Behaviour of Non-Newtonian Casson Fluid on MHD

IP 51031210 On Wed 21 Sep 2022 080116Copyright American Scientific Publishers

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ARTICLE

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Printed in the United States of America

Journal of NanofluidsVol 6 pp 946ndash955 2017(wwwaspbscomjon)

Behaviour of Non-Newtonian Casson Fluid on MHDFree Convective Flow Past a Vertically InclinedSurface Surrounded by Porous Medium IncludingThermal Diffusion A Finite Element TechniqueS V Sailaja1 B Shanker1 and R Srinivasa Raju2lowast

1Department of Mathematics Osmania University Hyderabad 500007 Telangana State India2Department of Mathematics GITAM University Hyderabad Campus Rudraram 502329 Telangana State India

The aim of this research paper is to study the influence of thermal diffusion on steady magnetohydrodynamicfree convection heat and mass transfer electrically conducting non-Newtonian Casson fluid flow over on anvertically inclined surface taken in to the account with constant heat flux Finite element method is employed tosolve the fundamental non-dimensional boundary layer partial differential equations for velocity temperature andconcentration distributions with the help of various dimensionless parameters The present results reveal thatthe velocity profiles are reduced with Casson fluid and angle of inclination parameters whereas the temperatureis decreased with increasing of Prandtl number It is also seen that the dimensionless velocity profiles areincreasing with increasing values of Grashof number for heat and mass transfer The intensification in valuesof Soret number produces a raise in the mass buoyancy force which results an increase in the value of velocityand concentration profiles An excellent agreement is observed between present numerical results with earlierexisted analytical solutions Hence finite difference method is stable convergent and superior than the otheranalytical and numerical methods

KEYWORDS MHD Thermal Diffusion Casson Fluid Porous Medium Finite Element Method

1 INTRODUCTIONIn fluid mechanics non-Newtonian fluid theory is one ofthe part based on the continuum theory that a fluid particlemay be considered as continuous in a structure Pseudoplastic time independent fluid is one of the non-newtonianfluids whose behaviour is that viscosity decreases withincreasing velocity gradient eg polymer solutions bloodetc Casson fluid is one of the pseudoplastic fluids thatmeans shear thinning fluids At low shear rates the shearthinning fluid is more viscous than the Newtonian fluidand at high shear rates it is less viscous So MHD flowwith Casson fluid is recently famous Casson1 presentedCasson fluid model for the prediction of the flow conductof pigment-oil suspensions Khalid et al2 investigated thefree convection flow of a Casson fluid over a fluctuatingvertical plat with constant wall temperature Mohyud-Dinand Khan3 have discussed on magnetic field and radia-tion effects on squeezing flow of a Casson fluid between

lowastAuthor to whom correspondence should be addressedEmail srivass999gmailcomReceived 3 March 2017Accepted 28 March 2017

parallel plates Abbasi and Shehzad4 studied the impactof Cattaneo-Christov heat flux model on three-dimensionalboundary layer flow of Maxwell fluid towards a bidirec-tional stretching surface Hayat et al5 studied the jointeffects of magnetic field and nanoparticles in the three-dimensional flow of Sisko fluid Abbasi et al6 studiedmagnetohydrodynamic doubly stratified flow of Maxwellnanofluid in presence of mixed convection with the effectsof thermophoresis Brownian motion and heat genera-tionabsorption Srinivasa Raju7 studied the properties ofCasson fluid flow on free convective flow past an infi-nite vertically inclined plate in presence of thermal radia-tion and magnetic field by applying finite element methodRamya et al8 discussed the behaviour of nanofluid onboundary layer viscous flow over a nonlinearly isother-mal stretching sheet in the presence of heat genera-tionabsorption heat transfer and slip boundary conditionsRamya et al9 found the numerical solutions of magne-tohydrodynamic boundary layer flow of nanofluid over astretching sheet in presence of chemical reaction and ther-mal radiation using Keller-box technique Keller-box solu-tions of magnetohydrodynamic boundary layer partial slipflow of nanofluids over a nonlinear stretching sheet with

946 J Nanofluids 2017 Vol 6 No 5 2169-432X20176946010 doi101166jon20171389



Sailaja et al Behaviour of Non-Newtonian Casson Fluid on MHD Free Convective Flow Past a Vertically Inclined Surface

ARTICLE

suctioninjection studied by Ramya et al10 The steady two-dimensional magnetohydrodynamic boundary layer flow ofviscous nano fluids over a nonlinear stretching sheet wasinvestigated numerically by Shravani et al11 Sailaja et al12

discussed the effects of double diffusive on magnetohy-drodynamic mixed convection flow towards a verticallyinclined plate embedded in porous medium in presence ofbiot number using finite element method Srinivasa Rajuet al13 studied the influence of angle of inclination onunsteady magnetohydrodynamic Casson fluid flow past avertical surface filled by porous medium in presence ofconstant heat flux chemical reaction and viscous dissi-pation by applying finite element method Nagendrammaand Leelaratnam14 reported the results of the steady MHDheat and mass transfer of a Casson nanofluid flow embed-ded in a porous medium with variable thermal con-ductivity velocity slip and suctionblowing parametersthrough a horizontal cylinder using Runge-Kutta fourthorder method with shooting technique Gnaneswara Reddyet al15 three-dimensional magneto hydrodynamic Cassonnanofluid flow heat and mass transfer over a stretchingsheet with thermal radiation heat sourcesink second-order velocity slip and double stratification using finitedifference Keller box technique Unsteady hydromagneticheat and mass transfer flow of a chemically reactiveCasson fluid past a stretching sheet with convective bound-ary condition was investigated by Mahanta et al16 byusing matlab bvp4c package Gnaneswara Reddy et al17

studied three-dimensional magneto hydrodynamic Cassonnanofluid flow and heat transfer past a stretching sheetwith thermal radiation heat sourcesink and double strat-ification by adapting the efficient finite difference Kellerbox technique Shima and Philip18 investigated the role ofthermal conductivity of dispersed nanomaterial on the ther-mal and rheological properties of metal and metal oxidenanofluids with average particles size stabilized with amonolayer of surfactant Angayarkanni and Philip19 stud-ied the thermal properties and internal microstructures ofn-hexadecane alkane containing nano-inclusions of coppernano-wire multi walled carbon nanotube and graphenenano-platelets of different volume fractions Buongiornoet al20 reported on the international nanofluid propertybenchmark exercise or inpbe in which the thermal con-ductivity of identical samples of colloidally stable dis-persions of nanoparticles or ldquonanofluidsrdquo was measuredby over 30 organizations worldwide using a variety ofexperimental methods The role of brownian motion inter-facial resistance morphology of suspended nanoparticlesand aggregating behavior were investigated both experi-mentally and theoretically by Philip Angayarkanni21

In all of the above-mentioned studies the case of thethermal-diffusion effect (Soret) has received little atten-tion An experimental investigation of this effect was firstperformed by Charles Soret in 1879 When mass trans-fer occurs simultaneously in a moving fluid the relation-ship between the fluxes and the driving potentials are of

a more integrate nature It has been observed that massflux (mass transfer per unit time and per unit area per-pendicular to the direction of transfer) can be generatednot only by the concentration gradients but also by thetemperature gradients Mass fluxes influenced by tem-perature gradients are known as the Soret (thermal dif-fusion) effect The Soret effect for example has beenutilized for isotope separation and in mixtures betweengases with very light molecular weight like Hydrogen andHelium The role of transportation of mass in control-ling temperature of a heated body is found to be moreeconomical than any other method Following Eckert andDrakersquos22 work several other investigators have carriedout model studies of the Soret effect in different types ofheat and mass transfer problems Anand Rao and Srini-vasa Raju23 found the numerical solutions of MHD flowand heat transfer along a flat plate embedded in porousmedium in presence of hall current Soret and Dufoureffects using finite element method Sheri and Raju24 stud-ied the influence of Soret on an unsteady magnetohydro-dynamics free convective flow past a semi-infinite verticalplate in the presence viscous dissipation The results ofthermal radiation and heat source on an unsteady MHDfree convective fluid flow over an infinite vertical platein occurrence of thermal diffusion and diffusion thermowere discussed by Raju et al25 Sahu and Rajput26 stud-ied the combined effects of thermal diffusion and chem-ical reaction on the steady free convection MHD flowthrough a porous medium bounded by an infinite verticalsurface with constant heat flux using two-term perturba-tion method Srinivasa Raju27 studied the combined effectsof thermal-diffusion and diffusion-thermo on unsteady freeconvection fluid flow past an infinite vertical porous platein presence of magnetic field and chemical reaction usingfinite element technique Srinivasa Raju et al28 found bothanalytical and numerical solutions of unsteady magneto-hydrodynamic free convective flow past an exponentiallymoving vertical plate with heat absorption and chemicalreaction Srinivasa Raju et al29 studied the application offinite element method to unsteady MHD free convectionflow past a vertically inclined porous plate including ther-mal diffusion and diffusion thermo effects Sami et al30

studied the MHD free convection flow in a porous mediumwith thermal diffusion and ramped wall temperature Theyobtained exact dimensionless solutions of momentum andenergy equations under Boussinesqrsquos approximation usingthe Laplace transforms Ahmad31 studied MHD transientfree convection and mass transfer flow of a viscous incom-pressible and electrically conducting fluid in the presenceof thermal diffusion and thermal radiation He obtainedexact solutions for velocity temperature and concentrationusing the Laplace transform method Lakshmi Narayanaand Murthy32 studied the Soret and Dufour effects on freeconvection heat and mass transfer from a horizontal flatplate in a Darcy porous medium

