906119908119906(119909

120597120579

120597119909

+ 120579) + V120597120579

120597119910

119909 =

120599

Pr119909

1205972

120579

1205971199102 (5)

and boundary conditions are

119906 (119909 119910 0) = 1199060(119909 119910) V (119909 119910 0) = V

0(119909 119910)

120579 (119909 119910 0) = 1205790(119909 119910) 119906 (119909 0 119905) = 1 V (119909 0 119905) = 0

(6a)

120597120579 (119909 0 119905)

120597119910

= minus

radicRe119909

119909

119906 (119909 119910 119905) = 0 120579 (119909 119910 119905) = 0

as 119910 997888rarr +infin

(6b)

3 Group Formulation of the Problem

31 Group Formulation The problem is solved by employinga two-parameter group transformation to the partial differ-ential equations of (3) to (5) This transformation decreasesthe three independent variables (119909 119910 119905) to one similarityvariable 120578(119909 119910 119905) and the governing equations of (3) to(5) are transformed to a system of ordinary differentialequations in terms of the similarity variable 120578This techniqueis based on a class of transformation namely 119866 includingtwo parameters (119886

1 1198862)

119866 119878 = 119862119878

(1198861 1198862) 119878 + 119870

119878

(1198861 1198862) (7)

where 119878 is representative for 119909 119910 119905 119906119908 V and 120579119862119904 and119870119904 are

real valued and at least differentiable in their real arguments(1198861 1198862) Dependent variables and related differentiates are as

follows via chain rule operations

119878119894= (

119862119878

119862119894)119878119894 119878

119894119895= (

119862119878

119862119894119862119895)119878119894119895

119894 = 119909 119910 119905 119895 = 119909 119910 119905

(8)

For instance (3) is transformed as follows

119906

120597119906119908

120597119909

+ 119906119908

120597119906

120597119909

+

120597V120597119910

= 1198671(1198861 1198862) [119906

120597119906119908

120597119909

+ 119906119908

120597119906

120597119909

+

120597V120597119910

]

(9)

where1198671(1198861 1198862) is constant If (7) and (8) are put in (9)

[

119862119906

119862119906119908

119862119909] 119906

120597119906119908

120597119909

+ [

119862119906

119862119906119908

119862119909] 119906119908

120597119906

120597119909

+ [

119862V

119862119910]

120597V120597119910

+ 1198771

= 1198671(1198861 1198862) [119906

120597119906119908

120597119909

+ 119906119908

120597119906

120597119909

+

120597V120597119910

]

(10)

and 1198771is [119862119906119908119870119906119862119909](120597119906

119908120597119909) + [119862

119906

119870119906119908119862119909

](120597119906120597119909) Forinvariant transformation 119877

1is equated to zero This is

satisfied by setting

119870119906

= 119870119906119908

= 0 (11)

Finally we get

[

119862119906

119862119906119908

119862119909] = [

119862V

119862119910] = 119867

1(1198861 1198862) (12)

Simultaneously after (4) and (5) transformation

119870119906

= 119870V= 119870119906119908

= 119870120579

= 119870119909

= 119870119910

= 0 (13)

[

119862119906

119862119906119908

119862119905] = [

(119862119906

)2

(119862119906119908)2

119862119909

] = [

119862119906

119862V119862119906119908

119862119910

]

=

(119862119906119908)119899

(119862119906

)119899

(119862119910)119899+1

= 119862120579

119862119909

= 1198672(1198861 1198862)

(14a)

[

119862120579

119862119909

119862119905] = [119862

119906

119862119906119908

119862120579

] = [

119862119909

119862V119862120579

119862119910

] = [

119862119909

119862120579

(119862119910)2]

= 1198673(1198861 1198862)

(14b)

and we have

119870119906

= 119870V= 119870119906119908

= 119870120579

= 0 (15)

Employing invariant transformation for boundary conditionsresults in

119870119905

= 119870119910

= 0 (16)

119862119906

= 1 (17)

Invoking (12) to (15) and (17) reduces to

119862119909

= 119862119910

119862119905

= (119862119910

)2(2minus119899)

119862119906119908

= 119862V=

1

(119862119910)119899(2minus119899)

119862120579

=

1

(119862119910)4(2minus119899)

(18)

Group 119866 is summarized as follows

119866 =

119866119904

119909 = 119862119910

119909

119910 = 119862119910

119910

119905 = (119862119910

)2(2minus119899)

119905

119906 = 119906

119906119908=

1

(119862119910)119899(2minus119899)

119906119908

V =1

(119862119910)119899(2minus119899)

V

120579 =

1

(119862119910)4(2minus119899)

120579

(19)

4 Journal of Applied Mathematics

This group converts invariantly the differential equations of(3) to (5) and the initial and boundary conditions (6a) and(6b)

32TheComplete Set of Absolute Invariants Thecomplete setof absolute invariants is as follows

(i) the absolute invariants of the independent variables(119909 119910 119905) are 120578 = 120578(119909 119910 119905)

(ii) the absolute invariants of the dependent variables(119906 V 119906

119908 120579) are

119892119895(119909 119910 119905 119906 119906

119908 V 120579) = 119865

119895(120578 (119909 119910 119905)) 119895 = 1 2 3 4

(20)

The basic theorem in group theory asserts that a function119892119895(119909 119910 119905 119906 119906

119908 V 120579) is an absolute invariant of a two-

parameter group if it satisfies the following two first-orderlinear differential equations

13

sum

119894=1

(120572119894119878119894+ 120572119894+1)

120597119892

120597119878119894

= 0

13

sum

119894=1

(120573119894119878119894+ 120573119894+1)

120597119892

120597119878119894

= 0 (21)

where 1205721= (120597119862

119909

1205971198861)(1198860

1 1198860

2) 1205722= (120597119870

119909

1205971198861)(1198860

1 1198860

2)

1205731= (120597119862

119909

1205971198862)(1198860

1 1198860

2) 1205732= (120597119870

119909

1205971198862)(1198860

1 1198860

2) and so

forth (11988601 1198860

2) are the identity elements of the group

The absolute invariant 120578(119909 119910 119905) of the independent vari-ables (119909 119910 119905) is determined using (21)

(1205721119909 + 1205722)

120597120578

120597119909

+ 1205723119910

120597120578

120597119910

+ 1205725119905

120597120578

120597119905

= 0

(1205731119909 + 1205732)

120597120578

120597119909

+ 1205733119910

120597120578

120597119910

+ 1205735119905

120597120578

120597119905

= 0

(22)

where 1205724= 1205734= 119870119910

= 0 and 1205726= 1205736= 119870119905

= 0Elimination of 119910(120597120578120597119910) and 120597120578120597119909 from (22) yields

(12058231119909 + 12058232)

120597120578

120597119909

+ 12058235119905

120597120578

120597119905

= 0 (23a)

(12058231119909 + 12058232) 119910

120597120578

120597119910

+ (12058215119909 + 12058225) 119905

120597120578

120597119905

= 0 (23b)

where 120582119894119895= 120572119894120573119895minus 120572119895120573119894and 119894 119895 = 1 2 3 4 5 Invoking (21)

and definitions of 120572 and 120573we get 1205725= 21205723and 120573

5= 21205733 thus

12058235= 12057231205735minus 12057251205733will assuage to zero Using (23a) we obtain

120597120578120597119909 = 0 indicating that 120578 is dependent on 119910 and 119905 Solving(23b) we get

120578 = 119910120587 (119905) (24)

where 120587(119905) = 119886119905119887 and 119886 and 119887 are arbitrary constantsSimultaneously the absolute invariants of the dependent

variables 119906 V 119906119908 and 120579 are obtained from the group transfor-

mation (19)

1198921(119909 119910 119905 119906) = 119906 (120578)

1198922(119909 119910 119905 119906) = 120579 (120578)

(25)

1198923(119909 119905 119906

119908) is obtained using (21) as

(1205721119909 + 1205722)

1205971198923

120597119909

+ (1205725119905 + 1205726)

1205971198923

120597119905

+ (12057211119906119908+ 12057212)

1205971198923

120597119906119908

= 0

(26a)

(1205731119909 + 1205732)

1205971198923

120597119909

+ (1205735119905 + 1205736)

1205971198923

120597119905

+ (12057311119906119908+ 12057312)

1205971198923

120597119906119908

= 0

(26b)

Eliminating terms of 1205971198923120597119909 and 120597119892

3120597119905 from ((26a) and

(26b)) results in

1198923(119909 119905 119906

119908) = 1206011(

119906119908

120596 (119909 119905)

) = 119864 (120578) (27)

Similarly for 1198924(119909 119905 V)

1198924(119909 119905 V) = 120601

2(

VΓ (119909 119905)

) = 119865 (120578) (28)

where 120596(119909 119905) Γ(119909 119905) 119864(120578) and 119865(120578) are functions to bedetermined Without loss of generality the 120601rsquos in (27) and(28) are selected to be identity functions hence the functions119906119908(119909 119905) and V(119909 119910 119905) are expressed in terms 119864(120578) and 119865(120578) as

119906119908(119909 119905) = 120596 (119909 119905) 119864 (120578) (29)

V (119909 119910 119905) = Γ (119909 119905) 119865 (120578) (30)

Since 120596(119909 119905) and 119906119908(119909 119905) are independent of y whereas 120578 is

function of y we conclude that 119864(120578) is equal to a constantsuch as 119864

0 Equation (29) yields

119906119908(119909 119905) = 119864

0120596 (119909 119905) (31)

Without loss of generality 1198640is equated to oneThe functions

of Γ(119909 119905) and120596(119909 119905)will be determined later on such that thegoverning equations of (3) to (5) reduce to a set of ordinarydifferential equations in 119864(120578) 119865(120578) and 119906(120578)

4 Reduction to a Set of Ordinary Problem

Now we define 120578 in general form of 120578 = 119910120587(119909 119905) Invoking(30) and (31) and (3) to (5)

119889119865

119889120578

+ 1198621120578

119889119906

119889120578

+ 1198622119906 = 0

119899(

119889119906

119889120578

)

119899minus1

1198892

119906

1198891205782minus [120578 (119862

3+ 1198624119906) + 119862

5119865]

119889119906

119889120578

minus 1198626119906 minus 11986271199062

+ 1198628120579 = 0

1198892

120579

1198891205782minus Pr [119889120579

119889120578

120578 (11986210+ 11986211119906) + 119862

12119865 minus 119862

9119906120579] = 0

(32)

Journal of Applied Mathematics 5

Constants of 1198621to 11986212are defined as follows

1198621=

120596

Γ1205872

120597120587

120597119909

1198622=

1

Γ120587

120597120596

120597119909

1198623=

1

120599120596119899minus1120587119899+2

120597120587

120597119905

1198624=

1

120599120596119899minus2120587119899+2

120597120587

120597119909

1198625=

Γ

120599120596119899minus1120587119899

1198626=

1

120599120596119899120587119899+1

120597120596

120597119905

1198627=

1

120599120596119899minus1120587119899+1

120597120596

120597119909

1198628=

119909

120599120596119899120587119899+1

119892120573119902119908

119896radicRe119909

1198629=

120596

1205991199091205872

11986210=

1

1205991205873

120597120587

120597119905

11986211=

120596

1205991205873

120597120587

120597119909

11986212=

Γ

120599120587

(33)

Now we obtain the exact value of constants as

1198627

1198628

= 1 997888rarr 120596 (119909) = (

119892120573119902119908

119896radicRe119909

)

