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Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2013 Article ID 380679 8 pageshttpdxdoiorg1011552013380679
Research ArticleHeat and Mass Transfer Effect on MHD Flow of a ViscoelasticFluid through a Porous Medium Bounded by an OscillatingPorous Plate in Slip Flow Regime
S N Sahoo
Department of Mathematics Institute of Technical Education and Research Siksha ldquoOrdquo Anusandhan University KhandagiriBhubaneswar Odisha 751030 India
Correspondence should be addressed to S N Sahoo sachisahooyahoocom
Received 30 March 2013 Accepted 18 June 2013
Academic Editor Jose C Merchuk
Copyright copy 2013 S N Sahoo This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Unsteady flow of an eclectically conducting and incompressible viscoelastic liquid of the Walter 1198611015840 model with simultaneous heatand mass transfer near an oscillating porous plate in slip flow regime under the influence of a transverse magnetic field of uniformstrength is presented The governing equations of the flow field are solved by a regular perturbation method for small elasticparameter and the expressions for the velocity temperature concentration skin friction 119862
119891 the heat flux in terms of the Nusselt
number N119906 and the rate of mass transfer in terms of the Sherwood number S
ℎare obtained The effects of the important flow
parameters on the dynamics are discussed Findings of the study reveal that the rarefaction parameter accelerates the fluid particlesin the flow domain Elastic parameter contributes to sudden fall of the velocity near the plate Magnetic force contributes to greaterskin friction as the time elapses Destructive reaction reduces whereas generative reaction enhances the concentration distribution
1 Introduction
Free convective flow in presence of heat source has beena subject of interest of many researchers because of itspossible application to geophysical sciences astrophysicalsciences and in cosmical studies Such flows arise eitherdue to unsteady motion of the boundary or the boundarytemperature The study of fluctuating flow is important inthe paper industry and many other technological fieldsThereforemany researchers have paid their attention towardsthe fluctuating flow of viscous incompressible fluid past aninfinite plate Singh et al [1] have analyzed the heat and masstransfer in MHD flow of a viscous fluids past a vertical plateunder oscillatory suction velocity Sharma and Singh [2] havereported the unsteady MHD-free convective flow and heattransfer along a vertical porous plate with variable suctionand internal heat generationThe problem of slip flow regimeis very important in this era of modern science technologyand vast ranging industrialization In many practical applica-tions the particle adjacent to a solid surface no longer takesthe velocity of the surface The particle at the surface has a
finite tangential velocity it slips along the surface The flowregime is called the slip flow regime and its effect cannotbe neglected The fluid slippage phenomenon at the solidboundaries appear in many applications such as microchan-nels or nanochannels and in application where a thin filmof light oils is attached to the moving plates or when thesurface is coated with special coating such as thickmonolayerof hydrophobic octadecyltrichlosilane that is lubrication ofmechanical device where a thin filmof lubricant is attached tothe surface slipping over one another or when the surfaces arecoated with special coating to minimize the friction betweenthem Singh and Gupta [3] have discussed the MHD-freeconvective flow of a viscous fluid through a porous mediumbounded by an oscillating porous plate in slip flow regimewith mass transfer Khandelwal and Jain [4] have analyzedthe unsteady MHD flow of a stratified fluid through porousmedium over a moving plate in slip flow regime Das et al[5] have studied the magnetohydrodynamic unsteady flowof a viscous stratified fluid through a porous medium pasta porous flat moving plate in the slip flow regime with
2 International Journal of Chemical Engineering
heat source The study of heat and mass transfer problemswith chemical reaction is of great practical importance toengineers and scientists because of their almost universaloccurrence in many branches of science and engineeringA few representative fields of interest in which combinedheat and mass transfer along with chemical reaction play animportant role are chemical process industries such as foodprocessing and polymer productionMahapatra et al [6] havestudied effects of chemical reaction on free convection flowthrough a porous medium bounded by a vertical surfaceMuthucumaraswamy [7] has studied effects of chemicalreaction on amoving isothermal vertical surfacewith suctionAl-Odat and Al-Azab [8] have studied influence of chemicalreaction on a transient MHD-free convection flow over amoving vertical plate
Viscoelastic fluid flow through porous media hasattracted the attention of scientists and engineers because ofits importance notably in the flow of the oil through porousrocks the extraction of energy from geothermal region andthe filtration of solids from liquids and drug permeationthrough human skin The flow through porous mediaalso occurs in the ground water hydrology irrigation anddrainage problems absorption and filtration processes in theground water hydrology irrigation and drainage problemsabsorption and filtration processes in chemical engineeringand soil erosion and tile drainage Chaudhary and Jain[9] have investigated the effects of the Hall current andradiation on MHD mixed convection flow of a viscoelasticfluid past and infinite vertical plate Sahoo et al [10] havediscussed the unsteady two-dimensional MHD flow andheat transfer of an elastic-viscous liquid past an infinite hotvertical porous surface bounded by porous medium withsourcesink Kumar and Chand [11] have studied the effect ofslip conditions and the Hall current on unsteady MHD flowof a viscoelastic fluid past an infinite vertical porous platethrough porous medium
The objective of the present study is to consider theviscoelastic fluid past a vertical plate in the slip flow regimepacked with uniform porous matrix in the presence of atransverse magnetic field In this paper we consider theproblem of Singh and Gupta [3] in viscoelastic fluid ofWalterrsquos1198611015840model taking into account the effect of heat sourceparameter and chemical reaction parameter
2 Formulation and Solution of the Problem
The physical configuration consists of an unsteady flow ofan electrically conducting and incompressible viscoelasticliquid of Walterrsquos 1198611015840 model with simultaneous heat and masstransfer near an oscillating infinite porous plate in slip flowregime with heat source and chemical reaction under theinfluence of a transverse magnetic field of uniform strengthThe 119910-axis is taken along the plate in vertical direction and119909-axis is perpendicular to it A uniform magnetic field ofstrength119861
0is applied in the direction of 119910-axis Let 119906 and V be
the velocity components in x- and y-directions respectivelyAs the plate is of infinite length all the variables in theproblem are functions of 119910 and 119905 Initially the plate and
fluid are at rest and then the plate is set to an oscillatorymotion The Reynolds number is assumed to be very smalland the induced magnetic field due to the flow is neglectedwith respect to the applied magnetic field The pressure 119901 inthe fluid is assumed to be constant If V
0represents the suction
or injection velocity at the plate the equation of continuity is
120597V
120597119910= 0 (1)
Under the condition 119910 = 0 V = minusV0everywhere We have
y
x998400
B0
x
Flow Diagram
Now the governing boundary layer equations of the flow fieldare
120597119906
120597119905minus V0
120597119906
120597119910= 120592
1205972119906
1205971199102minus 120590
1198612
0119906
120588minus120592119906
119870+ 119892120573 (119879 minus 119879
infin)
+ 119892120573 (119862 minus 119862infin) minus
1198700
120588(1205973119906
120597119905120597119910+ V0
1205973119906
1205971199103)
120597119879
120597119905minus V0
120597119879
120597119910= 120572
1205972119879
1205971199102minus 119878 (119879 minus 119879
infin)
120597119862
120597119905minus V0
120597119862
120597119910= 119863
1205972119862
1205971199102minus 119870119897(119862 minus 119862
infin)
(2)
The first order velocity slip boundary conditions of theproblemwhen the plate executes linear harmonic oscillationsin its own plane are given by
Equation (5) is of third order and two boundary condi-tions are available Due to inadequate boundary conditiona perturbation method has been applied with 119877
119888lt 1 as the
perturbation parameter This assumption is quite consistentas the model under consideration is valid only for slightlyelastic fluid
The problem of unsteady flow of an electrically conductingand incompressible viscoelastic liquid of Walterrsquos 1198611015840 modelwith heat andmass transfer near an oscillating infinite porous
International Journal of Chemical Engineering 5
plate in slip flow regime with heat source and chemicalreaction parameter under the influence of a transversemagnetic field of uniform strength has been considered Theeffects of the flow parameters such as the Prandtl numberPr porosity parameter 119870
119901 magnetic parameter M elastic
parameter 119877119888 heat source parameter S chemical reaction
and rarefaction parameter 119877 on the velocity field have beenstudied analytically and presented with the help of Figures1ndash4 The effects of the flow parameters on the temperaturefield and concentration distribution have been presented inFigures 5 and 6 respectively Further the effects of the flowparameters on the skin friction heat flux and rate of masstransfer have been discussed with the help of Tables 1ndash3 Fornumerical computation the values of 119866
119903are taken positive