J Nanofluids 6 946ndash955 2017 947



Behaviour of Non-Newtonian Casson Fluid on MHD Free Convective Flow Past a Vertically Inclined Surface Sailaja et al

ARTICLE

From the above discussion it is very much clear that theinfluence of Soret on MHD natural convective electricallyconducting non-Newtonian Casson fluid flow over on anvertically inclined porous plate taken in to the accountwith constant heat flux The system of coupled partial dif-ferential equations are solved numerically by using finiteelement method Numerical calculations are carried out forphysical parameters up to desired level of accuracy Theresults for shear stress and temperature gradient at wallare also calculated carefully as applications of both arevery important in engineering and industry It is hoped thatfindings from the present study will be useful in manytechnological and manufacturing processes

2 MATHEMATICAL ANALYSISConsider a steady free convective boundary layer flowof a viscous incompressible non-Newtonian Casson fluidwith variable fluid properties in the presence of a trans-verse magnetic field constant heat flux and Soret effectThe physical model and the coordinate system are shownin Figure 1 For this investigation we have to introducethe coordinate system xprime yprime with the length of the platealong xprime-axis in the upward vertical direction and yprime-axisis normal to the plate towards the fluid region The con-stant suction at the plate is parallel to yprime-axis The con-stant temperature T prime

w and concentration C primew higher than

the ambient temperature T prime and concentration C prime

ismaintained at the wall B0 is the uniform magnetic fieldwhich is applied normal to the plate It is assumed that thetransverse applied magnetic field and magnetic Reynoldrsquosnumber are assumed to be very small so that the induced

xprime

uCprimeinfinT prime

infin

T primew Cprime

w

Boundary layer

vo

g

BoO

Porous medium

yprime

α

v

Fig 1 Schematic view of flow configuration

magnetic field is negligible It is also assumed that inabsence of electric field there is no applied voltage TheBoussinesqrsquos approximation have been adopted for theflow because the fluid has constant kinematic viscosityand constant thermal conductivityThe rheological equation of state for an isotropic and

incompressible Casson fluid33 is given by

= 0+lowast (1)

equivalently

ij =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2(B+

pyradic2

)eij gt c

2(B+

pyradic2c

)eij lt c

(2)

where is shear stress 0 is Casson yield stress isdynamic viscosity lowast is shear rate = eijeij and eij is thei jth component of deformation rate is the productbased on the non-Newtonian fluid c is a critical value ofthis product B is plastic dynamic viscosity of the non-Newtonian fluid

py =B

radic2

(3)

denote the yield stress of fluid Some fluids require a grad-ually increasing shear stress to maintain a constant strainrate and are called Rheopectic In case of Casson fluid(Non-Newtonian) flow where gt c

= B +pyradic2

(4)

Substituting Eq (3) into Eq (4) then the kinematic vis-cosity can be written as

=

= B

(1+ 1

)(5)

Finally is the Casson fluid parameter and as rarr thegoverning equations of the Casson fluid model ( )given by Eqs (6)ndash(8) become the governing equationsof the Newtonian fluid model ( rarr Under theassumptions made above the governing partial differen-tial equations34ndash39 for the fully developed magnetohydro-dynamic free convective heat and mass transfer flow ofa viscous incompressible electrically conducting viscousdissipative and chemically reactive Casson fluid areContinuity Equation

vprime

yprime= 0 (6)

Momentum Equation

vprimeuprime

yprime=

(1+

)2uprime

yprime2+g13cosT primeminusT prime

+g13lowastcos

timesC prime minusC primeminus

(B2

0

)uprime minus

(

K prime

)uprime (7)

948 J Nanofluids 6 946ndash955 2017




ARTICLE

Energy Equation

vprimeT prime

yprime=(

Cp

)2T prime

yprime2(8)

Species Diffusion Equation

vprimeC prime

yprime=D

2C prime

yprime2+DT

2T prime

yprime2(9)

together with initial and boundary conditions

uprime = 0T prime

yprime= minus q

C prime = C prime

w at yprime = 0

uprime rarr 0 T prime rarr T prime C prime rarr C prime

as yprime rarr

⎫⎪⎬⎪⎭ (10)

Equation (6) gives

vprime = Constant=minusvo say (11)

where vprime is the suction velocity on the surface in steadystate such that vo gt 0

For non-dimensional coupled partial differential equa-tions we have to introduce the following dimensionlessvariables

u= uprime

vo y = yprimevo

= voT

prime minusT prime

q

= C prime minusC prime

C primew minusC prime

M = B20

v2o K = K primev2o

2

Gr = 2g13q

v4o Gc = g13lowastC prime

w minusC prime

v3o

Pr = Cp

Sc =

D

Sr = DT q

voCprimew minusC prime

Re = voxprime

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(12)

The above defined non-dimensionless variables in Eq (12)into Eqs (6)ndash(8) and we get(

1+ 1

)2u

y2= u

y+(M + 1

K

)uminusGrcos

minusGccos (13)

2

y2= Pr

y(14)

2

y2= Sc

yminus ScSr

(2

y2

)(15)

with connected initial and boundary conditions

u= 0

y=minus1 = 1 at y = 0

urarr 0 rarr 0 rarr 0 as y rarr

⎫⎪⎬⎪⎭ (16)

For the design of chemical engineering systems andpractical engineering applications the local skin-frictionNusselt number and Sherwood number important phys-ical parameters for this type of boundary layer flow

The Skin-friction at the plate which in the non-dimensional form is given by

Cf = primew

vov=(u

y

)y=0

(17)

The rate of heat transfer coefficient which in the non-dimensional form in terms of the Nusselt number isgiven by

Nu=minusxprime Tprimeyprimeyprime=0

T primewminusT prime

rArr NuReminus1=minus(

y

)y=0

(18)

The rate of mass transfer coefficient which in the non-dimensional form in terms of the Sherwood number isgiven by

Sh=minusxprime Cprimeyprimeyprime=0


rArr ShReminus1=minus(

y

)y=0

(19)

3 NUMERICAL SOLUTIONS BY FINITEELEMENT METHOD

The finite element method (Bathe40 and Reddy41 is apowerful technique for solving ordinary or partial differ-ential equations The basic concept of FEM is that thewhole domain is divided into smaller elements of finitedimensions called Finite Elements This method is themost versatile numerical technique in engineering analysisand has been employed to study diverse problems in heattransfer42 fluid mechanics43 chemical processing44 rigidbody dynamics45 and many other fields The steps involvedin the finite element analysis are as follows

31 Finite Element Technique311 Finite Element DiscretizationThe whole domain is divided into a finite number of sub-domains which is called the discretization of the domainEach sub domain is called an element The collection ofelements is called the finite-element mesh

312 Generation of the Element Equations(i) From the mesh a typical element is isolated and thevariational formulation of the given problem over the typ-ical element is constructed(ii) An approximate solution of the variational problem isassumed and the element equations are made by substitut-ing this solution in the above system(iii) The element matrix which is called stiffness matrixis constructed by using the element interpolation functions

313 Assembly of the Element EquationsThe algebraic equations so obtained are assembled byimposing the inter element continuity conditions Thisyields a large number of algebraic equations known asthe global finite element model which governs the wholedomain





ARTICLE

314 Imposition of the Boundary ConditionsOn the assembled equations the Dirichletrsquos and Neumannboundary conditions (16) are imposed

315 Solution of Assembled EquationsThe assembled equations so obtained can be solved by anyof the numerical technique viz Gauss elimination methodLU Decomposition method etc

32 Variational FormulationThe variational form associated with Eqs (13)ndash(15) overa typical linear element ye ye+1 is given by

int ye+1

ye

w1

[(1+ 1

)2u

y2minus(u

y

)minusNu+Grcos

+Gccos]dy = 0 (20)

int ye+1

ye

w2

[2

y2minus Pr

(

y

)]dy = 0 (21)

int ye+1

ye

w3

[2

y2minus Sc

(

y

)+ ScSr

(2

y2

)]dy = 0

(22)

Where N = M + 1K and w1 w2 w3 are arbitrary testfunctions and may be viewed as the variation in u respectively After reducing the order of integration andnon-linearity we arrive at the following system of equationsint ye+1

ye

[(1+ 1

)(w1

y

)(u

y

)+ w1

(u

y

)+Nw1u

minus Grw1cosminus Gcw1cos]dy

minus[w1

(1+ 1

)(u

y

)]ye+1

ye

= 0 (23)

int ye+1

ye

[Prw2

(

y

)+(w2

y

)(

y

)]dy

minus[w2

(

y

)]ye+1

ye

= 0 (24)

int ye+1

ye

[Scw3

(

y

)+(w3

y

)(

y

)

minus ScSrw3

(w3

y

)(

y

)]dy

minus[w3

(

y

)+ SrScw3

(

y

)]ye+1

ye

= 0 (25)