12

119909 (34)

From (31)

119906119908(119909) = (

119892120573119902119908

119896radicRe119909

)

12

119909 (35)

For 1198629= 1 we have

120587 =

1

12059912

(

119892120573119902119908

119896radicRe119909

)

14

(36)

According to the definition of similarity variable

120578 = 119910(

119892120573119902119908

1205992119896radicRe

119909

)

14

(37)

For 1198622= 1 we get

Γ = 12059912

(

119892120573119902119908

119896radicRe119909

)

14

(38)

Using (30) the horizontal component of velocity is

V (120578) = 12059912 (119892120573119902119908

119896radicRe119909

)

14

119865 (120578) (39)

As 120587 is invariant and 120596 is independent of time

1198621= 1198623= 1198624= 1198626= 11986210= 11986211= 0 (40)

On the other hand due to definitions of 120587 120596 and Γ

11986212= 1 119862

5= 1198627= 1198628= (120596120587)

1minus119899

(41)

Substituting the above constants in (32) they finally reduceto

119889119865

119889120578

+ 119906 = 0 (42)

119899(120596120587)119899minus1

(

119889119906

119889120578

)

119899minus1

1198892

119906

1198891205782minus 119865

119889119906

119889120578

minus 1199062

+ 120579 = 0 (43)

1198892

120579

1198891205782minus Pr(119889120579

119889120578

119865 + 119906120579) = 0 (44)

The new forms of boundary conditions are

119865 (0) = 0

119906 (0) = 1 119906 (infin) = 0

120597120579 (0)

120597120578

= minus1 120579 (infin) = 0

(45)

It can be proved that 120597120579(0)120597120578 = minus1 According to (2b) anddefinition of 120579 (2b) will change into

120597120579 (119909 0 119905)

120597119910

=

minusradicRe119909

119909

(46)

where Re119909= 119906119904119909120599 and 119906

119904is average velocity in y direction

Assuming 119906119904= 119906119908

Re119909= 1199092

1205872

(47)

Using chain rule operation

120597120579 (0)

120597120578

120597120578

120597119910

=

minusradicRe119909

119909

(48)

So according to Re119909and 120578 120597120579(0)120597120578 will be minus1

For instance assuming 119899 = 1 for a case ofNewtonian fluidflow (43) will reduce to

1198892

119906

1198891205782+

120578

2

119889119906

119889120578

minus 119906 = minus erfc(radicPr2

120578) (49)

where

119906 (0) = 1 119906 (infin) = 0 (50)

6 Journal of Applied Mathematics

Table 1 Comparison of present problem solution and case of Kassem [34] for Pr = 1 2 5 and 10 of Newtonian fluid

Pr Kassem [34] Present work120578 119906 (120578) Θ (120578) 119906 (120578) Θ (120578)

1

05 0669 0507 0667 05101 0391 0256 0381 026115 0165 0103 0161 01052 0000 0003 0003 0005

2

03 0736 0398 0742 040106 0515 0222 0520 022509 0315 0092 0317 009512 0143 0009 0150 0011

5

025 0745 0206 0750 020805 0525 0095 0531 0011075 0333 0040 0335 00421 0171 0017 0171 0017

10

02 0768 0124 0771 012504 0565 0046 0571 004806 0381 0015 0391 001708 0216 0007 0218 0008

0 05 1 15 20

02

04

06

08

1

12

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

u(120578)

Figure 2 Vertical velocity of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

5 Results and Discussion

In this section initially results of Newtonian fluid flow overstretching sheet are compared with Kassem [34] Table 1shows the results obtained by numerical solution of ODEequation (49) and [34] for power index 119899 equal to unityand Pr number of 1 There is a satisfying agreement amongthe present evaluations Complementary results for differentvalues of Pr are shown in Figures 2 and 3 It can beunderstood that as the fluid momentum diffusivity increaseswith respect to the thermal diffusivity large Pr numbersvertical velocity decreases In other words the velocity 119906(120578)inside the boundary layer decreases with the increase of fluidviscosity Also the temperature plot of boundary layer revealsa decrease of temperature at the wall temperature for larger

0 05 1 15 20

02

04

06

08

1

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

120579(120578)

Figure 3 Temperature of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

Pr So heat loss increases for larger Pr as the boundary layergets thinner

For non-Newtonian case of the fluid the vertical velocityand temperature of fluid adjacent to the stretching sheet areevaluated by solving (42) to (44) with the certain boundaryconditions of (45) Results of the shooting method solutionare illustrated in Figures 4 and 5 for Pr = 5 As the powerindex increases vertical velocity component encounters withresistance to rise due to increased value of shear stress Butfor lower power index vertical component of velocity movesfaster due to reduced apparent viscosity Similarly thinnerthermal boundary layer for higher power index will increasethe amount of heat loss

Journal of Applied Mathematics 7

0 2 4 6 8 100

02

04

06

08

1

12

n = 05n = 10n = 15

120578

u(120578)

Figure 4 Vertical velocity of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

0 05 1 15 2 250

02

04

06

120578

120579(120578)

n = 05n = 10n = 15

Figure 5 Temperature of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

6 Conclusion

Similarity solutions are performed for flow of power lawnon-Newtonian fluid film on unsteady stretching surfacesubjected to constant heat fluxThe nonlinear coupled partialdifferential equations (PDE) governing the flow and theboundary conditions are transformed to a set of ordinarydifferential equations (ODE) using two-parameter groupsThis technique reduces the number of independent variablesby two and conclusively the obtained ordinary differentialequations are solved numerically for the temperature andvelocity using the shooting method Results show that higherPr number and higher power index of non-Newtonian fluidencounter with difficulty to move as faster as lower Pr andpower index Enhanced amount of shear stress explains thereason of the predicted flow

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[2] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[3] R Usha and R Sridharan ldquoThe axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal of FluidsEngineering vol 117 no 1 pp 81ndash85 1995

[4] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[5] H I Andersson and T Ytrehus ldquoFalkner-Skan solution forgravity-driven film flowrdquo Journal of Applied Mechanics vol 52no 4 pp 783ndash786 1985

[6] H I Andersson and F Irgens ldquoGravity-driven laminar filmflow of power-law fluids along vertical wallsrdquo Journal of Non-Newtonian Fluid Mechanics vol 27 no 2 pp 153ndash172 1988

[7] D-Y Shang and H I Andersson ldquoHeat transfer in gravity-driven film flow of power-law fluidsrdquo International Journal ofHeat and Mass Transfer vol 42 no 11 pp 2085ndash2099 1998

[8] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[9] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[10] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching planewallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[11] R Tsai K H Huang and J S Huang ldquoFlow and heattransfer over an unsteady stretching surface with non-uniformheat sourcerdquo International Communications in Heat and MassTransfer vol 35 no 10 pp 1340ndash1343 2008

[12] B Sahoo and Y Do ldquoEffects of slip on sheet-driven flow andheat transfer of a third grade fluid past a stretching sheetrdquoInternational Communications in Heat and Mass Transfer vol37 no 8 pp 1064ndash1071 2010

[13] C R Illingworth ldquoUnsteady laminar flow of gas near aninfinite flat platerdquo Mathematical Proceedings of the CambridgePhilosophical Society vol 46 no 4 pp 603ndash613 1950

[14] R Siegel ldquoTransient free convection from a vertical flat platerdquoTransactions of the ASME vol 80 no 2 pp 347ndash359

[15] B Gebhart ldquoTransient natural convection from vertical ele-mentsrdquo Journal of Heat Transfer vol 83 pp 61ndash70 1961

[16] A J Chamkha ldquoSimilarity solution for thermal boundary layeron a stretched surface of a non-Newtonian fluidrdquo InternationalCommunications in Heat and Mass Transfer vol 24 no 5 pp643ndash652 1997

[17] J D Hellums and S W Churchill ldquoTransient and steady statefree and natural convection numerical solutions part Imdashthe

8 Journal of Applied Mathematics

isothermal vertical platerdquoAIChE Journal vol 8 no 5 pp 690ndash692 1962

[18] G D Callahan and W J Marner ldquoTransient free convectionwith mass transfer on an isothermal vertical flat platerdquo Inter-national Journal of Heat and Mass Transfer vol 19 no 2 pp165ndash174 1976

[19] T G Howell D R Jeng andK J deWitt ldquoMomentum and heattransfer on a continuous moving surface in a power law fluidrdquoInternational Journal of Heat and Mass Transfer vol 40 no 8pp 1853ndash1861 1997

[20] A J A Morgan ldquoThe reduction by one of the number ofindependent variables in some systems of partial differentialequationsrdquo The Quarterly Journal of Mathematics vol 3 no 1pp 250ndash259 1952

[21] A D Michal ldquoDifferential invariants and invariant partialdifferential equations under continuous transformation groupsin normed linear spacesrdquo Proceedings of the National Academyof Sciences of the United States of America vol 37 pp 623ndash6271951

[22] G Birkhoff ldquoMathematics for engineersrdquo Electrical Engineeringvol 67 no 12 pp 1185ndash1188 1948

[23] G Birkhoff Hydrodynamics A Study in Logic Fact and Simili-tude Princeton University Press Princeton NJ USA 1960

[24] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[25] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[26] W F Ames ldquoSimilarity for the nonlinear diffusion equationrdquoIndustrial amp Engineering Chemistry Fundamentals vol 4 no 1pp 72ndash76 1965

[27] W FAmesNonlinear Partial Differential Equations in Engineer-ing chapter 2 Academic Press New York NY USA 1972

[28] W F Ames and M C Nucci ldquoAnalysis of fluid equations bygroupmethodsrdquo Journal of EngineeringMathematics vol 20 no2 pp 181ndash187 1986

[29] M J Moran and R A Gaggioli ldquoSimilarity analysis of com-pressible boundary layer flows via group theoryrdquo TechnicalSummaryReport 838US ArmyMathematics ResearchCenterMadison Wis USA 1967

[30] M J Moran and R A Gaggioli ldquoSimilarity analysis via grouptheoryrdquo AIAA Journal vol 6 pp 2014ndash2016 1968

[31] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[32] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[33] M B Abd-el-Malek M M Kassem and M L MekkyldquoSimilarity solutions for unsteady free-convection flow from acontinuous moving vertical surfacerdquo Journal of Computationaland Applied Mathematics vol 164 no 1 pp 11ndash24 2004

[34] M Kassem ldquoGroup solution for unsteady free-convection flowfrom a vertical moving plate subjected to constant heat fluxrdquoJournal of Computational and Applied Mathematics vol 187 no1 pp 72ndash86 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

relationships 119906 = 119906lowast

119906119908and 120579(119909 119910 119905) = ((119879(119909 119910 119905) minus

119879infin)119902119908119909)119896radicRe

119909 Now (1) is rewritten in the form

119906

120597119906119908

120597119909

+ 119906119908

120597119906

120597119909

+

120597V120597119910

= 0 (3)

119906

120597119906119908

120597119905

+ 119906119908

120597119906

120597119905

+ 1199062

119908119906

120597119906

120597119909

+ 1199062

119906119908

120597119906119908

120597119909

+ V119906119908

120597119906

120597119910

= 120599119899119906119899

119908(

120597119906

120597119910

)

119899minus1

(

1205972

119906

1205971199102) +

119892120573119902119908

119896radicRe119909

120579119909

(4)