This indicates that the study has been carried out under theinfluence of the cooling of the plate Also we have takennt = 1205872 The interesting aspect of the problem is to studythe combined effect of the flow parameters with that of thefirst order velocity slip boundary condition when the plateexecutes linear harmonic oscillation in its own plane
Figure 1 shows the effect of the Prandtl number (Pr)permeability parameter (119870
119901) andmagnetic parameter (M) on
velocity profile For this figure we have taken that S = 1 119866119903
= 5 119866119898= 5 119878
119888= 024 119877
119888= 05 119870
119888= 2 R = 02 and V
0=
2 It is observed that the increase in the Prandtl number aswell as permeability parameter decreases the velocity of theflow field whereas increase in magnetic parameter increasesit Since Prandtl number is the ratio of kinematic viscosity tothermal diffusivity so as Pr increases the kinematic viscosityof the fluid dominates the thermal diffusivity of the fluidwhich leads to decreasing the velocity of the flow field Theapplication of transverse magnetic field sets up the Lorentzforce which enhances the fluid velocity
Figure 2 shows the effect of elastic parameter (119877119888) heat
source parameter (S) and chemical reaction (119870119888) parameter
on velocity frofile For this figure we have taken that Pr =071 119870
119901= 1M = 2 119866
119903= 5 119866
119898 = 5 119878
119888= 024 R = 02 and V
0
= 2 It is observed that the velocity of the flow field decreasesdue to the presence of elastic parameter chemical reactionparameter and heat source parameter For 119877
119888= S = 119870
119888= 0
the present work agrees with the work of Singh and Gupta[3]
Figure 3 depicts the effect of the thermal Grashof number(119866119903) the mass Grashof number (119866
119898) and the Schmidt
number (119878119888) on velocity frofile For this figure we have taken
that Pr = 071 S = 1 119870119901= 1 M = 2 119877
119888= 05 119870
119888= 2 R = 02
and V0= 2 It is observed that for the heavier species that is
with increasing 119878119888 the velocity decreases The velocity of the
flowfield decreases due to the increase in the thermalGrashofnumberMoreover buoyancy effect (119866
119898) due tomass transfer
enhances the velocityFigure 4 depicts the effect of suction parameter (V
0) and
rarefaction parameter (R) on velocity frofile For this figurewe have taken that Pr = 071 S = 1 119870
119901= 1 M = 2 119877
119888= 05
119870119888= 2 119866
119903= 5 119866
119898= 5 and 119878
119888= 024 It is observed that the
velocity of the flow field decreases due the presence of suction
0 05 1 15 2 25 3 35 4 45 5
1
0
minus1
minus2
minus3
minus4
minus5
u
y
Pr = 031Kp = 02M = 05
Pr = 071Kp = 02M = 1
Pr = 071Kp = 02M = 05
Pr = 071Kp = 04M = 05
Figure 1 Effect of Pr119870119901and119872 on velocity profile
0 05 1 15 2 25 3 35 4 45 5
05
0
minus05
minus1
minus15
minus2
u
y
Rc = 02 S = 05Kc = 1
Rc = 02 S = 05Kc = 2
Rc = 02 S = 1Kc = 1
Rc = 05 S = 05Kc = 1
Figure 2 Effect of 119877119888 S and 119870
119888on velocity profile
12
1
08
06
04
02
0
minus020 05 1 15 2 25 3 35 4 45 5
u
y
Gr = 25 Gm = 2 2
Gr = 2 Gm = 2 25
Sc =
Sc =
Gr = 2 Gm = 2 2Sc =
Gr = 2 Gm = 25 2Sc =
Figure 3 Effect of 119866119903 119866119898 and 119878
119888on velocity profile
6 International Journal of Chemical Engineering
06
04
02
0
minus02
minus04
minus06
minus08
minus1
minus12
0 = 15 R = 07
0 = 15 R = 05
0 = 2 R = 05
u
0 05 1 15 2 25 3 35 4 45 5y
Figure 4 Effect of V0and 119877 on velocity profile
parameter but the reverse effect is observed due the presenceof the rarefaction parameter
Figure 5 shows the effect of the Prandtl number heatsource parameter and suction parameter on the temperatureof the flow field It is observed that the temperature ofthe flow field diminishes as the Prandtl number increasesThis is consistent with the fact that the thermal boundarylayer thickness decreases with increasing Prandtl numberPresence of heat source reduces the temperature of the flowfield This may happen due the elastic property of the fluidIt is observed that temperature of the flow field diminishes asthe suction parameter increases
Figure 6 depicts the effect of the Schmidt number chemi-cal reaction parameter and suction parameter on concentra-tion distributionThe concentration distribution decreases atall points of the flow field with the increase in the Schmidtnumber This shows that the heavier diffusing species havea greater retarding effect on the concentration distributionof the flow field It is observed that a destructive reaction(119870119888gt 0) reduces the concentration distribution whereas
a generative reaction (119870119888
= 0) enhances it Also it isobserved that presence of suction parameter diminishes theconcentration distribution
The skin friction is an important phenomenon whichcharacterizes the frictional drag at the solid surface FromTable 1 it is observed that the skin friction decreases with theincrease in all the forcing forces but it is interesting to notethat the skin friction increases with the increase in magneticparameter
From Table 2 it is to note that all the entries are positiveIt is seen that the Prandtl number (Pr) heat source (S) andsuction parameter (V
0) increase the rate of heat transfer at the
surface of the plateFrom Table 3 it is to note that all the entries are positive
It is observed that Schmidt number (119878119888) chemical reaction
parameter (119870119888) and suction parameter (V
0) increase the rate
of mass transfer at the surface of the plate
1
09
08
07
06
05
04
03
02
01
0
Pr = 05 S = 05 0 = 02
Pr = 071 S = 05 0 = 02
Pr = 071 S = 05 0 = 05
Pr = 071 S = 1 0 = 02
0 05 1 15 2 25 3 35 4 45 5y
Pr = 071 S = 0 0 = 02
120579
Figure 5 Effect of Pr 119878 and V0on temperature profile
1
09
08
07
06
05
04
03
02
01
00 05 1 15 2 25 3 35 4 45 5
y
078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =
078Kc = 05 0 = 02Sc =
078Kc = 05 0 = 05Sc =
078Kc = 1 0 = 02Sc =120601
Figure 6 Effect of 119878119888 119870119888 and V
0on concentration profile
4 Conclusion
A theoretical study of unsteady MHD incompressible vis-coelastic liquid of Walterrsquos 1198611015840 model with heat and masstransfer near an oscillating infinite porous plate in slip flowregime under the influence of a transverse magnetic fieldof uniform strength is considered Some of the importantfindings of the problem are given in the following
(i) Presence of the Prandtl number decreases the velocityof the flow field whereas presence magnetic fieldincreases it
(ii) The velocity of the flow field decreases suddenly nearthe plate due to the presence of elastic parameter
(iii) The velocity of the flow field decreases due to theincrease in the thermal Grashof number
(iv) Thermal boundary layer thickness decreases withincreasing the Prandtl number
(v) Heavier diffusing species have a greater retardingeffect on the concentration distribution
Appendix
Consider the following
1198861=1
2[V0Pr + radic(V
0Pr)2 + 4119878Pr]
1198862=1
2[V0119878119888+ radic(V
0119878119888)2
+ 4119870119888119878119888]
1198863=1
2[V20+ radicV20+ 4119876] 119876 = 119872
2+
1
119870119901
1198864=1
2[V20+ radicV20+ 4 (119876 + 119894119899)]
1198601=
minus119866119903
1198862
1minus 1198861V0minus 119876
1198602=
minus119866119903
1198862
2minus 1198862V0minus 119876
1198603=
minus1
1 + 1198863119877[(1198861119877 + 1)119860
1+ (1198862119877 + 1)119860
2]
1198604=
1
1 + 1198771198864
1198605=
1198863
11198601V0
1198862
1minus 1198861V0minus 119876
1198606=
1198863
21198602V0
1198862
2minus 1198862V0minus 119876
1198607= 1198605+ 1198606minusV01198863
31198603119910
V0minus 21198863
1198608=
(minusV01198863
41198604+ 1198941198991198862
41198604) 119910
V0minus 21198864
(A1)
Nomenclature
119909 119910 Coordinate axes119906 V Velocity components in x- and y-directions119905 Time variable120583 Dynamic viscosity120592 Kinematic viscosity120572 Thermal conductivity119901 Pressure119892 Acceleration due to gravity120573 Coefficient of volume expansion120573 Coefficient of volume expansion with concentration1198800 Reference velocity
119879 Dimensional temperature120579 Nondimensional temperature119862 Dimensional concentration120601 Nondimensional concentrationPr The Prandtl number119866119903 The thermal Grashof number
119866119898 The mass Grashof number
119878119888 The Schmidt number
119872 Magnetic parameter119878 Heat sourcesink parameter119870119897 Dimensional chemical reaction parameter
119870119888 Nondimensional chemical reaction parameter
119879119908 Temperature at the wall
119879infin Temperature far away from the wall
119862119908 Concentration at the wall
119862infin Concentration far away from the wall
120590 Electric conductivity1198610 Uniform magnetic field
V0 Suctioninjection velocity
119863 Mass diffusion120588 Density119870 Dimensional porosity parameter119870119901 Nondimensional porosity parameter
1198700 Dimensional elastic parameter
119877119888 Nondimensional elastic parameter
119899 Frequency of oscillation
8 International Journal of Chemical Engineering
Acknowledgment
The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work
References
[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003
[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008
[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005
[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005
[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011
[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010
[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002
[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007
[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006
[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011
[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011