33 Finite Element FormulationThe finite element model from Eqs (23)ndash(25) by substi-tuting finite element approximations of the form

u=2sum

j=1

uej

ej =

2sumj=1

ej

ej =

2sumj=1

ej

ej (26)

With w1 = w2 = w3 = ej i = 12 where ue

j ej and

ej are the velocity temperature and concentration respec-

tively at the jth node of typical eth element ye ye+1 andej are the shape functions for this element ye ye+1 and

are taken as

e1 =

ye+1minus y

ye+1minus yeand e

2 =yminus ye

ye+1minus ye ye le y le ye+1

(27)The finite element model of the equations for eth elementthus formed is given by

⎡⎢⎢⎣K11 K12 K13

K21 K22 K23

K31 K32 K33

⎤⎥⎥⎦

⎡⎢⎢⎣ue

e

e

⎤⎥⎥⎦

+

⎡⎢⎢⎣M11 M12 M13

M21 M22 M23

M31 M32 M33

⎤⎥⎥⎦

⎡⎢⎢⎣uprimee

primee

primee

⎤⎥⎥⎦=

⎡⎢⎢⎣b1e

b2e

b3e

⎤⎥⎥⎦

(28)

Where Kmn Mmn and ue e e uprimee primeeprimee and bme mn= 123 are the set of matricesof order 2times2 and 2times1 respectively and

prime(dash) indicates

ddy These matrices are defined as

K11ij =

(1+ 1

)int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

K12ij =N

int ye+1

ye

ei

ej dy M12

ij =M13ij =0

K13ij =minusGr+Gccos

int ye+1

ye

ei

ej dy

M11ij =

int ye+1

ye

ei

ej dy K21

ij =0

K22ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy M33

ij =int ye+1

ye

ei

ej dy

K23ij =0 M21

ij =M23ij =0 M22

ij =int ye+1

ye

ei

ej dy

M31ij =M32

ij =0 K31ij =0

K32ij =minusSrSc

int ye+1

ye

(e

i

y

)(e

j

y

)dy

K33ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

b1ei =

[e

i

(1+ 1

)(u

y

)]ye+1

ye

b2ei =

[e

i

(

y

)]ye+1

ye

b3ei =

[e

i

(

y

)+SrSce

i

(

y

)]ye+1

ye

Each element matrix is of the order 8times 8 The wholedomain is divided into 100 linear elements of equal size





ARTICLE

Table I Grid invariance test for velocity temperature and concentration profiles

Mesh (grid) size= 00001 Mesh (grid) size = 0001 Mesh (grid) size = 001

u u u

0000000000 0200000003 1000000000 0000000000 0200000003 1000000000 0000000000 0200000003 10000000000992632389 0102704406 0660844505 0992708445 0102720678 0660844505 0992783785 0102736823 06608445051002345443 0051813565 0436002851 1002463222 0051835183 0436002851 1002579331 0051856663 04360029100776130080 0025564646 0286837012 0776260316 0025585003 0286837220 0776388943 0025605224 02868373990544180691 0012279855 0187711954 0544303596 0012295937 0187712133 0544424891 0012311925 01877123420362010717 0005716458 0121621393 0362115055 0005727679 0121621534 0362218052 0005738844 01216216610231195688 0002564892 0077278197 0231276751 0002571937 0077278346 0231356651 0002578950 00772784950140206009 0001097695 0047179937 0140263006 0001101665 0047180071 0140319273 0001105621 00471802020077051550 0000433813 0026324280 0077086285 0000435752 0026324367 0077120528 0000437686 00263244600032428648 0000137244 0011355995 0032444123 0000137947 0011356040 0032459375 0000138648 00113560850000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000

Table II Comparison of present numerical results with analyticalresults of Sahu and Rajput26 for different values of Gr and Gc

Analytical AnalyticalPresent results of Present results of

numerical Sahu and numerical of Sahu andresults Rajput26 results Rajput26

Gr Cf Gc Cf

50 547023659 5474180 50 547023659 5474180100 957214339 9558880 100 743624885 7424620150 1444108624 14439100 150 946225634 9449300

Table III Comparison of present numerical results with analyticalresults of Sahu and Rajput26 for different values of Pr and M



Pr Cf M Cf

071 547023659 5474180 10 646718246 6468820300 210639741 2086700 20 547023659 5474180700 160443852 1599660 30 422821349 4222860

after assembly of all the elements equations we obtain amatrix of the order 404times404 This system of equations asobtained after assembly of the element equations is non-linear Therefore an iterative scheme must be utilized in

the solution After imposing the boundary conditions onlya system of 397 equations remains for the solution whichis solved by the Gauss elimination method maintaining anaccuracy of 00001

4 STUDY OF GRID INDEPENDENCE OFFINITE ELEMENT METHOD

In general we have to study the grid indepen-dencydependency how should the mesh size be varied inorder to check the solution at different mesh (grid) sizesand get a range at which there is no variation in the solu-tion The numerical values of velocity temperature andconcentration profiles for different values of mesh (grid)size are shown in the following Table I From this tablewe observed that variation of velocity temperature andconcentration profiles are nearer for various mesh (grid)size Hence we conclude that the computational resultsare stable and converge

5 VALIDATION OF THE MODELIn order to validate the method used in this study andto judge the accuracy of the present analysis the skin-friction and rate of mass transfer coefficients results havebeen compared with the exact solutions with the previousresults of Sahu and Rajput26 for several special cases andthe results are found to be in good agreement The resultsare shown in Tables IIndashIV

Table IV Comparison of present numerical results with analytical results of Sahu and Rajput26 for different values of Sr and Sc

Analytical results of Sahu Analytical results of SahuPresent numerical results and Rajput26 Present numerical results and Rajput26

Sr Cf Sh Shlowast Sc Cf Sh Shlowast

00 482729914 278164475 4821830 2742820 022 652433842 046399421 6525790 045714005 515685510 226695420 5146530 2259890 066 605301168 085613463 6025210 084403110 547023659 178562243 5474180 1780800 078 594298856 093601433 5942040 0936298

Notes Here Cf and Shmdashpresent numerical results and and Shlowastmdashanalytical results of Sahu and Rajput26





ARTICLE

0

06

12

0 3 6 9

u

y

Gr = 10 20 30 40

Fig 2 Influence of Gr on velocity profiles

6 RESULTS AND DISCUSSIONIn this section the influence of various pertinent parame-ters such as Grashof number for heat transfer Gr Grashofnumber for mass transfer Gc Permeability parameter KMagnetic field parameter M Prandtl number Pr Schmidtnumber Sc Soret number Sr Casson fluid parameter and Angle of inclination parameter are deliberated inFigures 2 to 11 for velocity temperature and concentra-tion profiles For the numerical calculations of the velocitytemperature and concentration profiles the values of thePrandtl number were chosen for mercury (Pr= 0025) airat 25 C and one atmospheric pressure (Pr= 071) water(Pr = 700) and water at 4 C (Pr = 1162) and the val-ues of Sc were chosen for the gases representing diffusingchemical species of most common interest in air namelyhydrogen (Sc = 022) helium (Sc = 030) water vapour(Sc= 060) and ammonia (Sc= 078) To find solution ofthis problem an infinite vertical plate was placed in a finitelength in the flow Hence the entire problem in a finiteboundary was solved However in the graphs a span wisestep distance y of 0001 is used with ymax = 9 The veloc-ity temperature and concentration tend to zero as y tend

0

06

12

0 3 6 9

u

y

Gc = 10 20 30 40

Fig 3 Influence of Gc on velocity profiles

0

06

12

0 3 6 9

u

y

M = 10 20 30 40

Fig 4 Influence of M on velocity profiles

to 9 This is true for any value of y Thus finite lengthwas considered in this study

61 From Figure 2Figure 2 illustrates the variation of velocity of the flowfield for different values of Grashof number The ther-mal Grashof number characterizes the relative effect of thethermal buoyancy force to the viscous hydrodynamic forcein the boundary layer flow Increase of Gr number leads toa rise in the values of velocity owing to the assistance ofthermal buoyancy force which induces a favourable pres-sure gradient This implies that thermal buoyancy forcetends to accelerate velocity The fluid velocity attains a dis-tinctive maximum value in a region near the plate surfaceand then decays to the free stream value

62 From Figure 3The influence of Grashof number for mass transfer is illus-trated in Figure 3 The Grashof number for mass transferdefines the ratio of the species buoyancy force to the vis-cous hydrodynamic force As expected the fluid velocityincreases and the peak value is more distinctive due to

0

06

12

0 3 6 9

u

y

K = 10 20 30 40

Fig 5 Influence of K on velocity profiles





ARTICLE

0

01

02

0 3 6 9

θ

y

Pr = 071 10 70 1162

Fig 6 Influence of Pr on temperature profiles

increase in the species buoyancy force The velocity distri-bution attains a distinctive maximum value in the vicinityof the plate and then decreases properly to approach thefree stream value It is noticed that the velocity increaseswith increasing values of the Grashof number for masstransfer