119909

120597120579

120597119905

+ 119906119908119906(119909

120597120579

120597119909

+ 120579) + V120597120579

120597119910

119909 =

120599

Pr119909

1205972

120579

1205971199102 (5)

and boundary conditions are

119906 (119909 119910 0) = 1199060(119909 119910) V (119909 119910 0) = V

0(119909 119910)

120579 (119909 119910 0) = 1205790(119909 119910) 119906 (119909 0 119905) = 1 V (119909 0 119905) = 0

(6a)

120597120579 (119909 0 119905)

120597119910

= minus

radicRe119909

119909

119906 (119909 119910 119905) = 0 120579 (119909 119910 119905) = 0

as 119910 997888rarr +infin

(6b)

3 Group Formulation of the Problem

31 Group Formulation The problem is solved by employinga two-parameter group transformation to the partial differ-ential equations of (3) to (5) This transformation decreasesthe three independent variables (119909 119910 119905) to one similarityvariable 120578(119909 119910 119905) and the governing equations of (3) to(5) are transformed to a system of ordinary differentialequations in terms of the similarity variable 120578This techniqueis based on a class of transformation namely 119866 includingtwo parameters (119886

1 1198862)

119866 119878 = 119862119878

(1198861 1198862) 119878 + 119870

119878

(1198861 1198862) (7)

where 119878 is representative for 119909 119910 119905 119906119908 V and 120579119862119904 and119870119904 are

real valued and at least differentiable in their real arguments(1198861 1198862) Dependent variables and related differentiates are as

follows via chain rule operations

119878119894= (

119862119878

119862119894)119878119894 119878

119894119895= (

119862119878

119862119894119862119895)119878119894119895

119894 = 119909 119910 119905 119895 = 119909 119910 119905

(8)

For instance (3) is transformed as follows

119906

120597119906119908

120597119909

+ 119906119908

120597119906

120597119909

+

120597V120597119910

= 1198671(1198861 1198862) [119906

120597119906119908

120597119909

+ 119906119908

120597119906

120597119909

+

120597V120597119910

]

(9)

where1198671(1198861 1198862) is constant If (7) and (8) are put in (9)

[

119862119906

119862119906119908

119862119909] 119906

120597119906119908

120597119909

+ [

119862119906

119862119906119908

119862119909] 119906119908

120597119906

120597119909

+ [

119862V

119862119910]

120597V120597119910

+ 1198771

= 1198671(1198861 1198862) [119906

120597119906119908

120597119909

+ 119906119908

120597119906

120597119909

+

120597V120597119910

]

(10)

and 1198771is [119862119906119908119870119906119862119909](120597119906

119908120597119909) + [119862

119906

119870119906119908119862119909

](120597119906120597119909) Forinvariant transformation 119877

1is equated to zero This is

satisfied by setting

119870119906

= 119870119906119908

= 0 (11)

Finally we get

[

119862119906

119862119906119908

119862119909] = [

119862V

119862119910] = 119867

1(1198861 1198862) (12)

Simultaneously after (4) and (5) transformation

119870119906

= 119870V= 119870119906119908

= 119870120579

= 119870119909

= 119870119910

= 0 (13)

[

119862119906

119862119906119908

119862119905] = [

(119862119906

)2

(119862119906119908)2

119862119909

] = [

119862119906

119862V119862119906119908

119862119910

]

=

(119862119906119908)119899

(119862119906

)119899

(119862119910)119899+1

= 119862120579

119862119909

= 1198672(1198861 1198862)

(14a)

[

119862120579

119862119909

119862119905] = [119862

119906

119862119906119908

119862120579

] = [

119862119909

119862V119862120579

119862119910

] = [

119862119909

119862120579

(119862119910)2]

= 1198673(1198861 1198862)

(14b)

and we have

119870119906

= 119870V= 119870119906119908

= 119870120579

= 0 (15)

Employing invariant transformation for boundary conditionsresults in

119870119905

= 119870119910

= 0 (16)

119862119906

= 1 (17)

Invoking (12) to (15) and (17) reduces to

119862119909

= 119862119910

119862119905

= (119862119910

)2(2minus119899)

119862119906119908

= 119862V=

1

(119862119910)119899(2minus119899)

119862120579

=

1

(119862119910)4(2minus119899)

(18)

Group 119866 is summarized as follows

119866 =

119866119904

119909 = 119862119910

119909

119910 = 119862119910

119910

119905 = (119862119910

)2(2minus119899)

119905

119906 = 119906

119906119908=

1

(119862119910)119899(2minus119899)

119906119908

V =1

(119862119910)119899(2minus119899)

V

120579 =

1

(119862119910)4(2minus119899)

120579

(19)

4 Journal of Applied Mathematics

This group converts invariantly the differential equations of(3) to (5) and the initial and boundary conditions (6a) and(6b)

32TheComplete Set of Absolute Invariants Thecomplete setof absolute invariants is as follows

(i) the absolute invariants of the independent variables(119909 119910 119905) are 120578 = 120578(119909 119910 119905)

(ii) the absolute invariants of the dependent variables(119906 V 119906

119908 120579) are

119892119895(119909 119910 119905 119906 119906

119908 V 120579) = 119865

119895(120578 (119909 119910 119905)) 119895 = 1 2 3 4

(20)

The basic theorem in group theory asserts that a function119892119895(119909 119910 119905 119906 119906

119908 V 120579) is an absolute invariant of a two-

parameter group if it satisfies the following two first-orderlinear differential equations

13

sum

119894=1

(120572119894119878119894+ 120572119894+1)

120597119892

120597119878119894

= 0

13

sum

119894=1

(120573119894119878119894+ 120573119894+1)

120597119892

120597119878119894

= 0 (21)

where 1205721= (120597119862

119909

1205971198861)(1198860

1 1198860

2) 1205722= (120597119870

119909

1205971198861)(1198860

1 1198860

2)

1205731= (120597119862

119909

1205971198862)(1198860

1 1198860

2) 1205732= (120597119870

119909

1205971198862)(1198860

1 1198860

2) and so

forth (11988601 1198860

2) are the identity elements of the group

The absolute invariant 120578(119909 119910 119905) of the independent vari-ables (119909 119910 119905) is determined using (21)

(1205721119909 + 1205722)

120597120578

120597119909

+ 1205723119910

120597120578

120597119910

+ 1205725119905

120597120578

120597119905

= 0

(1205731119909 + 1205732)

120597120578

120597119909

+ 1205733119910

120597120578

120597119910

+ 1205735119905

120597120578

120597119905

= 0

(22)

where 1205724= 1205734= 119870119910

= 0 and 1205726= 1205736= 119870119905

= 0Elimination of 119910(120597120578120597119910) and 120597120578120597119909 from (22) yields

(12058231119909 + 12058232)

120597120578

120597119909

+ 12058235119905

120597120578

120597119905

= 0 (23a)

(12058231119909 + 12058232) 119910

120597120578

120597119910

+ (12058215119909 + 12058225) 119905

120597120578

120597119905

= 0 (23b)

where 120582119894119895= 120572119894120573119895minus 120572119895120573119894and 119894 119895 = 1 2 3 4 5 Invoking (21)

and definitions of 120572 and 120573we get 1205725= 21205723and 120573

5= 21205733 thus

12058235= 12057231205735minus 12057251205733will assuage to zero Using (23a) we obtain

120597120578120597119909 = 0 indicating that 120578 is dependent on 119910 and 119905 Solving(23b) we get

120578 = 119910120587 (119905) (24)

where 120587(119905) = 119886119905119887 and 119886 and 119887 are arbitrary constantsSimultaneously the absolute invariants of the dependent

variables 119906 V 119906119908 and 120579 are obtained from the group transfor-

mation (19)

1198921(119909 119910 119905 119906) = 119906 (120578)

1198922(119909 119910 119905 119906) = 120579 (120578)

(25)

1198923(119909 119905 119906

119908) is obtained using (21) as

(1205721119909 + 1205722)

1205971198923

120597119909

+ (1205725119905 + 1205726)

1205971198923

120597119905

+ (12057211119906119908+ 12057212)

1205971198923

120597119906119908

= 0

(26a)

(1205731119909 + 1205732)

1205971198923

120597119909

+ (1205735119905 + 1205736)

1205971198923

120597119905

+ (12057311119906119908+ 12057312)

1205971198923

120597119906119908

= 0

(26b)

Eliminating terms of 1205971198923120597119909 and 120597119892

3120597119905 from ((26a) and

(26b)) results in

1198923(119909 119905 119906

119908) = 1206011(

119906119908

120596 (119909 119905)

) = 119864 (120578) (27)

Similarly for 1198924(119909 119905 V)

1198924(119909 119905 V) = 120601

2(

VΓ (119909 119905)

) = 119865 (120578) (28)

where 120596(119909 119905) Γ(119909 119905) 119864(120578) and 119865(120578) are functions to bedetermined Without loss of generality the 120601rsquos in (27) and(28) are selected to be identity functions hence the functions119906119908(119909 119905) and V(119909 119910 119905) are expressed in terms 119864(120578) and 119865(120578) as

119906119908(119909 119905) = 120596 (119909 119905) 119864 (120578) (29)

V (119909 119910 119905) = Γ (119909 119905) 119865 (120578) (30)

Since 120596(119909 119905) and 119906119908(119909 119905) are independent of y whereas 120578 is

function of y we conclude that 119864(120578) is equal to a constantsuch as 119864

0 Equation (29) yields

119906119908(119909 119905) = 119864

0120596 (119909 119905) (31)

Without loss of generality 1198640is equated to oneThe functions

of Γ(119909 119905) and120596(119909 119905)will be determined later on such that thegoverning equations of (3) to (5) reduce to a set of ordinarydifferential equations in 119864(120578) 119865(120578) and 119906(120578)

4 Reduction to a Set of Ordinary Problem

Now we define 120578 in general form of 120578 = 119910120587(119909 119905) Invoking(30) and (31) and (3) to (5)

119889119865

119889120578

+ 1198621120578

119889119906

119889120578

+ 1198622119906 = 0

119899(

119889119906

119889120578

)

119899minus1

1198892

119906

1198891205782minus [120578 (119862

3+ 1198624119906) + 119862

5119865]

119889119906

119889120578

minus 1198626119906 minus 11986271199062

+ 1198628120579 = 0

1198892

120579

1198891205782minus Pr [119889120579

119889120578

120578 (11986210+ 11986211119906) + 119862

12119865 minus 119862

9119906120579] = 0

(32)

Journal of Applied Mathematics 5

Constants of 1198621to 11986212are defined as follows

1198621=

120596

Γ1205872

120597120587

120597119909

1198622=

1

Γ120587

120597120596

120597119909

1198623=

1

120599120596119899minus1120587119899+2

120597120587

120597119905

1198624=

1

120599120596119899minus2120587119899+2

120597120587

120597119909

1198625=

Γ

120599120596119899minus1120587119899

1198626=

1

120599120596119899120587119899+1

120597120596

120597119905

1198627=

1

120599120596119899minus1120587119899+1

120597120596

120597119909

1198628=

119909

120599120596119899120587119899+1

119892120573119902119908

119896radicRe119909

1198629=

120596

1205991199091205872

11986210=

1

1205991205873

120597120587

120597119905

11986211=

120596

1205991205873

120597120587

120597119909

11986212=

Γ

120599120587

(33)

Now we obtain the exact value of constants as

1198627

1198628

= 1 997888rarr 120596 (119909) = (

119892120573119902119908

119896radicRe119909

)

12

119909 (34)

From (31)