heat source The study of heat and mass transfer problemswith chemical reaction is of great practical importance toengineers and scientists because of their almost universaloccurrence in many branches of science and engineeringA few representative fields of interest in which combinedheat and mass transfer along with chemical reaction play animportant role are chemical process industries such as foodprocessing and polymer productionMahapatra et al [6] havestudied effects of chemical reaction on free convection flowthrough a porous medium bounded by a vertical surfaceMuthucumaraswamy [7] has studied effects of chemicalreaction on amoving isothermal vertical surfacewith suctionAl-Odat and Al-Azab [8] have studied influence of chemicalreaction on a transient MHD-free convection flow over amoving vertical plate
Viscoelastic fluid flow through porous media hasattracted the attention of scientists and engineers because ofits importance notably in the flow of the oil through porousrocks the extraction of energy from geothermal region andthe filtration of solids from liquids and drug permeationthrough human skin The flow through porous mediaalso occurs in the ground water hydrology irrigation anddrainage problems absorption and filtration processes in theground water hydrology irrigation and drainage problemsabsorption and filtration processes in chemical engineeringand soil erosion and tile drainage Chaudhary and Jain[9] have investigated the effects of the Hall current andradiation on MHD mixed convection flow of a viscoelasticfluid past and infinite vertical plate Sahoo et al [10] havediscussed the unsteady two-dimensional MHD flow andheat transfer of an elastic-viscous liquid past an infinite hotvertical porous surface bounded by porous medium withsourcesink Kumar and Chand [11] have studied the effect ofslip conditions and the Hall current on unsteady MHD flowof a viscoelastic fluid past an infinite vertical porous platethrough porous medium
The objective of the present study is to consider theviscoelastic fluid past a vertical plate in the slip flow regimepacked with uniform porous matrix in the presence of atransverse magnetic field In this paper we consider theproblem of Singh and Gupta [3] in viscoelastic fluid ofWalterrsquos1198611015840model taking into account the effect of heat sourceparameter and chemical reaction parameter
2 Formulation and Solution of the Problem
The physical configuration consists of an unsteady flow ofan electrically conducting and incompressible viscoelasticliquid of Walterrsquos 1198611015840 model with simultaneous heat and masstransfer near an oscillating infinite porous plate in slip flowregime with heat source and chemical reaction under theinfluence of a transverse magnetic field of uniform strengthThe 119910-axis is taken along the plate in vertical direction and119909-axis is perpendicular to it A uniform magnetic field ofstrength119861
0is applied in the direction of 119910-axis Let 119906 and V be
the velocity components in x- and y-directions respectivelyAs the plate is of infinite length all the variables in theproblem are functions of 119910 and 119905 Initially the plate and
fluid are at rest and then the plate is set to an oscillatorymotion The Reynolds number is assumed to be very smalland the induced magnetic field due to the flow is neglectedwith respect to the applied magnetic field The pressure 119901 inthe fluid is assumed to be constant If V
0represents the suction
or injection velocity at the plate the equation of continuity is
120597V
120597119910= 0 (1)
Under the condition 119910 = 0 V = minusV0everywhere We have
y
x998400
B0
x
Flow Diagram
Now the governing boundary layer equations of the flow fieldare
120597119906
120597119905minus V0
120597119906
120597119910= 120592
1205972119906
1205971199102minus 120590
1198612
0119906
120588minus120592119906
119870+ 119892120573 (119879 minus 119879
infin)
+ 119892120573 (119862 minus 119862infin) minus
1198700
120588(1205973119906
120597119905120597119910+ V0
1205973119906
1205971199103)
120597119879
120597119905minus V0
120597119879
120597119910= 120572
1205972119879
1205971199102minus 119878 (119879 minus 119879
infin)
120597119862
120597119905minus V0
120597119862
120597119910= 119863
1205972119862
1205971199102minus 119870119897(119862 minus 119862
infin)
(2)
The first order velocity slip boundary conditions of theproblemwhen the plate executes linear harmonic oscillationsin its own plane are given by
Equation (5) is of third order and two boundary condi-tions are available Due to inadequate boundary conditiona perturbation method has been applied with 119877
119888lt 1 as the
perturbation parameter This assumption is quite consistentas the model under consideration is valid only for slightlyelastic fluid
The problem of unsteady flow of an electrically conductingand incompressible viscoelastic liquid of Walterrsquos 1198611015840 modelwith heat andmass transfer near an oscillating infinite porous
International Journal of Chemical Engineering 5
plate in slip flow regime with heat source and chemicalreaction parameter under the influence of a transversemagnetic field of uniform strength has been considered Theeffects of the flow parameters such as the Prandtl numberPr porosity parameter 119870
119901 magnetic parameter M elastic
parameter 119877119888 heat source parameter S chemical reaction
and rarefaction parameter 119877 on the velocity field have beenstudied analytically and presented with the help of Figures1ndash4 The effects of the flow parameters on the temperaturefield and concentration distribution have been presented inFigures 5 and 6 respectively Further the effects of the flowparameters on the skin friction heat flux and rate of masstransfer have been discussed with the help of Tables 1ndash3 Fornumerical computation the values of 119866
119903are taken positive
This indicates that the study has been carried out under theinfluence of the cooling of the plate Also we have takennt = 1205872 The interesting aspect of the problem is to studythe combined effect of the flow parameters with that of thefirst order velocity slip boundary condition when the plateexecutes linear harmonic oscillation in its own plane
Figure 1 shows the effect of the Prandtl number (Pr)permeability parameter (119870
119901) andmagnetic parameter (M) on
velocity profile For this figure we have taken that S = 1 119866119903
= 5 119866119898= 5 119878
119888= 024 119877
119888= 05 119870
119888= 2 R = 02 and V
0=
2 It is observed that the increase in the Prandtl number aswell as permeability parameter decreases the velocity of theflow field whereas increase in magnetic parameter increasesit Since Prandtl number is the ratio of kinematic viscosity tothermal diffusivity so as Pr increases the kinematic viscosityof the fluid dominates the thermal diffusivity of the fluidwhich leads to decreasing the velocity of the flow field Theapplication of transverse magnetic field sets up the Lorentzforce which enhances the fluid velocity
Figure 2 shows the effect of elastic parameter (119877119888) heat
source parameter (S) and chemical reaction (119870119888) parameter
on velocity frofile For this figure we have taken that Pr =071 119870
119901= 1M = 2 119866
119903= 5 119866
119898 = 5 119878
119888= 024 R = 02 and V
0
= 2 It is observed that the velocity of the flow field decreasesdue to the presence of elastic parameter chemical reactionparameter and heat source parameter For 119877
119888= S = 119870
119888= 0
the present work agrees with the work of Singh and Gupta[3]
Figure 3 depicts the effect of the thermal Grashof number(119866119903) the mass Grashof number (119866
119898) and the Schmidt
number (119878119888) on velocity frofile For this figure we have taken
that Pr = 071 S = 1 119870119901= 1 M = 2 119877
119888= 05 119870
119888= 2 R = 02
and V0= 2 It is observed that for the heavier species that is
with increasing 119878119888 the velocity decreases The velocity of the
flowfield decreases due to the increase in the thermalGrashofnumberMoreover buoyancy effect (119866
119898) due tomass transfer
enhances the velocityFigure 4 depicts the effect of suction parameter (V
0) and
rarefaction parameter (R) on velocity frofile For this figurewe have taken that Pr = 071 S = 1 119870
119901= 1 M = 2 119877
119888= 05
119870119888= 2 119866
119903= 5 119866
119898= 5 and 119878
119888= 024 It is observed that the
velocity of the flow field decreases due the presence of suction
0 05 1 15 2 25 3 35 4 45 5
1
0
minus1
minus2
minus3
minus4
minus5
u
y
Pr = 031Kp = 02M = 05
Pr = 071Kp = 02M = 1
Pr = 071Kp = 02M = 05
Pr = 071Kp = 04M = 05
Figure 1 Effect of Pr119870119901and119872 on velocity profile
0 05 1 15 2 25 3 35 4 45 5
05
0
minus05
minus1
minus15
minus2
u
y
Rc = 02 S = 05Kc = 1
Rc = 02 S = 05Kc = 2
Rc = 02 S = 1Kc = 1
Rc = 05 S = 05Kc = 1
Figure 2 Effect of 119877119888 S and 119870
119888on velocity profile
12
1
08
06
04
02
0
minus020 05 1 15 2 25 3 35 4 45 5
u
y
Gr = 25 Gm = 2 2
Gr = 2 Gm = 2 25
Sc =
Sc =
Gr = 2 Gm = 2 2Sc =
Gr = 2 Gm = 25 2Sc =
Figure 3 Effect of 119866119903 119866119898 and 119878
119888on velocity profile
6 International Journal of Chemical Engineering
06
04
02
0
minus02
minus04
minus06
minus08
minus1
minus12
0 = 15 R = 07
0 = 15 R = 05
0 = 2 R = 05
u
0 05 1 15 2 25 3 35 4 45 5y
Figure 4 Effect of V0and 119877 on velocity profile
parameter but the reverse effect is observed due the presenceof the rarefaction parameter
Figure 5 shows the effect of the Prandtl number heatsource parameter and suction parameter on the temperatureof the flow field It is observed that the temperature ofthe flow field diminishes as the Prandtl number increasesThis is consistent with the fact that the thermal boundarylayer thickness decreases with increasing Prandtl numberPresence of heat source reduces the temperature of the flowfield This may happen due the elastic property of the fluidIt is observed that temperature of the flow field diminishes asthe suction parameter increases
Figure 6 depicts the effect of the Schmidt number chemi-cal reaction parameter and suction parameter on concentra-tion distributionThe concentration distribution decreases atall points of the flow field with the increase in the Schmidtnumber This shows that the heavier diffusing species havea greater retarding effect on the concentration distributionof the flow field It is observed that a destructive reaction(119870119888gt 0) reduces the concentration distribution whereas