63 From Figure 4From Figure 4 it is observed that for higher value of Mag-netic field parameter M magnitude of velocity profile andboundary layer thickness reduces Because an increase inmagnetic field up rises the opposite force to the flow direc-tion which is called resistive-type force (Lorentz force)which reduces the velocity profiles

64 From Figure 5The effect of Permeability parameter is presented in theFigure 5 From this figure we observe that the veloc-ity is increases with increasing values of K Physicallythis result can be achieved when the holes of the porousmedium may be neglected

0

05

1

0 3 6 9y

Sc = 022 030 060 078

φ

Fig 7 Influence of Sc on concentration profiles

0

06

12

0 3 6 9

u

y

Sr = 10 20 30 40

Fig 8 Influence of Sr on velocity profiles

65 From Figure 6It is clear from Figure 6 that temperature profile reducesfor higher value of Prandtl number Pr The Prandtl numberPr is contrariwise connected with thermal diffusivity Anincrease in Prandtl number Pr corresponds to decrease thethermal diffusivity which causes temperature of the fluidto reduce

66 From Figure 7The effect of increasing values of Sc is presented inFigure 7 As Sc increases there is a decrease in molecu-lar diffusivity which results in reduction in concentrationboundary layer thickness Because of this fact the concen-tration profile decreases with increasing values of Sc

67 From Figures 8 and 9Figures 8 and 9 illustrate the effect of Soret number Sr onvelocity and concentration profiles The Soret term definesthe effect of temperature gradients on the concentrationfield From these graphs it is observed that an increasingSr causes a rise in the velocity and concentration profilesthroughout the boundary layer

0

05

1

0 3 6 9y

Sr = 10 20 30 40φ

Fig 9 Influence of Sr on concentration profiles





ARTICLE

0

06

12

0 3 6 9

u

y

α = 30ordm 45ordm 60ordm 90ordm

Fig 10 Influence of on velocity profiles

0

06

12

0 3 6 9

u

y

γ = 01 03 07 10


68 From Figure 10The effect of angle of inclination of the surface on thevelocity field has been illustrated in Figure 10 It is seenthat as the angle of inclination of the surface increases thevelocity field decreases

69 From Figure 11Figure 11 shows the effect of the Casson fluid param-eter on velocity profiles It is observed that the Cassonfluid parameter creates a resistive-type force in the fluidflow Consequently the magnitude of the velocity profileand boundary layer thickness reduces for higher values ofCasson fluid parameter

7 CONCLUSIONSAn analysis is done to solve the MHD flow of Cassonfluid model over a vertically inclined plate in presence ofconstant heat flux and Soret effects Finite element methodis applied to solve the governing nonlinear coupled partialdifferential equations The main findings of this problemare listed below

1 The Grashof number has an accelerating effect on theflow velocity due to the enhancement in the buoyancyforce2 The momentum boundary layer thickness decreases forlarge values of Casson fluid parameter3 Permeability of the porous medium tends to acceleratethe velocity of the fluid throughout the boundary layerregion4 The effect of magnetic field parameter reduces thevelocity profiles5 The velocity and concentration profiles are increaseswith the increase of Soret parameter6 Schmidt number has proclivity to decline the concen-tration profiles7 The present numerical results have good agreementwith the earlier study by Sahu and Rajput26

NomenclatureList of VariablesC prime

Concentration of the fluid far away from the plate(Kg mminus3

C primew Concentration of the plate (Kg mminus3y Dimensionless displacement (m)

T prime Fluid temperature away from the plate (K)u Non-dimensional fluid velocity (m sminus1K prime Permeability of the fluid (m2Sh The local Sherwood numbervprime Velocity component (m sminus1uprime Velocity component in xprime-direction (m sminus1vo Constant velocity at the plate (m sminus1xprime Coordinate axis along the plate (m)yprime Co-ordinate axis normal to the plate (m)C prime Fluid Concentration (Kg mminus3T prime Fluid Temperature (K)T primew Fluid temperature at the wall (K)

Gr Grashof number for heat transferM Magnetic field parameter (or) Hartmann numberDT Mass diffusivity (m2 sminus1K Permeability parameter (m2D Solute mass diffusivity (m2 sminus1Cp Specific heat at constant pressure (J Kgminus1K)Nu The local Nusselt numberCf The local skin-friction (N mminus2B0 Uniform magnetic field (Tesla)Gc Grashof number for mass transferSr Soret numberg Acceleration of gravity 981 (m sminus2Pr Prandtl numberq Rate of heat transfer

Re Reynolds numberSc Schmidt number

Greek Symbols Thermal conductivity of the fluid (W mminus1Kminus1 Non-dimensional fluid Concentration (Kg mminus3





ARTICLE

Non-dimensional fluid temperature (K) primew Shear stress (N mminus213lowast Volumetric Coefficient of thermal expansion with

concentration (m3 Kgminus1 Angle of inclination of plate (degrees) Casson fluid parameter Electric conductivity of the fluid (s mminus1 Kinematic viscosity (m2 sminus1 Species concentration (Kg mminus3 The constant density (Kg mminus313 Volumetric coefficient of thermal expansion (Kminus1

Superscriptsprime Dimensionless Properties

Subscriptsw Conditions on the wall Free stream conditionsp Plate

References and Notes1 N Casson Rheology of Disperse Systems Pergamon New York

(1959) p 842 A Khalid I Khan A Khan and S Shafie Eng Sci Technol Int J

18 309 (2015)3 S Mohyud-Din and I Khan Aerosp Sci Technol 48 186

(2016)4 F Abbasi and S Shehzad J Mol Liq 220 848 (2016)5 T Hayat T Muhammad S Shehzad and A Alsaedi Adv Powder

Technol 27 504 (2016)6 F Abbasi S Shehzad T Hayat and B Ahmad J Magn Magn

Mater 404 159 (2016)7 R Srinivasa Raju Int J Eng Appl Sci 8 119 (2016)8 D Ramya R Srinivasa Raju J Anand Rao and M M Rashidi Int

J Nanosci Nanotech 12 251 (2016)9 D Ramya R Srinivasa Raju and J Anand Rao J Nanofluids 5 880

(2016)10 D Ramya R Srinivasa Raju and J Anand Rao J Nanofluids 6 541

(2017)11 I Shravani D Ramya and J Sucharitha J Nanofluids 6 567

(2017)12 S V Sailaja B Shanker and R Srinivasa Raju J Nanofluids 6 420

(2017)13 R Srinivasa Raju B Mahesh Reddy and G Jithender Reddy

J Nanofluids 6 1 (2017)14 V Nagendramma and A Leelaratnam J Nanofluids 6 410

(2017)15 M Gnaneswara Reddy J Manjula and P Padma J Nanofluids

6 436 (2017)16 G Mahanta M Das and S Shaw J Nanofluids 6 282 (2017)17 M Gnaneswara Reddy P Padma and G Rama Subba Reddy

J Nanofluids 6 71 (2017)18 P D Shima and J Philip Ind Eng Chem Res 53 980 (2014)19 S A Angayarkanni and J Philip J Appl Phys 118 094306

(2015)

20 J Buongiorno David C Venerus Naveen Prabhat T McKrellJ Townsend R Christianson Y V Tolmachev P Keblinski L-WHu J L Alvarado I C Bang S W Bishnoi M Bonetti F BotzA Cecere Y Chang G Chen H Chen S J Chung M K ChyuS K Das R D Paola Y Ding F Dubois G Dzido J EapenW Escher D Funfschilling Q Galand J Gao P E GharagozlooK E Goodson J G Gutierrez H Hong M Horton K S HwangC S Iorio S P Jang A B Jarzebski Y Jiang L Jin S KabelacAravind Kamath M A Kedzierski L G Kieng C Kim J-H KimS Kim S H Lee K C Leong I Manna B Michel R Ni H EPatel J Philip D Poulikakos C Reynaud R Savino P K SinghP Song Thirumalachari Sundararajan E Timofeeva T TritcakA N Turanov S Van Vaerenbergh D Wen Sanjeeva WitharanaC Yang W-H Yeh X-Z Zhao and S-Q Zhou J Appl Phys106 094312 (2009)

21 J Philip and A Angayarkanni Adv Colloid Interface Sci 225 146(2015)

22 E R G Eckert and R M Drake Analysis of Heat and Mass Trans-fer McGraw-Hill New York (1972)

23 J Anand Rao and R Srinivasa Raju J Ener Heat Mass Transf33 351 (2011)

24 S Siva Reddy and R Srinivasa Raju Int J Comput Methods EngSci Mech 16 132 (2015)

25 R S Raju K Sudhakar and M Rangamma J Inst Eng (India)Series C 94 175 (2013)

26 P K Sahu and U S Rajput Int J Math Sci Comput 3 44 (2013)27 R Srinivasa Raju J Inst Engineers (India) Series C 97 505

(2016)28 R Srinivasa Raju G Jithender Reddy J Anand Rao M M Rashidi

and Rama Subba Reddy Gorla Int J Thermal Sci 107 303 (2016)29 R Srinivasa Raju B Mahesh Reddy M M Rashidi and R S R