119906119908(119909) = (

119892120573119902119908

119896radicRe119909

)

12

119909 (35)

For 1198629= 1 we have

120587 =

1

12059912

(

119892120573119902119908

119896radicRe119909

)

14

(36)

According to the definition of similarity variable

120578 = 119910(

119892120573119902119908

1205992119896radicRe

119909

)

14

(37)

For 1198622= 1 we get

Γ = 12059912

(

119892120573119902119908

119896radicRe119909

)

14

(38)

Using (30) the horizontal component of velocity is

V (120578) = 12059912 (119892120573119902119908

119896radicRe119909

)

14

119865 (120578) (39)

As 120587 is invariant and 120596 is independent of time

1198621= 1198623= 1198624= 1198626= 11986210= 11986211= 0 (40)

On the other hand due to definitions of 120587 120596 and Γ

11986212= 1 119862

5= 1198627= 1198628= (120596120587)

1minus119899

(41)

Substituting the above constants in (32) they finally reduceto

119889119865

119889120578

+ 119906 = 0 (42)

119899(120596120587)119899minus1

(

119889119906

119889120578

)

119899minus1

1198892

119906

1198891205782minus 119865

119889119906

119889120578

minus 1199062

+ 120579 = 0 (43)

1198892

120579

1198891205782minus Pr(119889120579

119889120578

119865 + 119906120579) = 0 (44)

The new forms of boundary conditions are

119865 (0) = 0

119906 (0) = 1 119906 (infin) = 0

120597120579 (0)

120597120578

= minus1 120579 (infin) = 0

(45)

It can be proved that 120597120579(0)120597120578 = minus1 According to (2b) anddefinition of 120579 (2b) will change into

120597120579 (119909 0 119905)

120597119910

=

minusradicRe119909

119909

(46)

where Re119909= 119906119904119909120599 and 119906

119904is average velocity in y direction

Assuming 119906119904= 119906119908

Re119909= 1199092

1205872

(47)

Using chain rule operation

120597120579 (0)

120597120578

120597120578

120597119910

=

minusradicRe119909

119909

(48)

So according to Re119909and 120578 120597120579(0)120597120578 will be minus1

For instance assuming 119899 = 1 for a case ofNewtonian fluidflow (43) will reduce to

1198892

119906

1198891205782+

120578

2

119889119906

119889120578

minus 119906 = minus erfc(radicPr2

120578) (49)

where

119906 (0) = 1 119906 (infin) = 0 (50)

6 Journal of Applied Mathematics

Table 1 Comparison of present problem solution and case of Kassem [34] for Pr = 1 2 5 and 10 of Newtonian fluid

Pr Kassem [34] Present work120578 119906 (120578) Θ (120578) 119906 (120578) Θ (120578)

1

05 0669 0507 0667 05101 0391 0256 0381 026115 0165 0103 0161 01052 0000 0003 0003 0005

2

03 0736 0398 0742 040106 0515 0222 0520 022509 0315 0092 0317 009512 0143 0009 0150 0011

5

025 0745 0206 0750 020805 0525 0095 0531 0011075 0333 0040 0335 00421 0171 0017 0171 0017

10

02 0768 0124 0771 012504 0565 0046 0571 004806 0381 0015 0391 001708 0216 0007 0218 0008

0 05 1 15 20

02

04

06

08

1

12

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

u(120578)

Figure 2 Vertical velocity of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

5 Results and Discussion

In this section initially results of Newtonian fluid flow overstretching sheet are compared with Kassem [34] Table 1shows the results obtained by numerical solution of ODEequation (49) and [34] for power index 119899 equal to unityand Pr number of 1 There is a satisfying agreement amongthe present evaluations Complementary results for differentvalues of Pr are shown in Figures 2 and 3 It can beunderstood that as the fluid momentum diffusivity increaseswith respect to the thermal diffusivity large Pr numbersvertical velocity decreases In other words the velocity 119906(120578)inside the boundary layer decreases with the increase of fluidviscosity Also the temperature plot of boundary layer revealsa decrease of temperature at the wall temperature for larger

0 05 1 15 20

02

04

06

08

1

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

120579(120578)

Figure 3 Temperature of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

Pr So heat loss increases for larger Pr as the boundary layergets thinner

For non-Newtonian case of the fluid the vertical velocityand temperature of fluid adjacent to the stretching sheet areevaluated by solving (42) to (44) with the certain boundaryconditions of (45) Results of the shooting method solutionare illustrated in Figures 4 and 5 for Pr = 5 As the powerindex increases vertical velocity component encounters withresistance to rise due to increased value of shear stress Butfor lower power index vertical component of velocity movesfaster due to reduced apparent viscosity Similarly thinnerthermal boundary layer for higher power index will increasethe amount of heat loss

Journal of Applied Mathematics 7

0 2 4 6 8 100

02

04

06

08

1

12

n = 05n = 10n = 15

120578

u(120578)

Figure 4 Vertical velocity of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

0 05 1 15 2 250

02

04

06

120578

120579(120578)

n = 05n = 10n = 15

Figure 5 Temperature of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

6 Conclusion

Similarity solutions are performed for flow of power lawnon-Newtonian fluid film on unsteady stretching surfacesubjected to constant heat fluxThe nonlinear coupled partialdifferential equations (PDE) governing the flow and theboundary conditions are transformed to a set of ordinarydifferential equations (ODE) using two-parameter groupsThis technique reduces the number of independent variablesby two and conclusively the obtained ordinary differentialequations are solved numerically for the temperature andvelocity using the shooting method Results show that higherPr number and higher power index of non-Newtonian fluidencounter with difficulty to move as faster as lower Pr andpower index Enhanced amount of shear stress explains thereason of the predicted flow

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[2] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[3] R Usha and R Sridharan ldquoThe axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal of FluidsEngineering vol 117 no 1 pp 81ndash85 1995

[4] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[5] H I Andersson and T Ytrehus ldquoFalkner-Skan solution forgravity-driven film flowrdquo Journal of Applied Mechanics vol 52no 4 pp 783ndash786 1985

[6] H I Andersson and F Irgens ldquoGravity-driven laminar filmflow of power-law fluids along vertical wallsrdquo Journal of Non-Newtonian Fluid Mechanics vol 27 no 2 pp 153ndash172 1988

[7] D-Y Shang and H I Andersson ldquoHeat transfer in gravity-driven film flow of power-law fluidsrdquo International Journal ofHeat and Mass Transfer vol 42 no 11 pp 2085ndash2099 1998

[8] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[9] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[10] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching planewallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[11] R Tsai K H Huang and J S Huang ldquoFlow and heattransfer over an unsteady stretching surface with non-uniformheat sourcerdquo International Communications in Heat and MassTransfer vol 35 no 10 pp 1340ndash1343 2008

[12] B Sahoo and Y Do ldquoEffects of slip on sheet-driven flow andheat transfer of a third grade fluid past a stretching sheetrdquoInternational Communications in Heat and Mass Transfer vol37 no 8 pp 1064ndash1071 2010

[13] C R Illingworth ldquoUnsteady laminar flow of gas near aninfinite flat platerdquo Mathematical Proceedings of the CambridgePhilosophical Society vol 46 no 4 pp 603ndash613 1950

[14] R Siegel ldquoTransient free convection from a vertical flat platerdquoTransactions of the ASME vol 80 no 2 pp 347ndash359

[15] B Gebhart ldquoTransient natural convection from vertical ele-mentsrdquo Journal of Heat Transfer vol 83 pp 61ndash70 1961

[16] A J Chamkha ldquoSimilarity solution for thermal boundary layeron a stretched surface of a non-Newtonian fluidrdquo InternationalCommunications in Heat and Mass Transfer vol 24 no 5 pp643ndash652 1997

[17] J D Hellums and S W Churchill ldquoTransient and steady statefree and natural convection numerical solutions part Imdashthe

8 Journal of Applied Mathematics

isothermal vertical platerdquoAIChE Journal vol 8 no 5 pp 690ndash692 1962

[18] G D Callahan and W J Marner ldquoTransient free convectionwith mass transfer on an isothermal vertical flat platerdquo Inter-national Journal of Heat and Mass Transfer vol 19 no 2 pp165ndash174 1976

[19] T G Howell D R Jeng andK J deWitt ldquoMomentum and heattransfer on a continuous moving surface in a power law fluidrdquoInternational Journal of Heat and Mass Transfer vol 40 no 8pp 1853ndash1861 1997

[20] A J A Morgan ldquoThe reduction by one of the number ofindependent variables in some systems of partial differentialequationsrdquo The Quarterly Journal of Mathematics vol 3 no 1pp 250ndash259 1952

[21] A D Michal ldquoDifferential invariants and invariant partialdifferential equations under continuous transformation groupsin normed linear spacesrdquo Proceedings of the National Academyof Sciences of the United States of America vol 37 pp 623ndash6271951

[22] G Birkhoff ldquoMathematics for engineersrdquo Electrical Engineeringvol 67 no 12 pp 1185ndash1188 1948

[23] G Birkhoff Hydrodynamics A Study in Logic Fact and Simili-tude Princeton University Press Princeton NJ USA 1960

[24] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[25] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[26] W F Ames ldquoSimilarity for the nonlinear diffusion equationrdquoIndustrial amp Engineering Chemistry Fundamentals vol 4 no 1pp 72ndash76 1965

[27] W FAmesNonlinear Partial Differential Equations in Engineer-ing chapter 2 Academic Press New York NY USA 1972

[28] W F Ames and M C Nucci ldquoAnalysis of fluid equations bygroupmethodsrdquo Journal of EngineeringMathematics vol 20 no2 pp 181ndash187 1986

[29] M J Moran and R A Gaggioli ldquoSimilarity analysis of com-pressible boundary layer flows via group theoryrdquo TechnicalSummaryReport 838US ArmyMathematics ResearchCenterMadison Wis USA 1967

[30] M J Moran and R A Gaggioli ldquoSimilarity analysis via grouptheoryrdquo AIAA Journal vol 6 pp 2014ndash2016 1968

[31] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[32] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[33] M B Abd-el-Malek M M Kassem and M L MekkyldquoSimilarity solutions for unsteady free-convection flow from acontinuous moving vertical surfacerdquo Journal of Computationaland Applied Mathematics vol 164 no 1 pp 11ndash24 2004

[34] M Kassem ldquoGroup solution for unsteady free-convection flowfrom a vertical moving plate subjected to constant heat fluxrdquoJournal of Computational and Applied Mathematics vol 187 no1 pp 72ndash86 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

This group converts invariantly the differential equations of(3) to (5) and the initial and boundary conditions (6a) and(6b)

32TheComplete Set of Absolute Invariants Thecomplete setof absolute invariants is as follows

(i) the absolute invariants of the independent variables(119909 119910 119905) are 120578 = 120578(119909 119910 119905)

(ii) the absolute invariants of the dependent variables(119906 V 119906

119908 120579) are

119892119895(119909 119910 119905 119906 119906

119908 V 120579) = 119865

119895(120578 (119909 119910 119905)) 119895 = 1 2 3 4

(20)

The basic theorem in group theory asserts that a function119892119895(119909 119910 119905 119906 119906

119908 V 120579) is an absolute invariant of a two-

parameter group if it satisfies the following two first-orderlinear differential equations

13

sum

119894=1

(120572119894119878119894+ 120572119894+1)

120597119892

120597119878119894

= 0

13

sum

119894=1

(120573119894119878119894+ 120573119894+1)