a generative reaction (119870119888
= 0) enhances it Also it isobserved that presence of suction parameter diminishes theconcentration distribution
The skin friction is an important phenomenon whichcharacterizes the frictional drag at the solid surface FromTable 1 it is observed that the skin friction decreases with theincrease in all the forcing forces but it is interesting to notethat the skin friction increases with the increase in magneticparameter
From Table 2 it is to note that all the entries are positiveIt is seen that the Prandtl number (Pr) heat source (S) andsuction parameter (V
0) increase the rate of heat transfer at the
surface of the plateFrom Table 3 it is to note that all the entries are positive
It is observed that Schmidt number (119878119888) chemical reaction
parameter (119870119888) and suction parameter (V
0) increase the rate
of mass transfer at the surface of the plate
1
09
08
07
06
05
04
03
02
01
0
Pr = 05 S = 05 0 = 02
Pr = 071 S = 05 0 = 02
Pr = 071 S = 05 0 = 05
Pr = 071 S = 1 0 = 02
0 05 1 15 2 25 3 35 4 45 5y
Pr = 071 S = 0 0 = 02
120579
Figure 5 Effect of Pr 119878 and V0on temperature profile
1
09
08
07
06
05
04
03
02
01
00 05 1 15 2 25 3 35 4 45 5
y
078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =
078Kc = 05 0 = 02Sc =
078Kc = 05 0 = 05Sc =
078Kc = 1 0 = 02Sc =120601
Figure 6 Effect of 119878119888 119870119888 and V
0on concentration profile
4 Conclusion
A theoretical study of unsteady MHD incompressible vis-coelastic liquid of Walterrsquos 1198611015840 model with heat and masstransfer near an oscillating infinite porous plate in slip flowregime under the influence of a transverse magnetic fieldof uniform strength is considered Some of the importantfindings of the problem are given in the following
(i) Presence of the Prandtl number decreases the velocityof the flow field whereas presence magnetic fieldincreases it
(ii) The velocity of the flow field decreases suddenly nearthe plate due to the presence of elastic parameter
(iii) The velocity of the flow field decreases due to theincrease in the thermal Grashof number
(iv) Thermal boundary layer thickness decreases withincreasing the Prandtl number
(v) Heavier diffusing species have a greater retardingeffect on the concentration distribution
Appendix
Consider the following
1198861=1
2[V0Pr + radic(V
0Pr)2 + 4119878Pr]
1198862=1
2[V0119878119888+ radic(V
0119878119888)2
+ 4119870119888119878119888]
1198863=1
2[V20+ radicV20+ 4119876] 119876 = 119872
2+
1
119870119901
1198864=1
2[V20+ radicV20+ 4 (119876 + 119894119899)]
1198601=
minus119866119903
1198862
1minus 1198861V0minus 119876
1198602=
minus119866119903
1198862
2minus 1198862V0minus 119876
1198603=
minus1
1 + 1198863119877[(1198861119877 + 1)119860
1+ (1198862119877 + 1)119860
2]
1198604=
1
1 + 1198771198864
1198605=
1198863
11198601V0
1198862
1minus 1198861V0minus 119876
1198606=
1198863
21198602V0
1198862
2minus 1198862V0minus 119876
1198607= 1198605+ 1198606minusV01198863
31198603119910
V0minus 21198863
1198608=
(minusV01198863
41198604+ 1198941198991198862
41198604) 119910
V0minus 21198864
(A1)
Nomenclature
119909 119910 Coordinate axes119906 V Velocity components in x- and y-directions119905 Time variable120583 Dynamic viscosity120592 Kinematic viscosity120572 Thermal conductivity119901 Pressure119892 Acceleration due to gravity120573 Coefficient of volume expansion120573 Coefficient of volume expansion with concentration1198800 Reference velocity
119879 Dimensional temperature120579 Nondimensional temperature119862 Dimensional concentration120601 Nondimensional concentrationPr The Prandtl number119866119903 The thermal Grashof number
119866119898 The mass Grashof number
119878119888 The Schmidt number
119872 Magnetic parameter119878 Heat sourcesink parameter119870119897 Dimensional chemical reaction parameter
119870119888 Nondimensional chemical reaction parameter
119879119908 Temperature at the wall
119879infin Temperature far away from the wall
119862119908 Concentration at the wall
119862infin Concentration far away from the wall
120590 Electric conductivity1198610 Uniform magnetic field
V0 Suctioninjection velocity
119863 Mass diffusion120588 Density119870 Dimensional porosity parameter119870119901 Nondimensional porosity parameter
1198700 Dimensional elastic parameter
119877119888 Nondimensional elastic parameter
119899 Frequency of oscillation
8 International Journal of Chemical Engineering
Acknowledgment
The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work
References
[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003
[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008
[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005
[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005
[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011
[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010
[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002
[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007
[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006
[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011
[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011
Equation (5) is of third order and two boundary condi-tions are available Due to inadequate boundary conditiona perturbation method has been applied with 119877
119888lt 1 as the
perturbation parameter This assumption is quite consistentas the model under consideration is valid only for slightlyelastic fluid
The problem of unsteady flow of an electrically conductingand incompressible viscoelastic liquid of Walterrsquos 1198611015840 modelwith heat andmass transfer near an oscillating infinite porous
International Journal of Chemical Engineering 5
plate in slip flow regime with heat source and chemicalreaction parameter under the influence of a transversemagnetic field of uniform strength has been considered Theeffects of the flow parameters such as the Prandtl numberPr porosity parameter 119870
119901 magnetic parameter M elastic
parameter 119877119888 heat source parameter S chemical reaction
and rarefaction parameter 119877 on the velocity field have beenstudied analytically and presented with the help of Figures1ndash4 The effects of the flow parameters on the temperaturefield and concentration distribution have been presented inFigures 5 and 6 respectively Further the effects of the flowparameters on the skin friction heat flux and rate of masstransfer have been discussed with the help of Tables 1ndash3 Fornumerical computation the values of 119866
119903are taken positive
This indicates that the study has been carried out under theinfluence of the cooling of the plate Also we have takennt = 1205872 The interesting aspect of the problem is to studythe combined effect of the flow parameters with that of thefirst order velocity slip boundary condition when the plateexecutes linear harmonic oscillation in its own plane
Figure 1 shows the effect of the Prandtl number (Pr)permeability parameter (119870
119901) andmagnetic parameter (M) on
velocity profile For this figure we have taken that S = 1 119866119903
= 5 119866119898= 5 119878
119888= 024 119877
119888= 05 119870
119888= 2 R = 02 and V
0=
2 It is observed that the increase in the Prandtl number aswell as permeability parameter decreases the velocity of theflow field whereas increase in magnetic parameter increasesit Since Prandtl number is the ratio of kinematic viscosity tothermal diffusivity so as Pr increases the kinematic viscosityof the fluid dominates the thermal diffusivity of the fluidwhich leads to decreasing the velocity of the flow field Theapplication of transverse magnetic field sets up the Lorentzforce which enhances the fluid velocity
Figure 2 shows the effect of elastic parameter (119877119888) heat
source parameter (S) and chemical reaction (119870119888) parameter
on velocity frofile For this figure we have taken that Pr =071 119870
119901= 1M = 2 119866
119903= 5 119866
119898 = 5 119878
119888= 024 R = 02 and V
0
= 2 It is observed that the velocity of the flow field decreasesdue to the presence of elastic parameter chemical reactionparameter and heat source parameter For 119877
119888= S = 119870
119888= 0
the present work agrees with the work of Singh and Gupta[3]
Figure 3 depicts the effect of the thermal Grashof number(119866119903) the mass Grashof number (119866
119898) and the Schmidt
number (119878119888) on velocity frofile For this figure we have taken
that Pr = 071 S = 1 119870119901= 1 M = 2 119877
119888= 05 119870
119888= 2 R = 02
and V0= 2 It is observed that for the heavier species that is
with increasing 119878119888 the velocity decreases The velocity of the
flowfield decreases due to the increase in the thermalGrashofnumberMoreover buoyancy effect (119866
119898) due tomass transfer
enhances the velocityFigure 4 depicts the effect of suction parameter (V
0) and
rarefaction parameter (R) on velocity frofile For this figurewe have taken that Pr = 071 S = 1 119870
119901= 1 M = 2 119877
119888= 05
119870119888= 2 119866
119903= 5 119866
119898= 5 and 119878
119888= 024 It is observed that the
velocity of the flow field decreases due the presence of suction
0 05 1 15 2 25 3 35 4 45 5
1
0
minus1
minus2
minus3
minus4
minus5
u
y
Pr = 031Kp = 02M = 05
Pr = 071Kp = 02M = 1
Pr = 071Kp = 02M = 05
Pr = 071Kp = 04M = 05
Figure 1 Effect of Pr119870119901and119872 on velocity profile
0 05 1 15 2 25 3 35 4 45 5
05
0
minus05
minus1
minus15
minus2
u
y
Rc = 02 S = 05Kc = 1
Rc = 02 S = 05Kc = 2
Rc = 02 S = 1Kc = 1
Rc = 05 S = 05Kc = 1
Figure 2 Effect of 119877119888 S and 119870
119888on velocity profile
12
1
08
06
04
02
0
minus020 05 1 15 2 25 3 35 4 45 5
u
y
Gr = 25 Gm = 2 2
Gr = 2 Gm = 2 25
Sc =
Sc =
Gr = 2 Gm = 2 2Sc =
Gr = 2 Gm = 25 2Sc =
Figure 3 Effect of 119866119903 119866119898 and 119878
119888on velocity profile