Gorla J Porous Media 19 701 (2016)30 Ul Haq Sami I Khan F Ali and S Shafie J Phys Soc Japan

81 Article ID 044401 (2012)31 N Ahmad J Heat Transf 134 062701 (2012)32 P A Lakshmi Narayana and P V S N Murthy J Heat Transf

130 104504-1 (2008)33 R K Dash K N Mehta and G Jayaraman Int J Eng Sci

34 1145 (1996)34 W P Walwander T Y Chen and D F Cala Biorheology 12 111

(1975)35 S Mukhopadhyay Prativa Ranjan De K Bhattacharyya and G C

Layek Ain Shams Eng J 4 933 (2013)36 K Bhattacharyya Frontiers in Heat and Mass Transf 4 023003

(2013)37 T Hayat S A Shehzad A Alsaedi and M S Alhothuali Chin

Phys Lett 29 114704 (2012)38 K Bhattacharyya T Hayat and A Alsaedi Chin Phys B

22 024702 (2013)39 J Boyd J M Buick and S Green Phys Fluids 19 93 (2007)40 K J Bathe Finite Element Procedures Prentice-Hall New Jersey

(1996)41 J N Reddy An Introduction to the Finite Element Method

McGraw-Hill New York (1985)42 R Bhargava and P Rana Int J Appl Math Mech 7 29 (2011)43 Y Y Lin and S P Lo J Mat Proc Tech 140 646 (2003)44 W Dettmer and D Peric Comput Methods Appl Mech Eng

195 1633 (2006)45 A Hansbo and P Hansbo Comput Methods Appl Mech Eng

139 3523 (2004)





ARTICLE

suctioninjection studied by Ramya et al10 The steady two-dimensional magnetohydrodynamic boundary layer flow ofviscous nano fluids over a nonlinear stretching sheet wasinvestigated numerically by Shravani et al11 Sailaja et al12

discussed the effects of double diffusive on magnetohy-drodynamic mixed convection flow towards a verticallyinclined plate embedded in porous medium in presence ofbiot number using finite element method Srinivasa Rajuet al13 studied the influence of angle of inclination onunsteady magnetohydrodynamic Casson fluid flow past avertical surface filled by porous medium in presence ofconstant heat flux chemical reaction and viscous dissi-pation by applying finite element method Nagendrammaand Leelaratnam14 reported the results of the steady MHDheat and mass transfer of a Casson nanofluid flow embed-ded in a porous medium with variable thermal con-ductivity velocity slip and suctionblowing parametersthrough a horizontal cylinder using Runge-Kutta fourthorder method with shooting technique Gnaneswara Reddyet al15 three-dimensional magneto hydrodynamic Cassonnanofluid flow heat and mass transfer over a stretchingsheet with thermal radiation heat sourcesink second-order velocity slip and double stratification using finitedifference Keller box technique Unsteady hydromagneticheat and mass transfer flow of a chemically reactiveCasson fluid past a stretching sheet with convective bound-ary condition was investigated by Mahanta et al16 byusing matlab bvp4c package Gnaneswara Reddy et al17

studied three-dimensional magneto hydrodynamic Cassonnanofluid flow and heat transfer past a stretching sheetwith thermal radiation heat sourcesink and double strat-ification by adapting the efficient finite difference Kellerbox technique Shima and Philip18 investigated the role ofthermal conductivity of dispersed nanomaterial on the ther-mal and rheological properties of metal and metal oxidenanofluids with average particles size stabilized with amonolayer of surfactant Angayarkanni and Philip19 stud-ied the thermal properties and internal microstructures ofn-hexadecane alkane containing nano-inclusions of coppernano-wire multi walled carbon nanotube and graphenenano-platelets of different volume fractions Buongiornoet al20 reported on the international nanofluid propertybenchmark exercise or inpbe in which the thermal con-ductivity of identical samples of colloidally stable dis-persions of nanoparticles or ldquonanofluidsrdquo was measuredby over 30 organizations worldwide using a variety ofexperimental methods The role of brownian motion inter-facial resistance morphology of suspended nanoparticlesand aggregating behavior were investigated both experi-mentally and theoretically by Philip Angayarkanni21

In all of the above-mentioned studies the case of thethermal-diffusion effect (Soret) has received little atten-tion An experimental investigation of this effect was firstperformed by Charles Soret in 1879 When mass trans-fer occurs simultaneously in a moving fluid the relation-ship between the fluxes and the driving potentials are of

a more integrate nature It has been observed that massflux (mass transfer per unit time and per unit area per-pendicular to the direction of transfer) can be generatednot only by the concentration gradients but also by thetemperature gradients Mass fluxes influenced by tem-perature gradients are known as the Soret (thermal dif-fusion) effect The Soret effect for example has beenutilized for isotope separation and in mixtures betweengases with very light molecular weight like Hydrogen andHelium The role of transportation of mass in control-ling temperature of a heated body is found to be moreeconomical than any other method Following Eckert andDrakersquos22 work several other investigators have carriedout model studies of the Soret effect in different types ofheat and mass transfer problems Anand Rao and Srini-vasa Raju23 found the numerical solutions of MHD flowand heat transfer along a flat plate embedded in porousmedium in presence of hall current Soret and Dufoureffects using finite element method Sheri and Raju24 stud-ied the influence of Soret on an unsteady magnetohydro-dynamics free convective flow past a semi-infinite verticalplate in the presence viscous dissipation The results ofthermal radiation and heat source on an unsteady MHDfree convective fluid flow over an infinite vertical platein occurrence of thermal diffusion and diffusion thermowere discussed by Raju et al25 Sahu and Rajput26 stud-ied the combined effects of thermal diffusion and chem-ical reaction on the steady free convection MHD flowthrough a porous medium bounded by an infinite verticalsurface with constant heat flux using two-term perturba-tion method Srinivasa Raju27 studied the combined effectsof thermal-diffusion and diffusion-thermo on unsteady freeconvection fluid flow past an infinite vertical porous platein presence of magnetic field and chemical reaction usingfinite element technique Srinivasa Raju et al28 found bothanalytical and numerical solutions of unsteady magneto-hydrodynamic free convective flow past an exponentiallymoving vertical plate with heat absorption and chemicalreaction Srinivasa Raju et al29 studied the application offinite element method to unsteady MHD free convectionflow past a vertically inclined porous plate including ther-mal diffusion and diffusion thermo effects Sami et al30

studied the MHD free convection flow in a porous mediumwith thermal diffusion and ramped wall temperature Theyobtained exact dimensionless solutions of momentum andenergy equations under Boussinesqrsquos approximation usingthe Laplace transforms Ahmad31 studied MHD transientfree convection and mass transfer flow of a viscous incom-pressible and electrically conducting fluid in the presenceof thermal diffusion and thermal radiation He obtainedexact solutions for velocity temperature and concentrationusing the Laplace transform method Lakshmi Narayanaand Murthy32 studied the Soret and Dufour effects on freeconvection heat and mass transfer from a horizontal flatplate in a Darcy porous medium





ARTICLE






xprime

uCprimeinfinT prime

infin

T primew Cprime

w

Boundary layer

vo

g

BoO

Porous medium

yprime

α

v




= 0+lowast (1)

equivalently

ij =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2(B+

pyradic2

)eij gt c

2(B+

pyradic2c

)eij lt c

(2)


py =B

radic2

(3)


= B +pyradic2

(4)


=

= B

(1+ 1

)(5)


vprime

yprime= 0 (6)

Momentum Equation

vprimeuprime

yprime=

(1+

)2uprime


+g13lowastcos


(B2

0

)uprime minus

(

K prime

)uprime (7)





ARTICLE

Energy Equation

vprimeT prime

yprime=(

Cp

)2T prime

yprime2(8)


vprimeC prime

yprime=D

2C prime

yprime2+DT

2T prime

yprime2(9)


uprime = 0T prime

yprime= minus q

C prime = C prime

w at yprime = 0


as yprime rarr

⎫⎪⎬⎪⎭ (10)

Equation (6) gives




u= uprime

vo y = yprimevo

= voT

prime minusT prime

q



M = B20

v2o K = K primev2o

2

Gr = 2g13q


w minusC prime

v3o

Pr = Cp

Sc =

D

Sr = DT q


Re = voxprime

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(12)


1+ 1

)2u

y2= u

y+(M + 1

K

)uminusGrcos

minusGccos (13)

2

y2= Pr

y(14)

2

y2= Sc

yminus ScSr

(2

y2

)(15)


u= 0



⎫⎪⎬⎪⎭ (16)



Cf = primew

vov=(u

y

)y=0

(17)





y

)y=0

(18)





y

)y=0

(19)