120597119892

120597119878119894

= 0 (21)

where 1205721= (120597119862

119909

1205971198861)(1198860

1 1198860

2) 1205722= (120597119870

119909

1205971198861)(1198860

1 1198860

2)

1205731= (120597119862

119909

1205971198862)(1198860

1 1198860

2) 1205732= (120597119870

119909

1205971198862)(1198860

1 1198860

2) and so

forth (11988601 1198860

2) are the identity elements of the group

The absolute invariant 120578(119909 119910 119905) of the independent vari-ables (119909 119910 119905) is determined using (21)

(1205721119909 + 1205722)

120597120578

120597119909

+ 1205723119910

120597120578

120597119910

+ 1205725119905

120597120578

120597119905

= 0

(1205731119909 + 1205732)

120597120578

120597119909

+ 1205733119910

120597120578

120597119910

+ 1205735119905

120597120578

120597119905

= 0

(22)

where 1205724= 1205734= 119870119910

= 0 and 1205726= 1205736= 119870119905

= 0Elimination of 119910(120597120578120597119910) and 120597120578120597119909 from (22) yields

(12058231119909 + 12058232)

120597120578

120597119909

+ 12058235119905

120597120578

120597119905

= 0 (23a)

(12058231119909 + 12058232) 119910

120597120578

120597119910

+ (12058215119909 + 12058225) 119905

120597120578

120597119905

= 0 (23b)

where 120582119894119895= 120572119894120573119895minus 120572119895120573119894and 119894 119895 = 1 2 3 4 5 Invoking (21)

and definitions of 120572 and 120573we get 1205725= 21205723and 120573

5= 21205733 thus

12058235= 12057231205735minus 12057251205733will assuage to zero Using (23a) we obtain

120597120578120597119909 = 0 indicating that 120578 is dependent on 119910 and 119905 Solving(23b) we get

120578 = 119910120587 (119905) (24)

where 120587(119905) = 119886119905119887 and 119886 and 119887 are arbitrary constantsSimultaneously the absolute invariants of the dependent

variables 119906 V 119906119908 and 120579 are obtained from the group transfor-

mation (19)

1198921(119909 119910 119905 119906) = 119906 (120578)

1198922(119909 119910 119905 119906) = 120579 (120578)

(25)

1198923(119909 119905 119906

119908) is obtained using (21) as

(1205721119909 + 1205722)

1205971198923

120597119909

+ (1205725119905 + 1205726)

1205971198923

120597119905

+ (12057211119906119908+ 12057212)

1205971198923

120597119906119908

= 0

(26a)

(1205731119909 + 1205732)

1205971198923

120597119909

+ (1205735119905 + 1205736)

1205971198923

120597119905

+ (12057311119906119908+ 12057312)

1205971198923

120597119906119908

= 0

(26b)

Eliminating terms of 1205971198923120597119909 and 120597119892

3120597119905 from ((26a) and

(26b)) results in

1198923(119909 119905 119906

119908) = 1206011(

119906119908

120596 (119909 119905)

) = 119864 (120578) (27)

Similarly for 1198924(119909 119905 V)

1198924(119909 119905 V) = 120601

2(

VΓ (119909 119905)

) = 119865 (120578) (28)

where 120596(119909 119905) Γ(119909 119905) 119864(120578) and 119865(120578) are functions to bedetermined Without loss of generality the 120601rsquos in (27) and(28) are selected to be identity functions hence the functions119906119908(119909 119905) and V(119909 119910 119905) are expressed in terms 119864(120578) and 119865(120578) as

119906119908(119909 119905) = 120596 (119909 119905) 119864 (120578) (29)

V (119909 119910 119905) = Γ (119909 119905) 119865 (120578) (30)

Since 120596(119909 119905) and 119906119908(119909 119905) are independent of y whereas 120578 is

function of y we conclude that 119864(120578) is equal to a constantsuch as 119864

0 Equation (29) yields

119906119908(119909 119905) = 119864

0120596 (119909 119905) (31)

Without loss of generality 1198640is equated to oneThe functions

of Γ(119909 119905) and120596(119909 119905)will be determined later on such that thegoverning equations of (3) to (5) reduce to a set of ordinarydifferential equations in 119864(120578) 119865(120578) and 119906(120578)

4 Reduction to a Set of Ordinary Problem

Now we define 120578 in general form of 120578 = 119910120587(119909 119905) Invoking(30) and (31) and (3) to (5)

119889119865

119889120578

+ 1198621120578

119889119906

119889120578

+ 1198622119906 = 0

119899(

119889119906

119889120578

)

119899minus1

1198892

119906

1198891205782minus [120578 (119862

3+ 1198624119906) + 119862

5119865]

119889119906

119889120578

minus 1198626119906 minus 11986271199062

+ 1198628120579 = 0

1198892

120579

1198891205782minus Pr [119889120579

119889120578

120578 (11986210+ 11986211119906) + 119862

12119865 minus 119862

9119906120579] = 0

(32)

Journal of Applied Mathematics 5

Constants of 1198621to 11986212are defined as follows

1198621=

120596

Γ1205872

120597120587

120597119909

1198622=

1

Γ120587

120597120596

120597119909

1198623=

1

120599120596119899minus1120587119899+2

120597120587

120597119905

1198624=

1

120599120596119899minus2120587119899+2

120597120587

120597119909

1198625=

Γ

120599120596119899minus1120587119899

1198626=

1

120599120596119899120587119899+1

120597120596

120597119905

1198627=

1

120599120596119899minus1120587119899+1

120597120596

120597119909

1198628=

119909

120599120596119899120587119899+1

119892120573119902119908

119896radicRe119909

1198629=

120596

1205991199091205872

11986210=

1

1205991205873

120597120587

120597119905

11986211=

120596

1205991205873

120597120587

120597119909

11986212=

Γ

120599120587

(33)

Now we obtain the exact value of constants as

1198627

1198628

= 1 997888rarr 120596 (119909) = (

119892120573119902119908

119896radicRe119909

)

12

119909 (34)

From (31)

119906119908(119909) = (

119892120573119902119908

119896radicRe119909

)

12

119909 (35)

For 1198629= 1 we have

120587 =

1

12059912

(

119892120573119902119908

119896radicRe119909

)

14

(36)

According to the definition of similarity variable

120578 = 119910(

119892120573119902119908

1205992119896radicRe

119909

)

14

(37)

For 1198622= 1 we get

Γ = 12059912

(

119892120573119902119908

119896radicRe119909

)

14

(38)

Using (30) the horizontal component of velocity is

V (120578) = 12059912 (119892120573119902119908

119896radicRe119909

)

14

119865 (120578) (39)

As 120587 is invariant and 120596 is independent of time

1198621= 1198623= 1198624= 1198626= 11986210= 11986211= 0 (40)

On the other hand due to definitions of 120587 120596 and Γ

11986212= 1 119862

5= 1198627= 1198628= (120596120587)

1minus119899

(41)

Substituting the above constants in (32) they finally reduceto

119889119865

119889120578

+ 119906 = 0 (42)

119899(120596120587)119899minus1

(

119889119906

119889120578

)

119899minus1

1198892

119906

1198891205782minus 119865

119889119906

119889120578

minus 1199062

+ 120579 = 0 (43)

1198892

120579

1198891205782minus Pr(119889120579

119889120578

119865 + 119906120579) = 0 (44)

The new forms of boundary conditions are

119865 (0) = 0

119906 (0) = 1 119906 (infin) = 0

120597120579 (0)

120597120578

= minus1 120579 (infin) = 0

(45)

It can be proved that 120597120579(0)120597120578 = minus1 According to (2b) anddefinition of 120579 (2b) will change into

120597120579 (119909 0 119905)

120597119910

=

minusradicRe119909

119909

(46)

where Re119909= 119906119904119909120599 and 119906

119904is average velocity in y direction

Assuming 119906119904= 119906119908

Re119909= 1199092

1205872

(47)

Using chain rule operation

120597120579 (0)

120597120578

120597120578

120597119910

=

minusradicRe119909

119909

(48)

So according to Re119909and 120578 120597120579(0)120597120578 will be minus1

For instance assuming 119899 = 1 for a case ofNewtonian fluidflow (43) will reduce to

1198892

119906

1198891205782+

120578

2

119889119906

119889120578

minus 119906 = minus erfc(radicPr2

120578) (49)

where

119906 (0) = 1 119906 (infin) = 0 (50)

6 Journal of Applied Mathematics

Table 1 Comparison of present problem solution and case of Kassem [34] for Pr = 1 2 5 and 10 of Newtonian fluid

Pr Kassem [34] Present work120578 119906 (120578) Θ (120578) 119906 (120578) Θ (120578)

1

05 0669 0507 0667 05101 0391 0256 0381 026115 0165 0103 0161 01052 0000 0003 0003 0005

2

03 0736 0398 0742 040106 0515 0222 0520 022509 0315 0092 0317 009512 0143 0009 0150 0011

5

025 0745 0206 0750 020805 0525 0095 0531 0011075 0333 0040 0335 00421 0171 0017 0171 0017

10

02 0768 0124 0771 012504 0565 0046 0571 004806 0381 0015 0391 001708 0216 0007 0218 0008

0 05 1 15 20

02

04

06

08

1

12

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

u(120578)

Figure 2 Vertical velocity of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

5 Results and Discussion

In this section initially results of Newtonian fluid flow overstretching sheet are compared with Kassem [34] Table 1shows the results obtained by numerical solution of ODEequation (49) and [34] for power index 119899 equal to unityand Pr number of 1 There is a satisfying agreement amongthe present evaluations Complementary results for differentvalues of Pr are shown in Figures 2 and 3 It can beunderstood that as the fluid momentum diffusivity increaseswith respect to the thermal diffusivity large Pr numbersvertical velocity decreases In other words the velocity 119906(120578)inside the boundary layer decreases with the increase of fluidviscosity Also the temperature plot of boundary layer revealsa decrease of temperature at the wall temperature for larger

0 05 1 15 20

02

04

06

08

1

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

120579(120578)

Figure 3 Temperature of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

Pr So heat loss increases for larger Pr as the boundary layergets thinner

For non-Newtonian case of the fluid the vertical velocityand temperature of fluid adjacent to the stretching sheet areevaluated by solving (42) to (44) with the certain boundaryconditions of (45) Results of the shooting method solutionare illustrated in Figures 4 and 5 for Pr = 5 As the powerindex increases vertical velocity component encounters withresistance to rise due to increased value of shear stress Butfor lower power index vertical component of velocity movesfaster due to reduced apparent viscosity Similarly thinnerthermal boundary layer for higher power index will increasethe amount of heat loss

Journal of Applied Mathematics 7

0 2 4 6 8 100

02

04

06

08

1

12

n = 05n = 10n = 15

120578

u(120578)

Figure 4 Vertical velocity of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

0 05 1 15 2 250

02

04

06

120578

120579(120578)

n = 05n = 10n = 15

Figure 5 Temperature of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

6 Conclusion

Similarity solutions are performed for flow of power lawnon-Newtonian fluid film on unsteady stretching surfacesubjected to constant heat fluxThe nonlinear coupled partialdifferential equations (PDE) governing the flow and theboundary conditions are transformed to a set of ordinarydifferential equations (ODE) using two-parameter groupsThis technique reduces the number of independent variablesby two and conclusively the obtained ordinary differentialequations are solved numerically for the temperature andvelocity using the shooting method Results show that higherPr number and higher power index of non-Newtonian fluidencounter with difficulty to move as faster as lower Pr andpower index Enhanced amount of shear stress explains thereason of the predicted flow