6 International Journal of Chemical Engineering
06
04
02
0
minus02
minus04
minus06
minus08
minus1
minus12
0 = 15 R = 07
0 = 15 R = 05
0 = 2 R = 05
u
0 05 1 15 2 25 3 35 4 45 5y
Figure 4 Effect of V0and 119877 on velocity profile
parameter but the reverse effect is observed due the presenceof the rarefaction parameter
Figure 5 shows the effect of the Prandtl number heatsource parameter and suction parameter on the temperatureof the flow field It is observed that the temperature ofthe flow field diminishes as the Prandtl number increasesThis is consistent with the fact that the thermal boundarylayer thickness decreases with increasing Prandtl numberPresence of heat source reduces the temperature of the flowfield This may happen due the elastic property of the fluidIt is observed that temperature of the flow field diminishes asthe suction parameter increases
Figure 6 depicts the effect of the Schmidt number chemi-cal reaction parameter and suction parameter on concentra-tion distributionThe concentration distribution decreases atall points of the flow field with the increase in the Schmidtnumber This shows that the heavier diffusing species havea greater retarding effect on the concentration distributionof the flow field It is observed that a destructive reaction(119870119888gt 0) reduces the concentration distribution whereas
a generative reaction (119870119888
= 0) enhances it Also it isobserved that presence of suction parameter diminishes theconcentration distribution
The skin friction is an important phenomenon whichcharacterizes the frictional drag at the solid surface FromTable 1 it is observed that the skin friction decreases with theincrease in all the forcing forces but it is interesting to notethat the skin friction increases with the increase in magneticparameter
From Table 2 it is to note that all the entries are positiveIt is seen that the Prandtl number (Pr) heat source (S) andsuction parameter (V
0) increase the rate of heat transfer at the
surface of the plateFrom Table 3 it is to note that all the entries are positive
It is observed that Schmidt number (119878119888) chemical reaction
parameter (119870119888) and suction parameter (V
0) increase the rate
of mass transfer at the surface of the plate
1
09
08
07
06
05
04
03
02
01
0
Pr = 05 S = 05 0 = 02
Pr = 071 S = 05 0 = 02
Pr = 071 S = 05 0 = 05
Pr = 071 S = 1 0 = 02
0 05 1 15 2 25 3 35 4 45 5y
Pr = 071 S = 0 0 = 02
120579
Figure 5 Effect of Pr 119878 and V0on temperature profile
1
09
08
07
06
05
04
03
02
01
00 05 1 15 2 25 3 35 4 45 5
y
078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =
078Kc = 05 0 = 02Sc =
078Kc = 05 0 = 05Sc =
078Kc = 1 0 = 02Sc =120601
Figure 6 Effect of 119878119888 119870119888 and V
0on concentration profile
4 Conclusion
A theoretical study of unsteady MHD incompressible vis-coelastic liquid of Walterrsquos 1198611015840 model with heat and masstransfer near an oscillating infinite porous plate in slip flowregime under the influence of a transverse magnetic fieldof uniform strength is considered Some of the importantfindings of the problem are given in the following
(i) Presence of the Prandtl number decreases the velocityof the flow field whereas presence magnetic fieldincreases it
(ii) The velocity of the flow field decreases suddenly nearthe plate due to the presence of elastic parameter
(iii) The velocity of the flow field decreases due to theincrease in the thermal Grashof number
(iv) Thermal boundary layer thickness decreases withincreasing the Prandtl number
(v) Heavier diffusing species have a greater retardingeffect on the concentration distribution
Appendix
Consider the following
1198861=1
2[V0Pr + radic(V
0Pr)2 + 4119878Pr]
1198862=1
2[V0119878119888+ radic(V
0119878119888)2
+ 4119870119888119878119888]
1198863=1
2[V20+ radicV20+ 4119876] 119876 = 119872
2+
1
119870119901
1198864=1
2[V20+ radicV20+ 4 (119876 + 119894119899)]
1198601=
minus119866119903
1198862
1minus 1198861V0minus 119876
1198602=
minus119866119903
1198862
2minus 1198862V0minus 119876
1198603=
minus1
1 + 1198863119877[(1198861119877 + 1)119860
1+ (1198862119877 + 1)119860
2]
1198604=
1
1 + 1198771198864
1198605=
1198863
11198601V0
1198862
1minus 1198861V0minus 119876
1198606=
1198863
21198602V0
1198862
2minus 1198862V0minus 119876
1198607= 1198605+ 1198606minusV01198863
31198603119910
V0minus 21198863
1198608=
(minusV01198863
41198604+ 1198941198991198862
41198604) 119910
V0minus 21198864
(A1)
Nomenclature
119909 119910 Coordinate axes119906 V Velocity components in x- and y-directions119905 Time variable120583 Dynamic viscosity120592 Kinematic viscosity120572 Thermal conductivity119901 Pressure119892 Acceleration due to gravity120573 Coefficient of volume expansion120573 Coefficient of volume expansion with concentration1198800 Reference velocity
119879 Dimensional temperature120579 Nondimensional temperature119862 Dimensional concentration120601 Nondimensional concentrationPr The Prandtl number119866119903 The thermal Grashof number
119866119898 The mass Grashof number
119878119888 The Schmidt number
119872 Magnetic parameter119878 Heat sourcesink parameter119870119897 Dimensional chemical reaction parameter
119870119888 Nondimensional chemical reaction parameter
119879119908 Temperature at the wall
119879infin Temperature far away from the wall
119862119908 Concentration at the wall
119862infin Concentration far away from the wall
120590 Electric conductivity1198610 Uniform magnetic field
V0 Suctioninjection velocity
119863 Mass diffusion120588 Density119870 Dimensional porosity parameter119870119901 Nondimensional porosity parameter
1198700 Dimensional elastic parameter
119877119888 Nondimensional elastic parameter
119899 Frequency of oscillation
8 International Journal of Chemical Engineering
Acknowledgment
The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work
References
[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003
[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008
[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005
[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005
[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011
[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010
[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002
[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007
[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006
[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011
[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011
The problem of unsteady flow of an electrically conductingand incompressible viscoelastic liquid of Walterrsquos 1198611015840 modelwith heat andmass transfer near an oscillating infinite porous
International Journal of Chemical Engineering 5
plate in slip flow regime with heat source and chemicalreaction parameter under the influence of a transversemagnetic field of uniform strength has been considered Theeffects of the flow parameters such as the Prandtl numberPr porosity parameter 119870
119901 magnetic parameter M elastic
parameter 119877119888 heat source parameter S chemical reaction
and rarefaction parameter 119877 on the velocity field have beenstudied analytically and presented with the help of Figures1ndash4 The effects of the flow parameters on the temperaturefield and concentration distribution have been presented inFigures 5 and 6 respectively Further the effects of the flowparameters on the skin friction heat flux and rate of masstransfer have been discussed with the help of Tables 1ndash3 Fornumerical computation the values of 119866
119903are taken positive
This indicates that the study has been carried out under theinfluence of the cooling of the plate Also we have takennt = 1205872 The interesting aspect of the problem is to studythe combined effect of the flow parameters with that of thefirst order velocity slip boundary condition when the plateexecutes linear harmonic oscillation in its own plane
Figure 1 shows the effect of the Prandtl number (Pr)permeability parameter (119870
119901) andmagnetic parameter (M) on
velocity profile For this figure we have taken that S = 1 119866119903
= 5 119866119898= 5 119878
119888= 024 119877
119888= 05 119870
119888= 2 R = 02 and V
0=
2 It is observed that the increase in the Prandtl number aswell as permeability parameter decreases the velocity of theflow field whereas increase in magnetic parameter increasesit Since Prandtl number is the ratio of kinematic viscosity tothermal diffusivity so as Pr increases the kinematic viscosityof the fluid dominates the thermal diffusivity of the fluidwhich leads to decreasing the velocity of the flow field Theapplication of transverse magnetic field sets up the Lorentzforce which enhances the fluid velocity
Figure 2 shows the effect of elastic parameter (119877119888) heat
source parameter (S) and chemical reaction (119870119888) parameter
on velocity frofile For this figure we have taken that Pr =071 119870
119901= 1M = 2 119866
119903= 5 119866
119898 = 5 119878
119888= 024 R = 02 and V
0
= 2 It is observed that the velocity of the flow field decreasesdue to the presence of elastic parameter chemical reactionparameter and heat source parameter For 119877
119888= S = 119870
119888= 0
the present work agrees with the work of Singh and Gupta[3]
Figure 3 depicts the effect of the thermal Grashof number(119866119903) the mass Grashof number (119866
119898) and the Schmidt
number (119878119888) on velocity frofile For this figure we have taken
that Pr = 071 S = 1 119870119901= 1 M = 2 119877
119888= 05 119870
119888= 2 R = 02
and V0= 2 It is observed that for the heavier species that is
with increasing 119878119888 the velocity decreases The velocity of the
flowfield decreases due to the increase in the thermalGrashofnumberMoreover buoyancy effect (119866
119898) due tomass transfer
enhances the velocityFigure 4 depicts the effect of suction parameter (V
0) and
rarefaction parameter (R) on velocity frofile For this figurewe have taken that Pr = 071 S = 1 119870
119901= 1 M = 2 119877