ARTICLE




int ye+1

ye

w1

[(1+ 1

)2u

y2minus(u

y

)minusNu+Grcos

+Gccos]dy = 0 (20)

int ye+1

ye

w2

[2

y2minus Pr

(

y

)]dy = 0 (21)

int ye+1

ye

w3

[2

y2minus Sc

(

y

)+ ScSr

(2

y2

)]dy = 0

(22)


ye

[(1+ 1

)(w1

y

)(u

y

)+ w1

(u

y

)+Nw1u


minus[w1

(1+ 1

)(u

y

)]ye+1

ye

= 0 (23)

int ye+1

ye

[Prw2

(

y

)+(w2

y

)(

y

)]dy

minus[w2

(

y

)]ye+1

ye

= 0 (24)

int ye+1

ye

[Scw3

(

y

)+(w3

y

)(

y

)

minus ScSrw3

(w3

y

)(

y

)]dy

minus[w3

(

y

)+ SrScw3

(

y

)]ye+1

ye

= 0 (25)


u=2sum

j=1

uej

ej =

2sumj=1

ej

ej =

2sumj=1

ej

ej (26)


j ej and



are taken as

e1 =

ye+1minus y

ye+1minus yeand e

2 =yminus ye



⎡⎢⎢⎣K11 K12 K13

K21 K22 K23

K31 K32 K33

⎤⎥⎥⎦

⎡⎢⎢⎣ue

e

e

⎤⎥⎥⎦

+

⎡⎢⎢⎣M11 M12 M13

M21 M22 M23

M31 M32 M33

⎤⎥⎥⎦

⎡⎢⎢⎣uprimee

primee

primee

⎤⎥⎥⎦=

⎡⎢⎢⎣b1e

b2e

b3e

⎤⎥⎥⎦

(28)




K11ij =

(1+ 1

)int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

K12ij =N

int ye+1

ye

ei

ej dy M12

ij =M13ij =0


int ye+1

ye

ei

ej dy

M11ij =

int ye+1

ye

ei

ej dy K21

ij =0

K22ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy M33

ij =int ye+1

ye

ei

ej dy

K23ij =0 M21

ij =M23ij =0 M22

ij =int ye+1

ye

ei

ej dy

M31ij =M32

ij =0 K31ij =0

K32ij =minusSrSc

int ye+1

ye

(e

i

y

)(e

j

y

)dy

K33ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

b1ei =

[e

i

(1+ 1

)(u

y

)]ye+1

ye

b2ei =

[e

i

(

y

)]ye+1

ye

b3ei =

[e

i

(

y

)+SrSce

i

(

y

)]ye+1

ye






ARTICLE



u u u

0000000000 0200000003 1000000000 0000000000 0200000003 1000000000 0000000000 0200000003 10000000000992632389 0102704406 0660844505 0992708445 0102720678 0660844505 0992783785 0102736823 06608445051002345443 0051813565 0436002851 1002463222 0051835183 0436002851 1002579331 0051856663 04360029100776130080 0025564646 0286837012 0776260316 0025585003 0286837220 0776388943 0025605224 02868373990544180691 0012279855 0187711954 0544303596 0012295937 0187712133 0544424891 0012311925 01877123420362010717 0005716458 0121621393 0362115055 0005727679 0121621534 0362218052 0005738844 01216216610231195688 0002564892 0077278197 0231276751 0002571937 0077278346 0231356651 0002578950 00772784950140206009 0001097695 0047179937 0140263006 0001101665 0047180071 0140319273 0001105621 00471802020077051550 0000433813 0026324280 0077086285 0000435752 0026324367 0077120528 0000437686 00263244600032428648 0000137244 0011355995 0032444123 0000137947 0011356040 0032459375 0000138648 00113560850000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000




Gr Cf Gc Cf

50 547023659 5474180 50 547023659 5474180100 957214339 9558880 100 743624885 7424620150 1444108624 14439100 150 946225634 9449300




Pr Cf M Cf

071 547023659 5474180 10 646718246 6468820300 210639741 2086700 20 547023659 5474180700 160443852 1599660 30 422821349 4222860









00 482729914 278164475 4821830 2742820 022 652433842 046399421 6525790 045714005 515685510 226695420 5146530 2259890 066 605301168 085613463 6025210 084403110 547023659 178562243 5474180 1780800 078 594298856 093601433 5942040 0936298






ARTICLE

0

06

12

0 3 6 9

u

y

Gr = 10 20 30 40



0

06

12

0 3 6 9

u

y

Gc = 10 20 30 40


0

06

12

0 3 6 9

u

y

M = 10 20 30 40





0

06

12

0 3 6 9

u

y

K = 10 20 30 40






ARTICLE

0

01

02

0 3 6 9

θ

y

Pr = 071 10 70 1162





0

05

1

0 3 6 9y

Sc = 022 030 060 078

φ


0

06

12

0 3 6 9

u

y

Sr = 10 20 30 40





0

05

1

0 3 6 9y

Sr = 10 20 30 40φ






ARTICLE

0

06

12

0 3 6 9

u

y



0

06

12

0 3 6 9

u

y

γ = 01 03 07 10

















ARTICLE




















(2015)






















139 3523 (2004)





ARTICLE






xprime

uCprimeinfinT prime

infin

T primew Cprime

w

Boundary layer

vo

g

BoO

Porous medium

yprime

α

v




= 0+lowast (1)

equivalently

ij =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2(B+

pyradic2

)eij gt c

2(B+

pyradic2c

)eij lt c

(2)


py =B

radic2

(3)


= B +pyradic2

(4)


=

= B

(1+ 1

)(5)


vprime

yprime= 0 (6)

Momentum Equation

vprimeuprime

yprime=

(1+

)2uprime


+g13lowastcos


(B2

0

)uprime minus

(

K prime

)uprime (7)





ARTICLE

Energy Equation

vprimeT prime

yprime=(

Cp

)2T prime

yprime2(8)


vprimeC prime

yprime=D

2C prime

yprime2+DT

2T prime

yprime2(9)


uprime = 0T prime

yprime= minus q

C prime = C prime

w at yprime = 0


as yprime rarr

⎫⎪⎬⎪⎭ (10)

Equation (6) gives




u= uprime

vo y = yprimevo

= voT

prime minusT prime

q



M = B20

v2o K = K primev2o

2

Gr = 2g13q


w minusC prime

v3o

Pr = Cp

Sc =

D

Sr = DT q


Re = voxprime

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(12)


1+ 1

)2u

y2= u

y+(M + 1

K

)uminusGrcos

minusGccos (13)

2

y2= Pr

y(14)

2

y2= Sc

yminus ScSr

(2

y2

)(15)


u= 0



⎫⎪⎬⎪⎭ (16)



Cf = primew

vov=(u

y

)y=0

(17)





y

)y=0

(18)





y

)y=0

(19)










ARTICLE




int ye+1

ye

w1

[(1+ 1

)2u

y2minus(u

y

)minusNu+Grcos

+Gccos]dy = 0 (20)

int ye+1

ye

w2

[2

y2minus Pr

(

y

)]dy = 0 (21)

int ye+1

ye

w3

[2

y2minus Sc

(

y

)+ ScSr

(2

y2

)]dy = 0

(22)


ye

[(1+ 1

)(w1

y

)(u

y

)+ w1

(u

y

)+Nw1u


minus[w1

(1+ 1

)(u

y

)]ye+1

ye

= 0 (23)

int ye+1

ye

[Prw2

(

y

)+(w2

y

)(

y

)]dy

minus[w2

(

y

)]ye+1

ye

= 0 (24)

int ye+1

ye

[Scw3

(

y

)+(w3

y

)(

y

)

minus ScSrw3

(w3

y

)(

y

)]dy

minus[w3

(

y

)+ SrScw3

(

y

)]ye+1

ye

= 0 (25)


u=2sum

j=1

uej

ej =

2sumj=1

ej

ej =

2sumj=1

ej

ej (26)


j ej and



are taken as

e1 =

ye+1minus y

ye+1minus yeand e

2 =yminus ye



⎡⎢⎢⎣K11 K12 K13

K21 K22 K23

K31 K32 K33

⎤⎥⎥⎦

⎡⎢⎢⎣ue

e

e

⎤⎥⎥⎦

+

⎡⎢⎢⎣M11 M12 M13

M21 M22 M23

M31 M32 M33

⎤⎥⎥⎦

⎡⎢⎢⎣uprimee

primee

primee

⎤⎥⎥⎦=

⎡⎢⎢⎣b1e

b2e

b3e

⎤⎥⎥⎦

(28)




