THERMO- DIFFUSION AND DIFFUSION -THERMO EFFECTS ON UNSTEADY MHD FLOW PAST … · 2018. 9. 29. ·...

THERMO- DIFFUSION ANDDIFFUSION -THERMO EFFECTS ONUNSTEADY MHD FLOW PAST ANACCELARATED VERTICAL PLATE

WITH VISCOUSDISSIPATION-FINITE ELEMENT

STUDY

D.Mahendar1, B. Shankar Goud2,P. Srikanth Rao3

Department of Mathematics,1,3 BVRIT, Narsapur, Medak, India.

2JNTUH College of Engineering,Kukatpally, Hyderabad, India.

[email protected]

July 20, 2018

AbstractAn investigation of the effects of thermo-diffusion and

diffusion-thermo on an unsteady MHD natural convectionflow with heat and mass transfer of an electrically conduct-ing, viscous, incompressible fluid past an accelerated ver-tical plate in the presence of viscous dissipation. Consid-ered fluid is gray, absorbing-emitting radiation, but a non-scattering medium. The numerical solutions of the govern-ing nonlinear partial differential equations are obtained byfinite element method. The numerical values of fluid ve-locity, fluid temperature and species concentration are dis-played graphically. The values of skin friction coefficient,

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International Journal of Pure and Applied MathematicsVolume 120 No. 6 2018, 8165-8185ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

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Nusselt number and Sherwood number for various values ofpertinent flow parameters are presented through tables. Wehave shown that some results are in good agreement withearlier reported studies.

Key Words: Diffusion thermo, Thermo Diffusion, MHD,FEM, viscous dissipation.

1 INTRODUCTION

The study of the hydromagnetic flow of an electrically conductingfluid has many applications in science and engineering problemssuch as magnetohydrodynamic(MHD) generator , plasma studies,nuclear reactors, aerodynamic heating, etc. Elbasheshy [1] studiedMHD heat and mass transfer problem along a vertical plate underthe combined buoyancy effects on of thermal and spices diffusion.T.Arun kumar and L Anand Babu[2] has analyzed the study ofRadiation effect of MHD flow past an impulsive started verticalplate with variable temperature and uniform mass diffusion A fi-nite element method. Chamkha and Khaled [3] formulated similar-ity solutions for hydromagnetic simultaneous heat and mass trans-fer bynatural convection from an inclined plate with internal heatgeneration or absorption. D.Srinivasacharya and B.Mallikarjuna[4]studied Soret and dofour effects on mixed convection along a ver-tically away surface in porous medium with variable properties.J.Anand Rao and S.Shivaiah [5] analized chemical reaction effectson an unsteady MHD free convective flow past in infinite verticalplate with constant suction and heat source. Tsai and Huang [6]studied the heat and mass transfer for Soret and Dufour effectson Hiemenz flow through porous medium over a stretching surface.Kandasamy et al., [7] have studied the heat and mass transfer un-der a chemical reaction with a heat source. S.Siviah et.al [8] havestudied finite element analysis of chemical reaction and radiationeffects on isothermal vertical oscillating plate with variable massdiffusion. Exact solutions for MHD free convictive boundary layerflow past a porous vertical surface in the presence of chemical re-action, thermal radiation and suction were carried out by Raju etal. [19]. P.Chandra Reddy et al[10] have analyzed MHD convectivedouble diffusive laminar boundary layer flow past an acceleratedvertical plate using finite difference method.

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International Journal of Pure and Applied Mathematics Special Issue

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The object of the present paper is to analyze the Soret and Dufoureffects on MHD flow an electrically conducting, viscous and incom-pressible fluid past an infinite vertical porous plate in the presenceof viscous dissipation. The coupled nonlinear governing equationsalong with boundary conditions are solved by finite element methodand the results are presented through graphs.

2 MATHEMATICAL ANALYSIS

In this study, the problem under consideration is incompressible,viscous, radiating, heat absorbing, electrically conducting Newto-nian fluid flow past an infinite vertical porous plate in the activepresence of both diffusion thermo and thermo diffusion. Along theperpendicular direction to the plate, there is a magnetic field ofuniform strength applied. Along the vertically upward directionof the plate Xc - axis is taken and perpendicular to the plate Y c

-axis is assumed. At time tc£ 0, the fluid is at rest and the plate iskept at the temperature higher than ambient temperature TY c. Attime tc > 0, the plate starts accelerating linearly in its own planewith time and the temperature decreases with temperature T =

1(1+at)

. Also species concentration starts decreasing linearly withtime. An assumption is made that the effect of viscous dissipationis taken in account. Hence, with usual Boussineq’s and boundarylayer approximation.

Fig 1 Physical flow of the problem

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The flow modal is as follows:

With the initial and boundary conditions are as follows:

Where A =U20

u, the following non dimensional quantities are

defined as.