119888= 05
119870119888= 2 119866
119903= 5 119866
119898= 5 and 119878
119888= 024 It is observed that the
velocity of the flow field decreases due the presence of suction
0 05 1 15 2 25 3 35 4 45 5
1
0
minus1
minus2
minus3
minus4
minus5
u
y
Pr = 031Kp = 02M = 05
Pr = 071Kp = 02M = 1
Pr = 071Kp = 02M = 05
Pr = 071Kp = 04M = 05
Figure 1 Effect of Pr119870119901and119872 on velocity profile
0 05 1 15 2 25 3 35 4 45 5
05
0
minus05
minus1
minus15
minus2
u
y
Rc = 02 S = 05Kc = 1
Rc = 02 S = 05Kc = 2
Rc = 02 S = 1Kc = 1
Rc = 05 S = 05Kc = 1
Figure 2 Effect of 119877119888 S and 119870
119888on velocity profile
12
1
08
06
04
02
0
minus020 05 1 15 2 25 3 35 4 45 5
u
y
Gr = 25 Gm = 2 2
Gr = 2 Gm = 2 25
Sc =
Sc =
Gr = 2 Gm = 2 2Sc =
Gr = 2 Gm = 25 2Sc =
Figure 3 Effect of 119866119903 119866119898 and 119878
119888on velocity profile
6 International Journal of Chemical Engineering
06
04
02
0
minus02
minus04
minus06
minus08
minus1
minus12
0 = 15 R = 07
0 = 15 R = 05
0 = 2 R = 05
u
0 05 1 15 2 25 3 35 4 45 5y
Figure 4 Effect of V0and 119877 on velocity profile
parameter but the reverse effect is observed due the presenceof the rarefaction parameter
Figure 5 shows the effect of the Prandtl number heatsource parameter and suction parameter on the temperatureof the flow field It is observed that the temperature ofthe flow field diminishes as the Prandtl number increasesThis is consistent with the fact that the thermal boundarylayer thickness decreases with increasing Prandtl numberPresence of heat source reduces the temperature of the flowfield This may happen due the elastic property of the fluidIt is observed that temperature of the flow field diminishes asthe suction parameter increases
Figure 6 depicts the effect of the Schmidt number chemi-cal reaction parameter and suction parameter on concentra-tion distributionThe concentration distribution decreases atall points of the flow field with the increase in the Schmidtnumber This shows that the heavier diffusing species havea greater retarding effect on the concentration distributionof the flow field It is observed that a destructive reaction(119870119888gt 0) reduces the concentration distribution whereas
a generative reaction (119870119888
= 0) enhances it Also it isobserved that presence of suction parameter diminishes theconcentration distribution
The skin friction is an important phenomenon whichcharacterizes the frictional drag at the solid surface FromTable 1 it is observed that the skin friction decreases with theincrease in all the forcing forces but it is interesting to notethat the skin friction increases with the increase in magneticparameter
From Table 2 it is to note that all the entries are positiveIt is seen that the Prandtl number (Pr) heat source (S) andsuction parameter (V
0) increase the rate of heat transfer at the
surface of the plateFrom Table 3 it is to note that all the entries are positive
It is observed that Schmidt number (119878119888) chemical reaction
parameter (119870119888) and suction parameter (V
0) increase the rate
of mass transfer at the surface of the plate
1
09
08
07
06
05
04
03
02
01
0
Pr = 05 S = 05 0 = 02
Pr = 071 S = 05 0 = 02
Pr = 071 S = 05 0 = 05
Pr = 071 S = 1 0 = 02
0 05 1 15 2 25 3 35 4 45 5y
Pr = 071 S = 0 0 = 02
120579
Figure 5 Effect of Pr 119878 and V0on temperature profile
1
09
08
07
06
05
04
03
02
01
00 05 1 15 2 25 3 35 4 45 5
y
078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =
078Kc = 05 0 = 02Sc =
078Kc = 05 0 = 05Sc =
078Kc = 1 0 = 02Sc =120601
Figure 6 Effect of 119878119888 119870119888 and V
0on concentration profile
4 Conclusion
A theoretical study of unsteady MHD incompressible vis-coelastic liquid of Walterrsquos 1198611015840 model with heat and masstransfer near an oscillating infinite porous plate in slip flowregime under the influence of a transverse magnetic fieldof uniform strength is considered Some of the importantfindings of the problem are given in the following
(i) Presence of the Prandtl number decreases the velocityof the flow field whereas presence magnetic fieldincreases it
(ii) The velocity of the flow field decreases suddenly nearthe plate due to the presence of elastic parameter
(iii) The velocity of the flow field decreases due to theincrease in the thermal Grashof number
(iv) Thermal boundary layer thickness decreases withincreasing the Prandtl number
(v) Heavier diffusing species have a greater retardingeffect on the concentration distribution
Appendix
Consider the following
1198861=1
2[V0Pr + radic(V
0Pr)2 + 4119878Pr]
1198862=1
2[V0119878119888+ radic(V
0119878119888)2
+ 4119870119888119878119888]
1198863=1
2[V20+ radicV20+ 4119876] 119876 = 119872
2+
1
119870119901
1198864=1
2[V20+ radicV20+ 4 (119876 + 119894119899)]
1198601=
minus119866119903
1198862
1minus 1198861V0minus 119876
1198602=
minus119866119903
1198862
2minus 1198862V0minus 119876
1198603=
minus1
1 + 1198863119877[(1198861119877 + 1)119860
1+ (1198862119877 + 1)119860
2]
1198604=
1
1 + 1198771198864
1198605=
1198863
11198601V0
1198862
1minus 1198861V0minus 119876
1198606=
1198863
21198602V0
1198862
2minus 1198862V0minus 119876
1198607= 1198605+ 1198606minusV01198863
31198603119910
V0minus 21198863
1198608=
(minusV01198863
41198604+ 1198941198991198862
41198604) 119910
V0minus 21198864
(A1)
Nomenclature
119909 119910 Coordinate axes119906 V Velocity components in x- and y-directions119905 Time variable120583 Dynamic viscosity120592 Kinematic viscosity120572 Thermal conductivity119901 Pressure119892 Acceleration due to gravity120573 Coefficient of volume expansion120573 Coefficient of volume expansion with concentration1198800 Reference velocity
119879 Dimensional temperature120579 Nondimensional temperature119862 Dimensional concentration120601 Nondimensional concentrationPr The Prandtl number119866119903 The thermal Grashof number
119866119898 The mass Grashof number
119878119888 The Schmidt number
119872 Magnetic parameter119878 Heat sourcesink parameter119870119897 Dimensional chemical reaction parameter
119870119888 Nondimensional chemical reaction parameter
119879119908 Temperature at the wall
119879infin Temperature far away from the wall
119862119908 Concentration at the wall
119862infin Concentration far away from the wall
120590 Electric conductivity1198610 Uniform magnetic field
V0 Suctioninjection velocity
119863 Mass diffusion120588 Density119870 Dimensional porosity parameter119870119901 Nondimensional porosity parameter
1198700 Dimensional elastic parameter
119877119888 Nondimensional elastic parameter
119899 Frequency of oscillation
8 International Journal of Chemical Engineering
Acknowledgment
The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work
References
[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003
[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008
[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005
[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005
[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011
[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010
[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002
[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007
[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006
[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011
[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011
plate in slip flow regime with heat source and chemicalreaction parameter under the influence of a transversemagnetic field of uniform strength has been considered Theeffects of the flow parameters such as the Prandtl numberPr porosity parameter 119870
119901 magnetic parameter M elastic
parameter 119877119888 heat source parameter S chemical reaction
and rarefaction parameter 119877 on the velocity field have beenstudied analytically and presented with the help of Figures1ndash4 The effects of the flow parameters on the temperaturefield and concentration distribution have been presented inFigures 5 and 6 respectively Further the effects of the flowparameters on the skin friction heat flux and rate of masstransfer have been discussed with the help of Tables 1ndash3 Fornumerical computation the values of 119866
119903are taken positive
This indicates that the study has been carried out under theinfluence of the cooling of the plate Also we have takennt = 1205872 The interesting aspect of the problem is to studythe combined effect of the flow parameters with that of thefirst order velocity slip boundary condition when the plateexecutes linear harmonic oscillation in its own plane
Figure 1 shows the effect of the Prandtl number (Pr)permeability parameter (119870
119901) andmagnetic parameter (M) on
velocity profile For this figure we have taken that S = 1 119866119903
= 5 119866119898= 5 119878
119888= 024 119877
119888= 05 119870
119888= 2 R = 02 and V
0=
2 It is observed that the increase in the Prandtl number aswell as permeability parameter decreases the velocity of theflow field whereas increase in magnetic parameter increasesit Since Prandtl number is the ratio of kinematic viscosity tothermal diffusivity so as Pr increases the kinematic viscosityof the fluid dominates the thermal diffusivity of the fluidwhich leads to decreasing the velocity of the flow field Theapplication of transverse magnetic field sets up the Lorentzforce which enhances the fluid velocity
Figure 2 shows the effect of elastic parameter (119877119888) heat
source parameter (S) and chemical reaction (119870119888) parameter
on velocity frofile For this figure we have taken that Pr =071 119870
119901= 1M = 2 119866
119903= 5 119866
119898 = 5 119878
119888= 024 R = 02 and V
0
= 2 It is observed that the velocity of the flow field decreasesdue to the presence of elastic parameter chemical reactionparameter and heat source parameter For 119877
119888= S = 119870
119888= 0
the present work agrees with the work of Singh and Gupta[3]
Figure 3 depicts the effect of the thermal Grashof number(119866119903) the mass Grashof number (119866
119898) and the Schmidt