K11ij =

(1+ 1

)int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

K12ij =N

int ye+1

ye

ei

ej dy M12

ij =M13ij =0


int ye+1

ye

ei

ej dy

M11ij =

int ye+1

ye

ei

ej dy K21

ij =0

K22ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy M33

ij =int ye+1

ye

ei

ej dy

K23ij =0 M21

ij =M23ij =0 M22

ij =int ye+1

ye

ei

ej dy

M31ij =M32

ij =0 K31ij =0

K32ij =minusSrSc

int ye+1

ye

(e

i

y

)(e

j

y

)dy

K33ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

b1ei =

[e

i

(1+ 1

)(u

y

)]ye+1

ye

b2ei =

[e

i

(

y

)]ye+1

ye

b3ei =

[e

i

(

y

)+SrSce

i

(

y

)]ye+1

ye






ARTICLE



u u u

0000000000 0200000003 1000000000 0000000000 0200000003 1000000000 0000000000 0200000003 10000000000992632389 0102704406 0660844505 0992708445 0102720678 0660844505 0992783785 0102736823 06608445051002345443 0051813565 0436002851 1002463222 0051835183 0436002851 1002579331 0051856663 04360029100776130080 0025564646 0286837012 0776260316 0025585003 0286837220 0776388943 0025605224 02868373990544180691 0012279855 0187711954 0544303596 0012295937 0187712133 0544424891 0012311925 01877123420362010717 0005716458 0121621393 0362115055 0005727679 0121621534 0362218052 0005738844 01216216610231195688 0002564892 0077278197 0231276751 0002571937 0077278346 0231356651 0002578950 00772784950140206009 0001097695 0047179937 0140263006 0001101665 0047180071 0140319273 0001105621 00471802020077051550 0000433813 0026324280 0077086285 0000435752 0026324367 0077120528 0000437686 00263244600032428648 0000137244 0011355995 0032444123 0000137947 0011356040 0032459375 0000138648 00113560850000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000




Gr Cf Gc Cf

50 547023659 5474180 50 547023659 5474180100 957214339 9558880 100 743624885 7424620150 1444108624 14439100 150 946225634 9449300




Pr Cf M Cf

071 547023659 5474180 10 646718246 6468820300 210639741 2086700 20 547023659 5474180700 160443852 1599660 30 422821349 4222860









00 482729914 278164475 4821830 2742820 022 652433842 046399421 6525790 045714005 515685510 226695420 5146530 2259890 066 605301168 085613463 6025210 084403110 547023659 178562243 5474180 1780800 078 594298856 093601433 5942040 0936298






ARTICLE

0

06

12

0 3 6 9

u

y

Gr = 10 20 30 40



0

06

12

0 3 6 9

u

y

Gc = 10 20 30 40


0

06

12

0 3 6 9

u

y

M = 10 20 30 40





0

06

12

0 3 6 9

u

y

K = 10 20 30 40






ARTICLE

0

01

02

0 3 6 9

θ

y

Pr = 071 10 70 1162





0

05

1

0 3 6 9y

Sc = 022 030 060 078

φ


0

06

12

0 3 6 9

u

y

Sr = 10 20 30 40





0

05

1

0 3 6 9y

Sr = 10 20 30 40φ






ARTICLE

0

06

12

0 3 6 9

u

y



0

06

12

0 3 6 9

u

y

γ = 01 03 07 10

















ARTICLE




















(2015)






















139 3523 (2004)





ARTICLE

Energy Equation

vprimeT prime

yprime=(

Cp

)2T prime

yprime2(8)


vprimeC prime

yprime=D

2C prime

yprime2+DT

2T prime

yprime2(9)


uprime = 0T prime

yprime= minus q

C prime = C prime

w at yprime = 0


as yprime rarr

⎫⎪⎬⎪⎭ (10)

Equation (6) gives




u= uprime

vo y = yprimevo

= voT

prime minusT prime

q



M = B20

v2o K = K primev2o

2

Gr = 2g13q


w minusC prime

v3o

Pr = Cp

Sc =

D

Sr = DT q


Re = voxprime

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(12)


1+ 1

)2u

y2= u

y+(M + 1

K

)uminusGrcos

minusGccos (13)

2

y2= Pr

y(14)

2

y2= Sc

yminus ScSr

(2

y2

)(15)


u= 0



⎫⎪⎬⎪⎭ (16)



Cf = primew

vov=(u

y

)y=0

(17)





y

)y=0

(18)





y

)y=0

(19)










ARTICLE




int ye+1

ye

w1

[(1+ 1

)2u

y2minus(u

y

)minusNu+Grcos

+Gccos]dy = 0 (20)

int ye+1

ye

w2

[2

y2minus Pr

(

y

)]dy = 0 (21)

int ye+1

ye

w3

[2

y2minus Sc

(

y

)+ ScSr

(2

y2

)]dy = 0

(22)


ye

[(1+ 1

)(w1

y

)(u

y

)+ w1

(u

y

)+Nw1u


minus[w1

(1+ 1

)(u

y

)]ye+1

ye

= 0 (23)

int ye+1

ye

[Prw2

(

y

)+(w2

y

)(

y

)]dy

minus[w2

(

y

)]ye+1

ye

= 0 (24)

int ye+1

ye

[Scw3

(

y

)+(w3

y

)(

y

)

minus ScSrw3

(w3

y

)(

y

)]dy

minus[w3

(

y

)+ SrScw3

(

y

)]ye+1

ye

= 0 (25)


u=2sum

j=1

uej

ej =

2sumj=1

ej

ej =

2sumj=1

ej

ej (26)


j ej and



are taken as

e1 =

ye+1minus y

ye+1minus yeand e

2 =yminus ye



⎡⎢⎢⎣K11 K12 K13

K21 K22 K23

K31 K32 K33

⎤⎥⎥⎦

⎡⎢⎢⎣ue

e

e

⎤⎥⎥⎦

+

⎡⎢⎢⎣M11 M12 M13

M21 M22 M23

M31 M32 M33

⎤⎥⎥⎦

⎡⎢⎢⎣uprimee

primee

primee

⎤⎥⎥⎦=

⎡⎢⎢⎣b1e

b2e

b3e

⎤⎥⎥⎦

(28)




K11ij =

(1+ 1

)int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

K12ij =N

int ye+1

ye

ei

ej dy M12

ij =M13ij =0


int ye+1

ye

ei

ej dy

M11ij =

int ye+1

ye

ei

ej dy K21

ij =0

K22ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy M33

ij =int ye+1

ye

ei

ej dy

K23ij =0 M21

ij =M23ij =0 M22

ij =int ye+1

ye

ei

ej dy

M31ij =M32

ij =0 K31ij =0

K32ij =minusSrSc

int ye+1

ye

(e

i

y

)(e

j

y

)dy

K33ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

b1ei =

[e

i

(1+ 1

)(u

y

)]ye+1

ye

b2ei =

[e

i

(

y

)]ye+1

ye

b3ei =

[e

i

(

y

)+SrSce

i

(

y

)]ye+1

ye






ARTICLE



u u u

0000000000 0200000003 1000000000 0000000000 0200000003 1000000000 0000000000 0200000003 10000000000992632389 0102704406 0660844505 0992708445 0102720678 0660844505 0992783785 0102736823 06608445051002345443 0051813565 0436002851 1002463222 0051835183 0436002851 1002579331 0051856663 04360029100776130080 0025564646 0286837012 0776260316 0025585003 0286837220 0776388943 0025605224 02868373990544180691 0012279855 0187711954 0544303596 0012295937 0187712133 0544424891 0012311925 01877123420362010717 0005716458 0121621393 0362115055 0005727679 0121621534 0362218052 0005738844 01216216610231195688 0002564892 0077278197 0231276751 0002571937 0077278346 0231356651 0002578950 00772784950140206009 0001097695 0047179937 0140263006 0001101665 0047180071 0140319273 0001105621 00471802020077051550 0000433813 0026324280 0077086285 0000435752 0026324367 0077120528 0000437686 00263244600032428648 0000137244 0011355995 0032444123 0000137947 0011356040 0032459375 0000138648 00113560850000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000




Gr Cf Gc Cf

50 547023659 5474180 50 547023659 5474180100 957214339 9558880 100 743624885 7424620150 1444108624 14439100 150 946225634 9449300




Pr Cf M Cf

071 547023659 5474180 10 646718246 6468820300 210639741 2086700 20 547023659 5474180700 160443852 1599660 30 422821349 4222860









00 482729914 278164475 4821830 2742820 022 652433842 046399421 6525790 045714005 515685510 226695420 5146530 2259890 066 605301168 085613463 6025210 084403110 547023659 178562243 5474180 1780800 078 594298856 093601433 5942040 0936298






ARTICLE

0

06

12

0 3 6 9

u

y

Gr = 10 20 30 40



0

06

12

0 3 6 9

u

y

Gc = 10 20 30 40


0

06

12

0 3 6 9

u

y

M = 10 20 30 40





0

06

12

0 3 6 9

u

y

K = 10 20 30 40






ARTICLE

0

01

02

0 3 6 9

θ

y

Pr = 071 10 70 1162





0

05

1

0 3 6 9y

Sc = 022 030 060 078

φ


0

06

12

0 3 6 9

u

y

Sr = 10 20 30 40





0

05

1

0 3 6 9y

Sr = 10 20 30 40φ






ARTICLE

0

06

12

0 3 6 9

u

y



0

06

12

0 3 6 9

u

y

γ = 01 03 07 10

















ARTICLE




















(2015)






















139 3523 (2004)