With the non dimensional quantities equations (1), (2) and (3)changes to the following dimensionless form:

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where Gr, Gm, Pr, M, Sc, So, Du, Ec and K are the thermalGrashof number, mass Grashof number, Prandtl number, Magneticparameter, Schmidt number, Soret number, Dufour number, Eckertnumber and chemical reaction parameter respectively.With the initial and boundary conditions in nondimensionl formare :

3 SOLUTION OF THE PROBLEM

The linear functional for (6) over the typical element Yj ≤ y ≤ Ykfor the boundary value problem can be written as:

Where,

Omitting the first term in the above equation, we get

Let u(ε) = N (ε)φ(ε) be the finite element approximation solution( Yj ≤ y ≤ Yk ) where,

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are the basis functions. We obtain

Where dot denote the differentiation with respect to and writethe element equation for the elements Yi−1 ≤ y ≤ Yi and Yi ≤ y ≤Yi+1 assemble the elements equations, we obtain

Now put row corresponding to the node to zero, from the aboveequation the difference scheme with l(ε) = h is

Applying the trapezoidal rule, the following equations in Crank-Nicholson method are obtained:

Where,

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Similarly applying the applying Galerikin finite element method forequations (7) and (8) the following equations are obtained:

Where,

Here h,k are the mesh sizes along y- direction and t- direction re-spectively. Index i refers to the space and n refers to the time. Inequation (12), (13) and (14), taking i=1(1)m and using (9), thefollowing systems of equations are obtained:

Where Ai’s are the matrices of order m and XiBi’s column matriceshaving m- componenets. The solutions of the above system of equa-tions are obtained by using Thomas algorithm for the velocity(u),temperature(θ), concentration(c). Also numerical solutions are ob-tained by MATLAB-program. Computations are carried out untilthe steady state is reached. In order to prove the convergence ofthe Galerkin finite element method, the computations are carriedout for slight changed values of h,k by running same MATLAB-program, no significant changes was observed in the values of veloc-ity(u), temperature(θ), concentration(C). Hence, the finite elementmethod is stable and convergent.The skin friction, Nusselt number and Sherwood number are im-portant physical parameters for this type of boundary layer flowand are given by

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TABLE I Comparison of present skin friction (τ) results withprevious skin friction (τ ∗) results were obtained by P.Chandra

Reddy et al[10] and found to be in good agreement.

TABLE II Variations in Nusselt Number

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TABLE III Variations in Sherwood number

4 GRAPHS

Figure 2 Velocity profile for different values of M

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Figure 3 Velocity profile for different values ofGr

Figure 4 Velocity profile for different values of Gm

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Figure 5 Velocity profile for different values of K

Figure 6 Velocity profile for different values of Pr

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Figure 7 Velocity profile for different values of So

Figure 8 Velocity profile for different values of Du

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Figure 9 Velocity profile for different values Pr

Figure 10 Temperature profile for different values of So

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Figure 11 Temperature profile for different values of Du

Figure 12 Concentration profile for different values of Sc

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Figure 13 Temperature profile for different values of So

Figure 14 Concentration profile for different values of Du

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Figure 15 Temperature profile for different values of Ec

5 RESULTS AND DISCUSSION

The velocity profile , temperature and Concentration profile areillustrated in figures 1 to 16 for various parameters like, thermal-Grashof number(Gr), mass Grashof number (Gm), Prandtl num-ber(Pr), Magnetic parameter (M), Schmidt number (Sc), Soretnumber (So), Dufour number (Du), Eckert number (Ec) and chem-ical reaction parameter (K) respectively.Figures 2-8 exhibit the differences of the fluid velocity under the re-sults of different parameters. Figure 2 depicts the effect of magneticparameter on velocity profile. It is found that the velocity decreaseswith increasing values of magnetic parameter. It is known fact thatthe application of transverse magnetic field which is applied normalto the flow, results in a flow resistive force called the Lorentz forcewhich acts in the opposite direction of the flow. This force has theeffect of slowing the motion of the fluid The influence of Grashofnumber and modified Grashof number on velocity is described inFigures 3 and 4. An observation is made that velocity increaseson increase in either value which occurs due to the fact that fluid