number (119878119888) on velocity frofile For this figure we have taken
that Pr = 071 S = 1 119870119901= 1 M = 2 119877
119888= 05 119870
119888= 2 R = 02
and V0= 2 It is observed that for the heavier species that is
with increasing 119878119888 the velocity decreases The velocity of the
flowfield decreases due to the increase in the thermalGrashofnumberMoreover buoyancy effect (119866
119898) due tomass transfer
enhances the velocityFigure 4 depicts the effect of suction parameter (V
0) and
rarefaction parameter (R) on velocity frofile For this figurewe have taken that Pr = 071 S = 1 119870
119901= 1 M = 2 119877
119888= 05
119870119888= 2 119866
119903= 5 119866
119898= 5 and 119878
119888= 024 It is observed that the
velocity of the flow field decreases due the presence of suction
0 05 1 15 2 25 3 35 4 45 5
1
0
minus1
minus2
minus3
minus4
minus5
u
y
Pr = 031Kp = 02M = 05
Pr = 071Kp = 02M = 1
Pr = 071Kp = 02M = 05
Pr = 071Kp = 04M = 05
Figure 1 Effect of Pr119870119901and119872 on velocity profile
0 05 1 15 2 25 3 35 4 45 5
05
0
minus05
minus1
minus15
minus2
u
y
Rc = 02 S = 05Kc = 1
Rc = 02 S = 05Kc = 2
Rc = 02 S = 1Kc = 1
Rc = 05 S = 05Kc = 1
Figure 2 Effect of 119877119888 S and 119870
119888on velocity profile
12
1
08
06
04
02
0
minus020 05 1 15 2 25 3 35 4 45 5
u
y
Gr = 25 Gm = 2 2
Gr = 2 Gm = 2 25
Sc =
Sc =
Gr = 2 Gm = 2 2Sc =
Gr = 2 Gm = 25 2Sc =
Figure 3 Effect of 119866119903 119866119898 and 119878
119888on velocity profile
6 International Journal of Chemical Engineering
06
04
02
0
minus02
minus04
minus06
minus08
minus1
minus12
0 = 15 R = 07
0 = 15 R = 05
0 = 2 R = 05
u
0 05 1 15 2 25 3 35 4 45 5y
Figure 4 Effect of V0and 119877 on velocity profile
parameter but the reverse effect is observed due the presenceof the rarefaction parameter
Figure 5 shows the effect of the Prandtl number heatsource parameter and suction parameter on the temperatureof the flow field It is observed that the temperature ofthe flow field diminishes as the Prandtl number increasesThis is consistent with the fact that the thermal boundarylayer thickness decreases with increasing Prandtl numberPresence of heat source reduces the temperature of the flowfield This may happen due the elastic property of the fluidIt is observed that temperature of the flow field diminishes asthe suction parameter increases
Figure 6 depicts the effect of the Schmidt number chemi-cal reaction parameter and suction parameter on concentra-tion distributionThe concentration distribution decreases atall points of the flow field with the increase in the Schmidtnumber This shows that the heavier diffusing species havea greater retarding effect on the concentration distributionof the flow field It is observed that a destructive reaction(119870119888gt 0) reduces the concentration distribution whereas
a generative reaction (119870119888
= 0) enhances it Also it isobserved that presence of suction parameter diminishes theconcentration distribution
The skin friction is an important phenomenon whichcharacterizes the frictional drag at the solid surface FromTable 1 it is observed that the skin friction decreases with theincrease in all the forcing forces but it is interesting to notethat the skin friction increases with the increase in magneticparameter
From Table 2 it is to note that all the entries are positiveIt is seen that the Prandtl number (Pr) heat source (S) andsuction parameter (V
0) increase the rate of heat transfer at the
surface of the plateFrom Table 3 it is to note that all the entries are positive
It is observed that Schmidt number (119878119888) chemical reaction
parameter (119870119888) and suction parameter (V
0) increase the rate
of mass transfer at the surface of the plate
1
09
08
07
06
05
04
03
02
01
0
Pr = 05 S = 05 0 = 02
Pr = 071 S = 05 0 = 02
Pr = 071 S = 05 0 = 05
Pr = 071 S = 1 0 = 02
0 05 1 15 2 25 3 35 4 45 5y
Pr = 071 S = 0 0 = 02
120579
Figure 5 Effect of Pr 119878 and V0on temperature profile
1
09
08
07
06
05
04
03
02
01
00 05 1 15 2 25 3 35 4 45 5
y
078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =
078Kc = 05 0 = 02Sc =
078Kc = 05 0 = 05Sc =
078Kc = 1 0 = 02Sc =120601
Figure 6 Effect of 119878119888 119870119888 and V
0on concentration profile
4 Conclusion
A theoretical study of unsteady MHD incompressible vis-coelastic liquid of Walterrsquos 1198611015840 model with heat and masstransfer near an oscillating infinite porous plate in slip flowregime under the influence of a transverse magnetic fieldof uniform strength is considered Some of the importantfindings of the problem are given in the following
(i) Presence of the Prandtl number decreases the velocityof the flow field whereas presence magnetic fieldincreases it
(ii) The velocity of the flow field decreases suddenly nearthe plate due to the presence of elastic parameter
(iii) The velocity of the flow field decreases due to theincrease in the thermal Grashof number
(iv) Thermal boundary layer thickness decreases withincreasing the Prandtl number
(v) Heavier diffusing species have a greater retardingeffect on the concentration distribution
Appendix
Consider the following
1198861=1
2[V0Pr + radic(V
0Pr)2 + 4119878Pr]
1198862=1
2[V0119878119888+ radic(V
0119878119888)2
+ 4119870119888119878119888]
1198863=1
2[V20+ radicV20+ 4119876] 119876 = 119872
2+
1
119870119901
1198864=1
2[V20+ radicV20+ 4 (119876 + 119894119899)]
1198601=
minus119866119903
1198862
1minus 1198861V0minus 119876
1198602=
minus119866119903
1198862
2minus 1198862V0minus 119876
1198603=
minus1
1 + 1198863119877[(1198861119877 + 1)119860
1+ (1198862119877 + 1)119860
2]
1198604=
1
1 + 1198771198864
1198605=
1198863
11198601V0
1198862
1minus 1198861V0minus 119876
1198606=
1198863
21198602V0
1198862
2minus 1198862V0minus 119876
1198607= 1198605+ 1198606minusV01198863
31198603119910
V0minus 21198863
1198608=
(minusV01198863
41198604+ 1198941198991198862
41198604) 119910
V0minus 21198864
(A1)
Nomenclature
119909 119910 Coordinate axes119906 V Velocity components in x- and y-directions119905 Time variable120583 Dynamic viscosity120592 Kinematic viscosity120572 Thermal conductivity119901 Pressure119892 Acceleration due to gravity120573 Coefficient of volume expansion120573 Coefficient of volume expansion with concentration1198800 Reference velocity
119879 Dimensional temperature120579 Nondimensional temperature119862 Dimensional concentration120601 Nondimensional concentrationPr The Prandtl number119866119903 The thermal Grashof number
119866119898 The mass Grashof number
119878119888 The Schmidt number
119872 Magnetic parameter119878 Heat sourcesink parameter119870119897 Dimensional chemical reaction parameter
119870119888 Nondimensional chemical reaction parameter
119879119908 Temperature at the wall
119879infin Temperature far away from the wall
119862119908 Concentration at the wall
119862infin Concentration far away from the wall
120590 Electric conductivity1198610 Uniform magnetic field
V0 Suctioninjection velocity
119863 Mass diffusion120588 Density119870 Dimensional porosity parameter119870119901 Nondimensional porosity parameter
1198700 Dimensional elastic parameter
119877119888 Nondimensional elastic parameter
119899 Frequency of oscillation
8 International Journal of Chemical Engineering
Acknowledgment
The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work
References
[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003
[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008
[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005
[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005
[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011
[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010
[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002
[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007
[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006
[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011
[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011
Figure 4 Effect of V0and 119877 on velocity profile
parameter but the reverse effect is observed due the presenceof the rarefaction parameter
Figure 5 shows the effect of the Prandtl number heatsource parameter and suction parameter on the temperatureof the flow field It is observed that the temperature ofthe flow field diminishes as the Prandtl number increasesThis is consistent with the fact that the thermal boundarylayer thickness decreases with increasing Prandtl numberPresence of heat source reduces the temperature of the flowfield This may happen due the elastic property of the fluidIt is observed that temperature of the flow field diminishes asthe suction parameter increases
Figure 6 depicts the effect of the Schmidt number chemi-cal reaction parameter and suction parameter on concentra-tion distributionThe concentration distribution decreases atall points of the flow field with the increase in the Schmidtnumber This shows that the heavier diffusing species havea greater retarding effect on the concentration distributionof the flow field It is observed that a destructive reaction(119870119888gt 0) reduces the concentration distribution whereas
a generative reaction (119870119888
= 0) enhances it Also it isobserved that presence of suction parameter diminishes theconcentration distribution
The skin friction is an important phenomenon whichcharacterizes the frictional drag at the solid surface FromTable 1 it is observed that the skin friction decreases with theincrease in all the forcing forces but it is interesting to notethat the skin friction increases with the increase in magneticparameter
From Table 2 it is to note that all the entries are positiveIt is seen that the Prandtl number (Pr) heat source (S) andsuction parameter (V
0) increase the rate of heat transfer at the
surface of the plateFrom Table 3 it is to note that all the entries are positive
It is observed that Schmidt number (119878119888) chemical reaction
parameter (119870119888) and suction parameter (V
0) increase the rate
of mass transfer at the surface of the plate
1
09