ARTICLE




int ye+1

ye

w1

[(1+ 1

)2u

y2minus(u

y

)minusNu+Grcos

+Gccos]dy = 0 (20)

int ye+1

ye

w2

[2

y2minus Pr

(

y

)]dy = 0 (21)

int ye+1

ye

w3

[2

y2minus Sc

(

y

)+ ScSr

(2

y2

)]dy = 0

(22)


ye

[(1+ 1

)(w1

y

)(u

y

)+ w1

(u

y

)+Nw1u


minus[w1

(1+ 1

)(u

y

)]ye+1

ye

= 0 (23)

int ye+1

ye

[Prw2

(

y

)+(w2

y

)(

y

)]dy

minus[w2

(

y

)]ye+1

ye

= 0 (24)

int ye+1

ye

[Scw3

(

y

)+(w3

y

)(

y

)

minus ScSrw3

(w3

y

)(

y

)]dy

minus[w3

(

y

)+ SrScw3

(

y

)]ye+1

ye

= 0 (25)


u=2sum

j=1

uej

ej =

2sumj=1

ej

ej =

2sumj=1

ej

ej (26)


j ej and



are taken as

e1 =

ye+1minus y

ye+1minus yeand e

2 =yminus ye



⎡⎢⎢⎣K11 K12 K13

K21 K22 K23

K31 K32 K33

⎤⎥⎥⎦

⎡⎢⎢⎣ue

e

e

⎤⎥⎥⎦

+

⎡⎢⎢⎣M11 M12 M13

M21 M22 M23

M31 M32 M33

⎤⎥⎥⎦

⎡⎢⎢⎣uprimee

primee

primee

⎤⎥⎥⎦=

⎡⎢⎢⎣b1e

b2e

b3e

⎤⎥⎥⎦

(28)




K11ij =

(1+ 1

)int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

K12ij =N

int ye+1

ye

ei

ej dy M12

ij =M13ij =0


int ye+1

ye

ei

ej dy

M11ij =

int ye+1

ye

ei

ej dy K21

ij =0

K22ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy M33

ij =int ye+1

ye

ei

ej dy

K23ij =0 M21

ij =M23ij =0 M22

ij =int ye+1

ye

ei

ej dy

M31ij =M32

ij =0 K31ij =0

K32ij =minusSrSc

int ye+1

ye

(e

i

y

)(e

j

y

)dy

K33ij =

int ye+1

ye

[(e

i

y

)(e

j

y

)]dy

b1ei =

[e

i

(1+ 1

)(u

y

)]ye+1

ye

b2ei =

[e

i

(

y

)]ye+1

ye

b3ei =

[e

i

(

y

)+SrSce

i

(

y

)]ye+1

ye






ARTICLE



u u u

0000000000 0200000003 1000000000 0000000000 0200000003 1000000000 0000000000 0200000003 10000000000992632389 0102704406 0660844505 0992708445 0102720678 0660844505 0992783785 0102736823 06608445051002345443 0051813565 0436002851 1002463222 0051835183 0436002851 1002579331 0051856663 04360029100776130080 0025564646 0286837012 0776260316 0025585003 0286837220 0776388943 0025605224 02868373990544180691 0012279855 0187711954 0544303596 0012295937 0187712133 0544424891 0012311925 01877123420362010717 0005716458 0121621393 0362115055 0005727679 0121621534 0362218052 0005738844 01216216610231195688 0002564892 0077278197 0231276751 0002571937 0077278346 0231356651 0002578950 00772784950140206009 0001097695 0047179937 0140263006 0001101665 0047180071 0140319273 0001105621 00471802020077051550 0000433813 0026324280 0077086285 0000435752 0026324367 0077120528 0000437686 00263244600032428648 0000137244 0011355995 0032444123 0000137947 0011356040 0032459375 0000138648 00113560850000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000




Gr Cf Gc Cf

50 547023659 5474180 50 547023659 5474180100 957214339 9558880 100 743624885 7424620150 1444108624 14439100 150 946225634 9449300




Pr Cf M Cf

071 547023659 5474180 10 646718246 6468820300 210639741 2086700 20 547023659 5474180700 160443852 1599660 30 422821349 4222860









00 482729914 278164475 4821830 2742820 022 652433842 046399421 6525790 045714005 515685510 226695420 5146530 2259890 066 605301168 085613463 6025210 084403110 547023659 178562243 5474180 1780800 078 594298856 093601433 5942040 0936298






ARTICLE

0

06

12

0 3 6 9

u

y

Gr = 10 20 30 40



0

06

12

0 3 6 9

u

y

Gc = 10 20 30 40


0

06

12

0 3 6 9

u

y

M = 10 20 30 40





0

06

12

0 3 6 9

u

y

K = 10 20 30 40






ARTICLE

0

01

02

0 3 6 9

θ

y

Pr = 071 10 70 1162





0

05

1

0 3 6 9y

Sc = 022 030 060 078

φ


0

06

12

0 3 6 9

u

y

Sr = 10 20 30 40





0

05

1

0 3 6 9y

Sr = 10 20 30 40φ






ARTICLE

0

06

12

0 3 6 9

u

y



0

06

12

0 3 6 9

u

y

γ = 01 03 07 10

















ARTICLE




















(2015)






















139 3523 (2004)





ARTICLE



u u u

0000000000 0200000003 1000000000 0000000000 0200000003 1000000000 0000000000 0200000003 10000000000992632389 0102704406 0660844505 0992708445 0102720678 0660844505 0992783785 0102736823 06608445051002345443 0051813565 0436002851 1002463222 0051835183 0436002851 1002579331 0051856663 04360029100776130080 0025564646 0286837012 0776260316 0025585003 0286837220 0776388943 0025605224 02868373990544180691 0012279855 0187711954 0544303596 0012295937 0187712133 0544424891 0012311925 01877123420362010717 0005716458 0121621393 0362115055 0005727679 0121621534 0362218052 0005738844 01216216610231195688 0002564892 0077278197 0231276751 0002571937 0077278346 0231356651 0002578950 00772784950140206009 0001097695 0047179937 0140263006 0001101665 0047180071 0140319273 0001105621 00471802020077051550 0000433813 0026324280 0077086285 0000435752 0026324367 0077120528 0000437686 00263244600032428648 0000137244 0011355995 0032444123 0000137947 0011356040 0032459375 0000138648 00113560850000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000




Gr Cf Gc Cf

50 547023659 5474180 50 547023659 5474180100 957214339 9558880 100 743624885 7424620150 1444108624 14439100 150 946225634 9449300




Pr Cf M Cf

071 547023659 5474180 10 646718246 6468820300 210639741 2086700 20 547023659 5474180700 160443852 1599660 30 422821349 4222860









00 482729914 278164475 4821830 2742820 022 652433842 046399421 6525790 045714005 515685510 226695420 5146530 2259890 066 605301168 085613463 6025210 084403110 547023659 178562243 5474180 1780800 078 594298856 093601433 5942040 0936298






ARTICLE

0

06

12

0 3 6 9

u

y

Gr = 10 20 30 40



0

06

12

0 3 6 9

u

y

Gc = 10 20 30 40


0

06

12

0 3 6 9

u

y

M = 10 20 30 40





0

06

12

0 3 6 9

u

y

K = 10 20 30 40






ARTICLE

0

01

02

0 3 6 9

θ

y

Pr = 071 10 70 1162





0

05

1

0 3 6 9y

Sc = 022 030 060 078

φ


0

06

12

0 3 6 9

u

y

Sr = 10 20 30 40





0

05

1

0 3 6 9y

Sr = 10 20 30 40φ






ARTICLE

0

06

12

0 3 6 9

u

y



0

06

12

0 3 6 9

u

y

γ = 01 03 07 10

















ARTICLE




















(2015)






















139 3523 (2004)





ARTICLE

0

06

12

0 3 6 9

u

y

Gr = 10 20 30 40



0

06

12

0 3 6 9

u

y

Gc = 10 20 30 40


0

06

12

0 3 6 9

u

y

M = 10 20 30 40





0

06

12

0 3 6 9

u

y

K = 10 20 30 40






ARTICLE

0

01

02

0 3 6 9

θ

y

Pr = 071 10 70 1162





0

05

1

0 3 6 9y

Sc = 022 030 060 078

φ


0

06

12

0 3 6 9

u

y

Sr = 10 20 30 40





0

05

1

0 3 6 9y

Sr = 10 20 30 40φ






ARTICLE

0

06

12

0 3 6 9

u

y



0

06

12

0 3 6 9

u

y

γ = 01 03 07 10

















ARTICLE




















(2015)






















139 3523 (2004)





ARTICLE

0

01

02

0 3 6 9

θ

y

Pr = 071 10 70 1162





0

05

1

0 3 6 9y

Sc = 022 030 060 078

φ


0

06

12

0 3 6 9

u

y

Sr = 10 20 30 40





0

05

1

0 3 6 9y

Sr = 10 20 30 40φ






ARTICLE

0

06

12

0 3 6 9

u

y



0

06

12

0 3 6 9

u

y

γ = 01 03 07 10

















ARTICLE




















(2015)






















139 3523 (2004)





ARTICLE

0

06

12

0 3 6 9

u

y



0

06

12

0 3 6 9

u

y

γ = 01 03 07 10

















ARTICLE




















(2015)






















139 3523 (2004)





ARTICLE




















(2015)






















139 3523 (2004)


Behaviour of Non-Newtonian Casson Fluid on MHD

Documents