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velocity is enhanced with the buoyancy acting on the fluid particlesdue to gravitational forces.From figure 5 illustrating the effect of porous permeability parame-ter on velocity, it can be made clear that increase in porosity param-eter values induces velocity. To explain physically, gradual increasein the permeability of porous medium creates a rise in the flow offluid through it. It can also be stated that when the holes of theporous medium become large, the acting resistance on the mediummay be neglected.From figure 6, velocity distribution and its effects by Prandtl num-ber state that the velocity tends to decrease with an increase inPrandtl number. This can be clearly explained stating that thefluid of low Prandtl number has high thermal diffusivity thereby itacquires a higher temperature in steady state. This in turn createsmore buoyancy force i.e. higher velocity of fluid as ccompared tothat of high Prandtl numbered fluid.Figure 7 shows the effect of Soret number on velocity boundary-layer stating that the velocity boundary layer thickness goes higherwith an increase in the value of Soret number. Figure 8 shows theeffect of Dufour number on velocity boundary layer that its thick-ness increases with an increase in the Dufour number. With theassumptions made in the problem, maximum temperature of thefluid is found at the plate surface and it tends to decrease exponen-tially away from the plate to reach the free stream zero value.Temperature is affected by varying Prandtl number as shown in fig-ure 9. Surface temperature is observed to reduce with an increase inPrandtl number. This occurs because reduced fluid velocity wouldmean heat is not convicted readily and hence surface temperaturedecreases.Figures 10 and 11 describes the temperature affected by Soret num-ber, it temperature increases with increasing values of Soret numberand Dufour number.Concentration is affected by Schmidt number; this is shown in Fig-ure 12, describes the Schmidt number increase leads to a decreasetrend in the concentration field. Alternatively, far away from theplate, such significance is not found to exist.Figure 13 illustrate the variation of the concentration distributionwith the effect of Soret number. It is noticed that the concentra-tion boundary layer thickness increases with an increase in Soret

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number. Figure 14 demonstrate the variation of the concentrationprofile with the Dufour number. It is seen that the concentrationboundary layer thickness decreases with an increase in the Dufournumber. Figure 15 shows that the temperature of the fluid increaseswith an increase in Eckert number.

6 CONCLUSION

The non-dimensional governing equations of the problem are solvedby using finite element method. The variations in the velocity, tem-perature and concentration with the effects of various parametersencountered in the problem are studied through graphs. Also theeffects some of the above parameters on skin friction, Nusselt num-ber and Sherwood number are observed. The following are some ofthe notable conclusions:

• The fluid velocity increases with an increase of and.

• The fluid velocity reduces for increasing values of and .

• Temperature profiles decreases with an increase of , whereasit increases in the case of and.

• With an increase of values results in rising of the concentra-tion, but it decreases under the influence of the and .

• Skin friction decreases for increasing values of both Soret andDufour numbers.

• Nusselt number increases with increasing values of Dufournumbers.

• The rate of mass transfer is enhanced with increasing valuesof Schmidt number and decreasing values of Soret number.

References

[1] ElbasheshyEMS. Heat and mass transfer problems along a ver-tical plate and concentration in the presence of magnetic field.Int. J. of Engineering Science, 34(5), pp.515-22, 1997.

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[2] T.Arun kumar and L Anand Babu. study of Radiation effectof MHD flow past an impulsive started vertical plate with vari-able temperature and uniform mass Diffusion A finite elementmethod. Ind.J.Sci.and tech., 1(3), pp.3-9, 2013.

[3] A.J. Chamkha and A.R.A.Khaled. Similarity solutions for hy-dromagnetic simultaneous heat and mass transfer by naturalconvection from an inclined plate with internal heat generationor absorption. Heat Mass Transfer, 37, pp.117-123, 2001.

[4] D.Srinivasacharya., B. Mallikarjuna., R.Bhuvanavijaya. Soretand Dufour effects on mixed convection along a vertically awaysurface in porous medium with variable properties. Ain ShamsEngineering Journal, 6, pp.553-564,2015.

[5] J. Anand Rao and S. Shivaiah. Chemical reaction effects onan unsteady MHD free convective flow past an infinite verticalporous plate with constant suction and heat source. Int. J. ofApplied Mathematics and Mechanics, 7(8), pp. 98118, 2011.

[6] R. Tsai, J.S. Huang. Heat and mass transfer for Soret and Du-four’s effects on Hiemenz flow through porous medium onto astretching surface. Int. J. Heat Mass Transf , 52, pp.23992406,2009.

[7] R. Kandasamy, K. Periasamy and K. K. S. Prabhu. Effects ofchemical reaction, heat and mass transfer along a wedge withheat source and Concentration in the presence of suction orinjection. Int.J. of Heat and Mass transfer, 48, pp. 1376-1388,2005.

[8] S. Sivaiah,. G. Murali., and M. C. K. Reddy. finite elementanalysis of chemical reaction and radiation effects on isother-mal vertical oscillating plate with variable mass diffusion.ISRN Mathematical Physics Volume 2012, pp.1-14.

[9] Raju, M.C., Reddy, N.A., Varma, S.V.K. Analytical study ofMHD free convictive,dissipative boundary layer flow past aporous vertical surface in the presence of thermal radiation,chemical reaction and constant suction. Ain Shams Engineer-ing Journal, 5 (4), pp.1361-1369,2014.

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[10] P. Chandra Reddy, M.C.Raju and G.S.S. Raju. Magnetohydro-dynamic convective double diffusive laminar boundary layerflow past an accelerated vertical plate. International Journalof Engineering Research in Africa, Vol. 20, pp 80-92,2016

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