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[2] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[3] R Usha and R Sridharan ldquoThe axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal of FluidsEngineering vol 117 no 1 pp 81ndash85 1995

[4] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[5] H I Andersson and T Ytrehus ldquoFalkner-Skan solution forgravity-driven film flowrdquo Journal of Applied Mechanics vol 52no 4 pp 783ndash786 1985

[6] H I Andersson and F Irgens ldquoGravity-driven laminar filmflow of power-law fluids along vertical wallsrdquo Journal of Non-Newtonian Fluid Mechanics vol 27 no 2 pp 153ndash172 1988

[7] D-Y Shang and H I Andersson ldquoHeat transfer in gravity-driven film flow of power-law fluidsrdquo International Journal ofHeat and Mass Transfer vol 42 no 11 pp 2085ndash2099 1998

[8] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[9] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[10] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching planewallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[11] R Tsai K H Huang and J S Huang ldquoFlow and heattransfer over an unsteady stretching surface with non-uniformheat sourcerdquo International Communications in Heat and MassTransfer vol 35 no 10 pp 1340ndash1343 2008

[12] B Sahoo and Y Do ldquoEffects of slip on sheet-driven flow andheat transfer of a third grade fluid past a stretching sheetrdquoInternational Communications in Heat and Mass Transfer vol37 no 8 pp 1064ndash1071 2010

[13] C R Illingworth ldquoUnsteady laminar flow of gas near aninfinite flat platerdquo Mathematical Proceedings of the CambridgePhilosophical Society vol 46 no 4 pp 603ndash613 1950

[14] R Siegel ldquoTransient free convection from a vertical flat platerdquoTransactions of the ASME vol 80 no 2 pp 347ndash359

[15] B Gebhart ldquoTransient natural convection from vertical ele-mentsrdquo Journal of Heat Transfer vol 83 pp 61ndash70 1961

[16] A J Chamkha ldquoSimilarity solution for thermal boundary layeron a stretched surface of a non-Newtonian fluidrdquo InternationalCommunications in Heat and Mass Transfer vol 24 no 5 pp643ndash652 1997

[17] J D Hellums and S W Churchill ldquoTransient and steady statefree and natural convection numerical solutions part Imdashthe

8 Journal of Applied Mathematics

isothermal vertical platerdquoAIChE Journal vol 8 no 5 pp 690ndash692 1962

[18] G D Callahan and W J Marner ldquoTransient free convectionwith mass transfer on an isothermal vertical flat platerdquo Inter-national Journal of Heat and Mass Transfer vol 19 no 2 pp165ndash174 1976

[19] T G Howell D R Jeng andK J deWitt ldquoMomentum and heattransfer on a continuous moving surface in a power law fluidrdquoInternational Journal of Heat and Mass Transfer vol 40 no 8pp 1853ndash1861 1997

[20] A J A Morgan ldquoThe reduction by one of the number ofindependent variables in some systems of partial differentialequationsrdquo The Quarterly Journal of Mathematics vol 3 no 1pp 250ndash259 1952

[21] A D Michal ldquoDifferential invariants and invariant partialdifferential equations under continuous transformation groupsin normed linear spacesrdquo Proceedings of the National Academyof Sciences of the United States of America vol 37 pp 623ndash6271951

[22] G Birkhoff ldquoMathematics for engineersrdquo Electrical Engineeringvol 67 no 12 pp 1185ndash1188 1948

[23] G Birkhoff Hydrodynamics A Study in Logic Fact and Simili-tude Princeton University Press Princeton NJ USA 1960

[24] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[25] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[26] W F Ames ldquoSimilarity for the nonlinear diffusion equationrdquoIndustrial amp Engineering Chemistry Fundamentals vol 4 no 1pp 72ndash76 1965

[27] W FAmesNonlinear Partial Differential Equations in Engineer-ing chapter 2 Academic Press New York NY USA 1972

[28] W F Ames and M C Nucci ldquoAnalysis of fluid equations bygroupmethodsrdquo Journal of EngineeringMathematics vol 20 no2 pp 181ndash187 1986

[29] M J Moran and R A Gaggioli ldquoSimilarity analysis of com-pressible boundary layer flows via group theoryrdquo TechnicalSummaryReport 838US ArmyMathematics ResearchCenterMadison Wis USA 1967

[30] M J Moran and R A Gaggioli ldquoSimilarity analysis via grouptheoryrdquo AIAA Journal vol 6 pp 2014ndash2016 1968

[31] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[32] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[33] M B Abd-el-Malek M M Kassem and M L MekkyldquoSimilarity solutions for unsteady free-convection flow from acontinuous moving vertical surfacerdquo Journal of Computationaland Applied Mathematics vol 164 no 1 pp 11ndash24 2004

[34] M Kassem ldquoGroup solution for unsteady free-convection flowfrom a vertical moving plate subjected to constant heat fluxrdquoJournal of Computational and Applied Mathematics vol 187 no1 pp 72ndash86 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Constants of 1198621to 11986212are defined as follows

1198621=

120596

Γ1205872

120597120587

120597119909

1198622=

1

Γ120587

120597120596

120597119909

1198623=

1

120599120596119899minus1120587119899+2

120597120587

120597119905

1198624=

1

120599120596119899minus2120587119899+2

120597120587

120597119909

1198625=

Γ

120599120596119899minus1120587119899

1198626=

1

120599120596119899120587119899+1

120597120596

120597119905

1198627=

1

120599120596119899minus1120587119899+1

120597120596

120597119909

1198628=

119909

120599120596119899120587119899+1

119892120573119902119908

119896radicRe119909

1198629=

120596

1205991199091205872

11986210=

1

1205991205873

120597120587

120597119905

11986211=

120596

1205991205873

120597120587

120597119909

11986212=

Γ

120599120587

(33)

Now we obtain the exact value of constants as

1198627

1198628

= 1 997888rarr 120596 (119909) = (

119892120573119902119908

119896radicRe119909

)

12

119909 (34)

From (31)

119906119908(119909) = (

119892120573119902119908

119896radicRe119909

)

12

119909 (35)

For 1198629= 1 we have

120587 =

1

12059912

(

119892120573119902119908

119896radicRe119909

)

14

(36)

According to the definition of similarity variable

120578 = 119910(

119892120573119902119908

1205992119896radicRe

119909

)

14

(37)

For 1198622= 1 we get

Γ = 12059912

(

119892120573119902119908

119896radicRe119909

)

14

(38)

Using (30) the horizontal component of velocity is

V (120578) = 12059912 (119892120573119902119908

119896radicRe119909

)

14

119865 (120578) (39)

As 120587 is invariant and 120596 is independent of time

1198621= 1198623= 1198624= 1198626= 11986210= 11986211= 0 (40)

On the other hand due to definitions of 120587 120596 and Γ

11986212= 1 119862

5= 1198627= 1198628= (120596120587)

1minus119899

(41)

Substituting the above constants in (32) they finally reduceto

119889119865

119889120578

+ 119906 = 0 (42)

119899(120596120587)119899minus1

(

119889119906

119889120578

)

119899minus1

1198892

119906

1198891205782minus 119865

119889119906

119889120578

minus 1199062

+ 120579 = 0 (43)

1198892

120579

1198891205782minus Pr(119889120579

119889120578

119865 + 119906120579) = 0 (44)

The new forms of boundary conditions are

119865 (0) = 0

119906 (0) = 1 119906 (infin) = 0

120597120579 (0)

120597120578

= minus1 120579 (infin) = 0

(45)

It can be proved that 120597120579(0)120597120578 = minus1 According to (2b) anddefinition of 120579 (2b) will change into

120597120579 (119909 0 119905)

120597119910

=

minusradicRe119909

119909

(46)

where Re119909= 119906119904119909120599 and 119906

119904is average velocity in y direction

Assuming 119906119904= 119906119908

Re119909= 1199092

1205872

(47)

Using chain rule operation

120597120579 (0)

120597120578

120597120578

120597119910

=

minusradicRe119909

119909

(48)

So according to Re119909and 120578 120597120579(0)120597120578 will be minus1

For instance assuming 119899 = 1 for a case ofNewtonian fluidflow (43) will reduce to

1198892

119906

1198891205782+

120578

2

119889119906

119889120578

minus 119906 = minus erfc(radicPr2

120578) (49)

where

119906 (0) = 1 119906 (infin) = 0 (50)

6 Journal of Applied Mathematics

Table 1 Comparison of present problem solution and case of Kassem [34] for Pr = 1 2 5 and 10 of Newtonian fluid

Pr Kassem [34] Present work120578 119906 (120578) Θ (120578) 119906 (120578) Θ (120578)

1

05 0669 0507 0667 05101 0391 0256 0381 026115 0165 0103 0161 01052 0000 0003 0003 0005

2

03 0736 0398 0742 040106 0515 0222 0520 022509 0315 0092 0317 009512 0143 0009 0150 0011

5

025 0745 0206 0750 020805 0525 0095 0531 0011075 0333 0040 0335 00421 0171 0017 0171 0017

10

02 0768 0124 0771 012504 0565 0046 0571 004806 0381 0015 0391 001708 0216 0007 0218 0008

0 05 1 15 20

02

04

06

08

1

12

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

u(120578)

Figure 2 Vertical velocity of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

5 Results and Discussion

In this section initially results of Newtonian fluid flow overstretching sheet are compared with Kassem [34] Table 1shows the results obtained by numerical solution of ODEequation (49) and [34] for power index 119899 equal to unityand Pr number of 1 There is a satisfying agreement amongthe present evaluations Complementary results for differentvalues of Pr are shown in Figures 2 and 3 It can beunderstood that as the fluid momentum diffusivity increaseswith respect to the thermal diffusivity large Pr numbersvertical velocity decreases In other words the velocity 119906(120578)inside the boundary layer decreases with the increase of fluidviscosity Also the temperature plot of boundary layer revealsa decrease of temperature at the wall temperature for larger

0 05 1 15 20

02

04

06

08

1

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

120579(120578)

Figure 3 Temperature of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

Pr So heat loss increases for larger Pr as the boundary layergets thinner

For non-Newtonian case of the fluid the vertical velocityand temperature of fluid adjacent to the stretching sheet areevaluated by solving (42) to (44) with the certain boundaryconditions of (45) Results of the shooting method solutionare illustrated in Figures 4 and 5 for Pr = 5 As the powerindex increases vertical velocity component encounters withresistance to rise due to increased value of shear stress Butfor lower power index vertical component of velocity movesfaster due to reduced apparent viscosity Similarly thinnerthermal boundary layer for higher power index will increasethe amount of heat loss

Journal of Applied Mathematics 7

0 2 4 6 8 100

02

04

06

08

1

12

n = 05n = 10n = 15

120578

u(120578)

Figure 4 Vertical velocity of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

0 05 1 15 2 250

02

04

06

120578

120579(120578)

n = 05n = 10n = 15

Figure 5 Temperature of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

6 Conclusion

Similarity solutions are performed for flow of power lawnon-Newtonian fluid film on unsteady stretching surfacesubjected to constant heat fluxThe nonlinear coupled partialdifferential equations (PDE) governing the flow and theboundary conditions are transformed to a set of ordinarydifferential equations (ODE) using two-parameter groupsThis technique reduces the number of independent variablesby two and conclusively the obtained ordinary differentialequations are solved numerically for the temperature andvelocity using the shooting method Results show that higherPr number and higher power index of non-Newtonian fluidencounter with difficulty to move as faster as lower Pr andpower index Enhanced amount of shear stress explains thereason of the predicted flow