08
07
06
05
04
03
02
01
0
Pr = 05 S = 05 0 = 02
Pr = 071 S = 05 0 = 02
Pr = 071 S = 05 0 = 05
Pr = 071 S = 1 0 = 02
0 05 1 15 2 25 3 35 4 45 5y
Pr = 071 S = 0 0 = 02
120579
Figure 5 Effect of Pr 119878 and V0on temperature profile
1
09
08
07
06
05
04
03
02
01
00 05 1 15 2 25 3 35 4 45 5
y
078Kc = 0 0 = 02Sc =030Kc = 05 0 = 02Sc =
078Kc = 05 0 = 02Sc =
078Kc = 05 0 = 05Sc =
078Kc = 1 0 = 02Sc =120601
Figure 6 Effect of 119878119888 119870119888 and V
0on concentration profile
4 Conclusion
A theoretical study of unsteady MHD incompressible vis-coelastic liquid of Walterrsquos 1198611015840 model with heat and masstransfer near an oscillating infinite porous plate in slip flowregime under the influence of a transverse magnetic fieldof uniform strength is considered Some of the importantfindings of the problem are given in the following
(i) Presence of the Prandtl number decreases the velocityof the flow field whereas presence magnetic fieldincreases it
(ii) The velocity of the flow field decreases suddenly nearthe plate due to the presence of elastic parameter
(iii) The velocity of the flow field decreases due to theincrease in the thermal Grashof number
(iv) Thermal boundary layer thickness decreases withincreasing the Prandtl number
(v) Heavier diffusing species have a greater retardingeffect on the concentration distribution
Appendix
Consider the following
1198861=1
2[V0Pr + radic(V
0Pr)2 + 4119878Pr]
1198862=1
2[V0119878119888+ radic(V
0119878119888)2
+ 4119870119888119878119888]
1198863=1
2[V20+ radicV20+ 4119876] 119876 = 119872
2+
1
119870119901
1198864=1
2[V20+ radicV20+ 4 (119876 + 119894119899)]
1198601=
minus119866119903
1198862
1minus 1198861V0minus 119876
1198602=
minus119866119903
1198862
2minus 1198862V0minus 119876
1198603=
minus1
1 + 1198863119877[(1198861119877 + 1)119860
1+ (1198862119877 + 1)119860
2]
1198604=
1
1 + 1198771198864
1198605=
1198863
11198601V0
1198862
1minus 1198861V0minus 119876
1198606=
1198863
21198602V0
1198862
2minus 1198862V0minus 119876
1198607= 1198605+ 1198606minusV01198863
31198603119910
V0minus 21198863
1198608=
(minusV01198863
41198604+ 1198941198991198862
41198604) 119910
V0minus 21198864
(A1)
Nomenclature
119909 119910 Coordinate axes119906 V Velocity components in x- and y-directions119905 Time variable120583 Dynamic viscosity120592 Kinematic viscosity120572 Thermal conductivity119901 Pressure119892 Acceleration due to gravity120573 Coefficient of volume expansion120573 Coefficient of volume expansion with concentration1198800 Reference velocity
119879 Dimensional temperature120579 Nondimensional temperature119862 Dimensional concentration120601 Nondimensional concentrationPr The Prandtl number119866119903 The thermal Grashof number
119866119898 The mass Grashof number
119878119888 The Schmidt number
119872 Magnetic parameter119878 Heat sourcesink parameter119870119897 Dimensional chemical reaction parameter
119870119888 Nondimensional chemical reaction parameter
119879119908 Temperature at the wall
119879infin Temperature far away from the wall
119862119908 Concentration at the wall
119862infin Concentration far away from the wall
120590 Electric conductivity1198610 Uniform magnetic field
V0 Suctioninjection velocity
119863 Mass diffusion120588 Density119870 Dimensional porosity parameter119870119901 Nondimensional porosity parameter
1198700 Dimensional elastic parameter
119877119888 Nondimensional elastic parameter
119899 Frequency of oscillation
8 International Journal of Chemical Engineering
Acknowledgment
The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work
References
[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003
[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008
[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005
[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005
[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011
[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010
[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002
[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007
[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006
[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011
[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011
(iv) Thermal boundary layer thickness decreases withincreasing the Prandtl number
(v) Heavier diffusing species have a greater retardingeffect on the concentration distribution
Appendix
Consider the following
1198861=1
2[V0Pr + radic(V
0Pr)2 + 4119878Pr]
1198862=1
2[V0119878119888+ radic(V
0119878119888)2
+ 4119870119888119878119888]
1198863=1
2[V20+ radicV20+ 4119876] 119876 = 119872
2+
1
119870119901
1198864=1
2[V20+ radicV20+ 4 (119876 + 119894119899)]
1198601=
minus119866119903
1198862
1minus 1198861V0minus 119876
1198602=
minus119866119903
1198862
2minus 1198862V0minus 119876
1198603=
minus1
1 + 1198863119877[(1198861119877 + 1)119860
1+ (1198862119877 + 1)119860
2]
1198604=
1
1 + 1198771198864
1198605=
1198863
11198601V0
1198862
1minus 1198861V0minus 119876
1198606=
1198863
21198602V0
1198862
2minus 1198862V0minus 119876
1198607= 1198605+ 1198606minusV01198863
31198603119910
V0minus 21198863
1198608=
(minusV01198863
41198604+ 1198941198991198862
41198604) 119910
V0minus 21198864
(A1)
Nomenclature
119909 119910 Coordinate axes119906 V Velocity components in x- and y-directions119905 Time variable120583 Dynamic viscosity120592 Kinematic viscosity120572 Thermal conductivity119901 Pressure119892 Acceleration due to gravity120573 Coefficient of volume expansion120573 Coefficient of volume expansion with concentration1198800 Reference velocity
119879 Dimensional temperature120579 Nondimensional temperature119862 Dimensional concentration120601 Nondimensional concentrationPr The Prandtl number119866119903 The thermal Grashof number
119866119898 The mass Grashof number
119878119888 The Schmidt number
119872 Magnetic parameter119878 Heat sourcesink parameter119870119897 Dimensional chemical reaction parameter
119870119888 Nondimensional chemical reaction parameter
119879119908 Temperature at the wall
119879infin Temperature far away from the wall
119862119908 Concentration at the wall
119862infin Concentration far away from the wall
120590 Electric conductivity1198610 Uniform magnetic field
V0 Suctioninjection velocity
119863 Mass diffusion120588 Density119870 Dimensional porosity parameter119870119901 Nondimensional porosity parameter
1198700 Dimensional elastic parameter
119877119888 Nondimensional elastic parameter
119899 Frequency of oscillation
8 International Journal of Chemical Engineering
Acknowledgment
The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work
References
[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003
[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008
[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005
[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005
[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011
[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010
[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002
[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007
[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006
[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011
[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011
The author wish to express his special thanks to ProfessorG C Dash S ldquoOrdquo A University Bhubaneswar Odisha Indiafor his valuable suggesion and constant encouragement tocomplete the work
References
[1] A K Singh A K Singh and N P Singh ldquoHeat and masstransfer in MHD flow of a viscous fluid past a vertical plateunder oscillatory suction velocityrdquo Indian Journal of Pure andApplied Mathematics vol 34 no 3 pp 429ndash442 2003
[2] P R Sharma and G Singh ldquoUnsteady MHD free convectiveflow and and heat transfer along a vertical porous plate withvariable suction and internal heat generationrdquo InternationalJournal of Applied Mathematics and Mechanics vol 4 no 5 pp1ndash8 2008
[3] P Singh and C B Gupta ldquoMHD free convective flow of viscousfluid through a porous medium bounded by an oscillaingporous plate in slip flow regime with mass transferrdquo IndianJournal of Theoretical Physics vol 53 no 2 pp 111ndash120 2005
[4] A K Khandelwal and N C Jain ldquoUnsteady MHD flow ofstratified fluid through porous medium over a moving plate inslip flow regimerdquo Indian Journal of Theoretical Physics vol 53no 1 pp 25ndash35 2005
[5] S S Das L K Mishra and P K Mishra ldquoEffect of heat sourceon MHD free convection flow past an oscillating porous platein the slip flow regimerdquo International Journal of Energy andEnvironment vol 2 no 5 pp 945ndash952 2011
[6] N Mahapatra G C Dash S Panda and M Acharya ldquoEffectsof chemical reaction on free convection flow through a porousmedium bounded by a vertical surfacerdquo Journal of EngineeringPhysics andThermophysics vol 83 no 1 pp 130ndash140 2010
[7] R Muthucumaraswamy ldquoEffects of a chemical reaction on amoving isothermal vertical surface with suctionrdquoActaMechan-ica vol 155 no 1-2 pp 65ndash70 2002
[8] M Q Al-Odat and T A Al-Azab ldquoInfluence of chemicalreaction on a transient MHD free convection flow over amoving vertical platerdquoThe Journal of Engineering Research vol12 no 3 pp 15ndash21 2007
[9] R C Chaudhary and P Jain ldquoHall effect on MHD mixedconvection flow of a visco-elastic fluid past and infinite verticalplate with mass transfer and radiationrdquoTheoretical and AppliedMechanics vol 33 no 4 pp 281ndash309 2006
[10] S N Sahoo J P Panda and G C Dash ldquoUnsteady twodimensional MHD flow and heat transfer of an elastic-viscousliquid past an infinite hot vertical porous surface bounded byporous medium with sourcesinkrdquo AMSE France vol 80 no2 pp 26ndash42 2011
[11] R Kumar and K Chand ldquoEffect of slip conditions and Hallcurrent on unsteady MHD flow of a viscoelastic fluid pastan infinite vertical porous plate through porous mediumrdquoInternational Journal of Engineering Science and Technology vol3 no 4 pp 3124ndash3133 2011