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[2] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[3] R Usha and R Sridharan ldquoThe axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal of FluidsEngineering vol 117 no 1 pp 81ndash85 1995

[4] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[5] H I Andersson and T Ytrehus ldquoFalkner-Skan solution forgravity-driven film flowrdquo Journal of Applied Mechanics vol 52no 4 pp 783ndash786 1985

[6] H I Andersson and F Irgens ldquoGravity-driven laminar filmflow of power-law fluids along vertical wallsrdquo Journal of Non-Newtonian Fluid Mechanics vol 27 no 2 pp 153ndash172 1988

[7] D-Y Shang and H I Andersson ldquoHeat transfer in gravity-driven film flow of power-law fluidsrdquo International Journal ofHeat and Mass Transfer vol 42 no 11 pp 2085ndash2099 1998

[8] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[9] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[10] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching planewallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[11] R Tsai K H Huang and J S Huang ldquoFlow and heattransfer over an unsteady stretching surface with non-uniformheat sourcerdquo International Communications in Heat and MassTransfer vol 35 no 10 pp 1340ndash1343 2008

[12] B Sahoo and Y Do ldquoEffects of slip on sheet-driven flow andheat transfer of a third grade fluid past a stretching sheetrdquoInternational Communications in Heat and Mass Transfer vol37 no 8 pp 1064ndash1071 2010

[13] C R Illingworth ldquoUnsteady laminar flow of gas near aninfinite flat platerdquo Mathematical Proceedings of the CambridgePhilosophical Society vol 46 no 4 pp 603ndash613 1950

[14] R Siegel ldquoTransient free convection from a vertical flat platerdquoTransactions of the ASME vol 80 no 2 pp 347ndash359

[15] B Gebhart ldquoTransient natural convection from vertical ele-mentsrdquo Journal of Heat Transfer vol 83 pp 61ndash70 1961

[16] A J Chamkha ldquoSimilarity solution for thermal boundary layeron a stretched surface of a non-Newtonian fluidrdquo InternationalCommunications in Heat and Mass Transfer vol 24 no 5 pp643ndash652 1997

[17] J D Hellums and S W Churchill ldquoTransient and steady statefree and natural convection numerical solutions part Imdashthe

8 Journal of Applied Mathematics

isothermal vertical platerdquoAIChE Journal vol 8 no 5 pp 690ndash692 1962

[18] G D Callahan and W J Marner ldquoTransient free convectionwith mass transfer on an isothermal vertical flat platerdquo Inter-national Journal of Heat and Mass Transfer vol 19 no 2 pp165ndash174 1976

[19] T G Howell D R Jeng andK J deWitt ldquoMomentum and heattransfer on a continuous moving surface in a power law fluidrdquoInternational Journal of Heat and Mass Transfer vol 40 no 8pp 1853ndash1861 1997

[20] A J A Morgan ldquoThe reduction by one of the number ofindependent variables in some systems of partial differentialequationsrdquo The Quarterly Journal of Mathematics vol 3 no 1pp 250ndash259 1952

[21] A D Michal ldquoDifferential invariants and invariant partialdifferential equations under continuous transformation groupsin normed linear spacesrdquo Proceedings of the National Academyof Sciences of the United States of America vol 37 pp 623ndash6271951

[22] G Birkhoff ldquoMathematics for engineersrdquo Electrical Engineeringvol 67 no 12 pp 1185ndash1188 1948

[23] G Birkhoff Hydrodynamics A Study in Logic Fact and Simili-tude Princeton University Press Princeton NJ USA 1960

[24] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[25] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[26] W F Ames ldquoSimilarity for the nonlinear diffusion equationrdquoIndustrial amp Engineering Chemistry Fundamentals vol 4 no 1pp 72ndash76 1965

[27] W FAmesNonlinear Partial Differential Equations in Engineer-ing chapter 2 Academic Press New York NY USA 1972

[28] W F Ames and M C Nucci ldquoAnalysis of fluid equations bygroupmethodsrdquo Journal of EngineeringMathematics vol 20 no2 pp 181ndash187 1986

[29] M J Moran and R A Gaggioli ldquoSimilarity analysis of com-pressible boundary layer flows via group theoryrdquo TechnicalSummaryReport 838US ArmyMathematics ResearchCenterMadison Wis USA 1967

[30] M J Moran and R A Gaggioli ldquoSimilarity analysis via grouptheoryrdquo AIAA Journal vol 6 pp 2014ndash2016 1968

[31] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[32] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[33] M B Abd-el-Malek M M Kassem and M L MekkyldquoSimilarity solutions for unsteady free-convection flow from acontinuous moving vertical surfacerdquo Journal of Computationaland Applied Mathematics vol 164 no 1 pp 11ndash24 2004

[34] M Kassem ldquoGroup solution for unsteady free-convection flowfrom a vertical moving plate subjected to constant heat fluxrdquoJournal of Computational and Applied Mathematics vol 187 no1 pp 72ndash86 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Table 1 Comparison of present problem solution and case of Kassem [34] for Pr = 1 2 5 and 10 of Newtonian fluid

Pr Kassem [34] Present work120578 119906 (120578) Θ (120578) 119906 (120578) Θ (120578)

1

05 0669 0507 0667 05101 0391 0256 0381 026115 0165 0103 0161 01052 0000 0003 0003 0005

2

03 0736 0398 0742 040106 0515 0222 0520 022509 0315 0092 0317 009512 0143 0009 0150 0011

5

025 0745 0206 0750 020805 0525 0095 0531 0011075 0333 0040 0335 00421 0171 0017 0171 0017

10

02 0768 0124 0771 012504 0565 0046 0571 004806 0381 0015 0391 001708 0216 0007 0218 0008

0 05 1 15 20

02

04

06

08

1

12

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

u(120578)

Figure 2 Vertical velocity of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

5 Results and Discussion

In this section initially results of Newtonian fluid flow overstretching sheet are compared with Kassem [34] Table 1shows the results obtained by numerical solution of ODEequation (49) and [34] for power index 119899 equal to unityand Pr number of 1 There is a satisfying agreement amongthe present evaluations Complementary results for differentvalues of Pr are shown in Figures 2 and 3 It can beunderstood that as the fluid momentum diffusivity increaseswith respect to the thermal diffusivity large Pr numbersvertical velocity decreases In other words the velocity 119906(120578)inside the boundary layer decreases with the increase of fluidviscosity Also the temperature plot of boundary layer revealsa decrease of temperature at the wall temperature for larger

0 05 1 15 20

02

04

06

08

1

120578

Pr = 1

Pr = 2

Pr = 5

Pr = 10

120579(120578)

Figure 3 Temperature of Newtonian fluid at the vicinity ofstretching sheet for different values of Pr

Pr So heat loss increases for larger Pr as the boundary layergets thinner

For non-Newtonian case of the fluid the vertical velocityand temperature of fluid adjacent to the stretching sheet areevaluated by solving (42) to (44) with the certain boundaryconditions of (45) Results of the shooting method solutionare illustrated in Figures 4 and 5 for Pr = 5 As the powerindex increases vertical velocity component encounters withresistance to rise due to increased value of shear stress Butfor lower power index vertical component of velocity movesfaster due to reduced apparent viscosity Similarly thinnerthermal boundary layer for higher power index will increasethe amount of heat loss

Journal of Applied Mathematics 7

0 2 4 6 8 100

02

04

06

08

1

12

n = 05n = 10n = 15

120578

u(120578)

Figure 4 Vertical velocity of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

0 05 1 15 2 250

02

04

06

120578

120579(120578)

n = 05n = 10n = 15

Figure 5 Temperature of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

6 Conclusion

Similarity solutions are performed for flow of power lawnon-Newtonian fluid film on unsteady stretching surfacesubjected to constant heat fluxThe nonlinear coupled partialdifferential equations (PDE) governing the flow and theboundary conditions are transformed to a set of ordinarydifferential equations (ODE) using two-parameter groupsThis technique reduces the number of independent variablesby two and conclusively the obtained ordinary differentialequations are solved numerically for the temperature andvelocity using the shooting method Results show that higherPr number and higher power index of non-Newtonian fluidencounter with difficulty to move as faster as lower Pr andpower index Enhanced amount of shear stress explains thereason of the predicted flow

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[2] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[3] R Usha and R Sridharan ldquoThe axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal of FluidsEngineering vol 117 no 1 pp 81ndash85 1995

[4] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[5] H I Andersson and T Ytrehus ldquoFalkner-Skan solution forgravity-driven film flowrdquo Journal of Applied Mechanics vol 52no 4 pp 783ndash786 1985

[6] H I Andersson and F Irgens ldquoGravity-driven laminar filmflow of power-law fluids along vertical wallsrdquo Journal of Non-Newtonian Fluid Mechanics vol 27 no 2 pp 153ndash172 1988

[7] D-Y Shang and H I Andersson ldquoHeat transfer in gravity-driven film flow of power-law fluidsrdquo International Journal ofHeat and Mass Transfer vol 42 no 11 pp 2085ndash2099 1998

[8] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[9] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[10] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching planewallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[11] R Tsai K H Huang and J S Huang ldquoFlow and heattransfer over an unsteady stretching surface with non-uniformheat sourcerdquo International Communications in Heat and MassTransfer vol 35 no 10 pp 1340ndash1343 2008

[12] B Sahoo and Y Do ldquoEffects of slip on sheet-driven flow andheat transfer of a third grade fluid past a stretching sheetrdquoInternational Communications in Heat and Mass Transfer vol37 no 8 pp 1064ndash1071 2010

[13] C R Illingworth ldquoUnsteady laminar flow of gas near aninfinite flat platerdquo Mathematical Proceedings of the CambridgePhilosophical Society vol 46 no 4 pp 603ndash613 1950

[14] R Siegel ldquoTransient free convection from a vertical flat platerdquoTransactions of the ASME vol 80 no 2 pp 347ndash359

[15] B Gebhart ldquoTransient natural convection from vertical ele-mentsrdquo Journal of Heat Transfer vol 83 pp 61ndash70 1961

[16] A J Chamkha ldquoSimilarity solution for thermal boundary layeron a stretched surface of a non-Newtonian fluidrdquo InternationalCommunications in Heat and Mass Transfer vol 24 no 5 pp643ndash652 1997

[17] J D Hellums and S W Churchill ldquoTransient and steady statefree and natural convection numerical solutions part Imdashthe

8 Journal of Applied Mathematics

isothermal vertical platerdquoAIChE Journal vol 8 no 5 pp 690ndash692 1962

[18] G D Callahan and W J Marner ldquoTransient free convectionwith mass transfer on an isothermal vertical flat platerdquo Inter-national Journal of Heat and Mass Transfer vol 19 no 2 pp165ndash174 1976

[19] T G Howell D R Jeng andK J deWitt ldquoMomentum and heattransfer on a continuous moving surface in a power law fluidrdquoInternational Journal of Heat and Mass Transfer vol 40 no 8pp 1853ndash1861 1997

[20] A J A Morgan ldquoThe reduction by one of the number ofindependent variables in some systems of partial differentialequationsrdquo The Quarterly Journal of Mathematics vol 3 no 1pp 250ndash259 1952

[21] A D Michal ldquoDifferential invariants and invariant partialdifferential equations under continuous transformation groupsin normed linear spacesrdquo Proceedings of the National Academyof Sciences of the United States of America vol 37 pp 623ndash6271951

[22] G Birkhoff ldquoMathematics for engineersrdquo Electrical Engineeringvol 67 no 12 pp 1185ndash1188 1948

[23] G Birkhoff Hydrodynamics A Study in Logic Fact and Simili-tude Princeton University Press Princeton NJ USA 1960

[24] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[25] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[26] W F Ames ldquoSimilarity for the nonlinear diffusion equationrdquoIndustrial amp Engineering Chemistry Fundamentals vol 4 no 1pp 72ndash76 1965

[27] W FAmesNonlinear Partial Differential Equations in Engineer-ing chapter 2 Academic Press New York NY USA 1972

[28] W F Ames and M C Nucci ldquoAnalysis of fluid equations bygroupmethodsrdquo Journal of EngineeringMathematics vol 20 no2 pp 181ndash187 1986

[29] M J Moran and R A Gaggioli ldquoSimilarity analysis of com-pressible boundary layer flows via group theoryrdquo TechnicalSummaryReport 838US ArmyMathematics ResearchCenterMadison Wis USA 1967

[30] M J Moran and R A Gaggioli ldquoSimilarity analysis via grouptheoryrdquo AIAA Journal vol 6 pp 2014ndash2016 1968

[31] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[32] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[33] M B Abd-el-Malek M M Kassem and M L MekkyldquoSimilarity solutions for unsteady free-convection flow from acontinuous moving vertical surfacerdquo Journal of Computationaland Applied Mathematics vol 164 no 1 pp 11ndash24 2004

[34] M Kassem ldquoGroup solution for unsteady free-convection flowfrom a vertical moving plate subjected to constant heat fluxrdquoJournal of Computational and Applied Mathematics vol 187 no1 pp 72ndash86 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

0 2 4 6 8 100

02

04

06

08

1

12

n = 05n = 10n = 15

120578

u(120578)

Figure 4 Vertical velocity of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

0 05 1 15 2 250

02

04

06

120578

120579(120578)

n = 05n = 10n = 15

Figure 5 Temperature of non-Newtonian fluid at the vicinity ofstretching sheet for constant values of Pr = 5

6 Conclusion

Similarity solutions are performed for flow of power lawnon-Newtonian fluid film on unsteady stretching surfacesubjected to constant heat fluxThe nonlinear coupled partialdifferential equations (PDE) governing the flow and theboundary conditions are transformed to a set of ordinarydifferential equations (ODE) using two-parameter groupsThis technique reduces the number of independent variablesby two and conclusively the obtained ordinary differentialequations are solved numerically for the temperature andvelocity using the shooting method Results show that higherPr number and higher power index of non-Newtonian fluidencounter with difficulty to move as faster as lower Pr andpower index Enhanced amount of shear stress explains thereason of the predicted flow

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

[2] H I Andersson J BAarseth andB SDandapat ldquoHeat transferin a liquid film on an unsteady stretching surfacerdquo InternationalJournal of Heat andMass Transfer vol 43 no 1 pp 69ndash74 2000

[3] R Usha and R Sridharan ldquoThe axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal of FluidsEngineering vol 117 no 1 pp 81ndash85 1995

[4] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[5] H I Andersson and T Ytrehus ldquoFalkner-Skan solution forgravity-driven film flowrdquo Journal of Applied Mechanics vol 52no 4 pp 783ndash786 1985

[6] H I Andersson and F Irgens ldquoGravity-driven laminar filmflow of power-law fluids along vertical wallsrdquo Journal of Non-Newtonian Fluid Mechanics vol 27 no 2 pp 153ndash172 1988

[7] D-Y Shang and H I Andersson ldquoHeat transfer in gravity-driven film flow of power-law fluidsrdquo International Journal ofHeat and Mass Transfer vol 42 no 11 pp 2085ndash2099 1998

[8] C-H Chen ldquoHeat transfer in a power-law fluid film over aunsteady stretching sheetrdquo Heat and Mass Transfer vol 39 no8-9 pp 791ndash796 2003

[9] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[10] A Mehmood and A Ali ldquoAnalytic solution of generalizedthree-dimensional flow and heat transfer over a stretching planewallrdquo International Communications in Heat andMass Transfervol 33 no 10 pp 1243ndash1252 2006

[11] R Tsai K H Huang and J S Huang ldquoFlow and heattransfer over an unsteady stretching surface with non-uniformheat sourcerdquo International Communications in Heat and MassTransfer vol 35 no 10 pp 1340ndash1343 2008

[12] B Sahoo and Y Do ldquoEffects of slip on sheet-driven flow andheat transfer of a third grade fluid past a stretching sheetrdquoInternational Communications in Heat and Mass Transfer vol37 no 8 pp 1064ndash1071 2010

[13] C R Illingworth ldquoUnsteady laminar flow of gas near aninfinite flat platerdquo Mathematical Proceedings of the CambridgePhilosophical Society vol 46 no 4 pp 603ndash613 1950

[14] R Siegel ldquoTransient free convection from a vertical flat platerdquoTransactions of the ASME vol 80 no 2 pp 347ndash359

[15] B Gebhart ldquoTransient natural convection from vertical ele-mentsrdquo Journal of Heat Transfer vol 83 pp 61ndash70 1961

[16] A J Chamkha ldquoSimilarity solution for thermal boundary layeron a stretched surface of a non-Newtonian fluidrdquo InternationalCommunications in Heat and Mass Transfer vol 24 no 5 pp643ndash652 1997

[17] J D Hellums and S W Churchill ldquoTransient and steady statefree and natural convection numerical solutions part Imdashthe

8 Journal of Applied Mathematics

isothermal vertical platerdquoAIChE Journal vol 8 no 5 pp 690ndash692 1962

[18] G D Callahan and W J Marner ldquoTransient free convectionwith mass transfer on an isothermal vertical flat platerdquo Inter-national Journal of Heat and Mass Transfer vol 19 no 2 pp165ndash174 1976

[19] T G Howell D R Jeng andK J deWitt ldquoMomentum and heattransfer on a continuous moving surface in a power law fluidrdquoInternational Journal of Heat and Mass Transfer vol 40 no 8pp 1853ndash1861 1997

[20] A J A Morgan ldquoThe reduction by one of the number ofindependent variables in some systems of partial differentialequationsrdquo The Quarterly Journal of Mathematics vol 3 no 1pp 250ndash259 1952

[21] A D Michal ldquoDifferential invariants and invariant partialdifferential equations under continuous transformation groupsin normed linear spacesrdquo Proceedings of the National Academyof Sciences of the United States of America vol 37 pp 623ndash6271951

[22] G Birkhoff ldquoMathematics for engineersrdquo Electrical Engineeringvol 67 no 12 pp 1185ndash1188 1948

[23] G Birkhoff Hydrodynamics A Study in Logic Fact and Simili-tude Princeton University Press Princeton NJ USA 1960

[24] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[25] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[26] W F Ames ldquoSimilarity for the nonlinear diffusion equationrdquoIndustrial amp Engineering Chemistry Fundamentals vol 4 no 1pp 72ndash76 1965

[27] W FAmesNonlinear Partial Differential Equations in Engineer-ing chapter 2 Academic Press New York NY USA 1972

[28] W F Ames and M C Nucci ldquoAnalysis of fluid equations bygroupmethodsrdquo Journal of EngineeringMathematics vol 20 no2 pp 181ndash187 1986

[29] M J Moran and R A Gaggioli ldquoSimilarity analysis of com-pressible boundary layer flows via group theoryrdquo TechnicalSummaryReport 838US ArmyMathematics ResearchCenterMadison Wis USA 1967

[30] M J Moran and R A Gaggioli ldquoSimilarity analysis via grouptheoryrdquo AIAA Journal vol 6 pp 2014ndash2016 1968

[31] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[32] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[33] M B Abd-el-Malek M M Kassem and M L MekkyldquoSimilarity solutions for unsteady free-convection flow from acontinuous moving vertical surfacerdquo Journal of Computationaland Applied Mathematics vol 164 no 1 pp 11ndash24 2004

[34] M Kassem ldquoGroup solution for unsteady free-convection flowfrom a vertical moving plate subjected to constant heat fluxrdquoJournal of Computational and Applied Mathematics vol 187 no1 pp 72ndash86 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

isothermal vertical platerdquoAIChE Journal vol 8 no 5 pp 690ndash692 1962

[18] G D Callahan and W J Marner ldquoTransient free convectionwith mass transfer on an isothermal vertical flat platerdquo Inter-national Journal of Heat and Mass Transfer vol 19 no 2 pp165ndash174 1976

[19] T G Howell D R Jeng andK J deWitt ldquoMomentum and heattransfer on a continuous moving surface in a power law fluidrdquoInternational Journal of Heat and Mass Transfer vol 40 no 8pp 1853ndash1861 1997

[20] A J A Morgan ldquoThe reduction by one of the number ofindependent variables in some systems of partial differentialequationsrdquo The Quarterly Journal of Mathematics vol 3 no 1pp 250ndash259 1952

[21] A D Michal ldquoDifferential invariants and invariant partialdifferential equations under continuous transformation groupsin normed linear spacesrdquo Proceedings of the National Academyof Sciences of the United States of America vol 37 pp 623ndash6271951

[22] G Birkhoff ldquoMathematics for engineersrdquo Electrical Engineeringvol 67 no 12 pp 1185ndash1188 1948

[23] G Birkhoff Hydrodynamics A Study in Logic Fact and Simili-tude Princeton University Press Princeton NJ USA 1960

[24] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[25] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[26] W F Ames ldquoSimilarity for the nonlinear diffusion equationrdquoIndustrial amp Engineering Chemistry Fundamentals vol 4 no 1pp 72ndash76 1965

[27] W FAmesNonlinear Partial Differential Equations in Engineer-ing chapter 2 Academic Press New York NY USA 1972

[28] W F Ames and M C Nucci ldquoAnalysis of fluid equations bygroupmethodsrdquo Journal of EngineeringMathematics vol 20 no2 pp 181ndash187 1986

[29] M J Moran and R A Gaggioli ldquoSimilarity analysis of com-pressible boundary layer flows via group theoryrdquo TechnicalSummaryReport 838US ArmyMathematics ResearchCenterMadison Wis USA 1967

[30] M J Moran and R A Gaggioli ldquoSimilarity analysis via grouptheoryrdquo AIAA Journal vol 6 pp 2014ndash2016 1968

[31] M J Moran and R A Gaggioli ldquoA new systematic formalismfor similarity analysis with application to boundary layerflowsrdquo Technical Summary Report 918 US ArmyMathematicsResearch Center 1968

[32] M J Moran and R A Gaggioli ldquoReduction of the numberof variables in systems of partial differential equations withauxiliary conditionsrdquo SIAM Journal on Applied Mathematicsvol 16 no 1 pp 202ndash215 1968

[33] M B Abd-el-Malek M M Kassem and M L MekkyldquoSimilarity solutions for unsteady free-convection flow from acontinuous moving vertical surfacerdquo Journal of Computationaland Applied Mathematics vol 164 no 1 pp 11ndash24 2004

[34] M Kassem ldquoGroup solution for unsteady free-convection flowfrom a vertical moving plate subjected to constant heat fluxrdquoJournal of Computational and Applied Mathematics vol 187 no1 pp 72ndash86 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of