Radiation Effect on MHD Slip Flow past a Stretching Sheet with Variable Viscosity and Heat Source/Sink

Transcription

1 International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 3, Issue 5, May 05, PP 8-7 ISSN X (Print) & ISSN (Online) Radiation Effect on MHD Sli Flow ast a Stretching Sheet with Variable Viscosity and Heat Source/Sink R.L.V.Renuka Devi Det. of Mathematics Sri Venkateswara University, Tiruati, India rlvrenukadevi@gmail.com A. Neeraja Det. of Mathematics, Sri Adithya Engineering College East Godavari, India dr.neeraja7@gmail.com N. Bhaskar Reddy Det. of Mathematics Sri Venkateswara University, Tiruati, India nbrsvu@gmail.com Abstract: This aer focuses on a steady two-dimensional sli flow of a viscous incomressible electrically conducting and radiating fluid ast a linearly stretching sheet with temerature deendent viscosity is taking into account. The governing boundary layer equations are solved by using Runge-Kutta fourth order technique along with shooting method. The influence of various governing arameters on the fluid velocity, temerature, concentration, skin-friction coefficient, Nusselt number and Sherwood number are comuted and discussed in detail. Keywords: Radiation, Heat and Mass Transfer,Heat Source/Sink, Magnetic field, Sli Flow, variable viscosity.. INTRODUCTION Boundary layer flow over a moving continuous and linearly stretching surface is a significant tye of flow which has considerable ractical alications in engineering, electrochemistry and olymer rocessing, for examle, materials manufactured by extrusion rocesses and heat treated materials travelling between a feed roll and a windu roll or on a conveyor belt ossess the characteristics of a moving continuous surface. To be more secific, it may be ointed out that many metallurgical rocesses involve the cooling of continuous stris or filaments by drawing them through a quiescent fluid and that in the rocess of drawing, these stris are sometimes stretched. It may be made of drawing, annealing and tinning of coer wires. In all the cases the roerties of the final roduct deend to a great extent on the rate of cooling. By drawing such stris in an electrically conducting fluid subjected to a magnetic field, the rate of cooling can be controlled and a final roduct of desired characteristics can be achieved. Flow and heat transfer of a viscous fluid ast a stretching sheet is a significant roblem with industrial heat transfer alications. The steady boundary layer flow of an incomressible viscous fluid due a linearly stretching sheet was investigated by Crane []. He obtained an exact similarity solution. The ioneering work of Crane [] was extended by Pavlov []. Po and Na [3] discussed the unsteady flow due to a stretching sheet. Anderson et al. [4] described the heat transfer in unsteady liquid film over a stretching surface. Many recent studies have been focused on the roblem of magnetic field effect on laminar mixed convection boundary layer flow over a vertical non-linear stretching sheet [5-7]. Habibi Matin et al. [8] studied the mixed convection MHD flow of nanofluid over a non-linear stretching sheet with effects of viscous dissiation and variable magnetic field. Hamad et al. [9] investigated magnetic field effects on free convection flow of a nanofluid ast a vertical semi-infinite flat late. Kandasamy et al [0] resented the Scaling grou transformation for MHD boundary-layer flow of a nanofluid ast a vertical stretching surface in the resence of suction/injection. In all the above mentioned flow roblems, the thermohysical roerties of fluid were assumed to be constant. However, it is noticed that these roerties, esecially the fluid viscosity, may change with temerature. In order to aroriately model the flow and heat transfer henomena, it becomes essential to consider the variation of fluid viscosity due to temerature. Lai and Kulacki [] considered the effects of variable viscosity on convective heat transfer along a vertical surface in orous medium. Po et al. [] discussed the influence of variable viscosity on laminar boundary layer flow and heat transfer due to a continuously moving fiat late. El-Aziz [3] studied the flow, heat and mass transfer characteristics of a viscous electrically conducting fluid having temerature ARC Page 8

2 R.L.V.Renuka Devi deendent viscosity and thermal conductivity ast a continuously stretching surface, taking into account of the effect of Ohmic heating. Further, some very imortant investigations regarding the variable viscosity effects on the flow and heat transfer over stretching sheet under different hysical conditions were made by Pantokratoras [4], Mukhoadhyay [5, 6]. The non-adherence of the fluid to a solid boundary, known as velocity sli, is a henomenon that has been observed under certain circumstances. Fluid in micro electro mechanical systems encounters the sli at the boundary. In the revious investigations, it is assumed that the flow field obeys the no-sli condition at the boundary. But, this no-sli boundary condition needs to be relaced by artial sli boundary condition in some ractical roblems. Beavers and Joseh [7] considered the fluid flow over a ermeable wall using the sli boundary condition. The effects of sli at the boundary on the flow of Newtonian fluid over a stretching sheet were studied by Anderson [8] and Wang [9]. Ariel et al. [0] analyzed the flow of a viscoelastic fluid over a stretching sheet with artial sli. Ariel [] also studied the sli effects on the two dimensional stagnation oint flow of an elastoviscous fluid. Bhattacharyya et al. [] showed the sli effects on the dual solutions of stagnation-oint flow and heat transfer towards a shrinking sheet. Recently, Bhattacharyya et al. [3] studied the boundary layer sli flow and heat transfer ast a stretching sheet with temerature deendent viscosity. Mukhoadhyay et al. [4] analyzed the effects of temerature deendent viscosity on MHD boundary layer flow and heat transfer over stretching sheet. An extensive literature that deals with flows in the resence of radiation is now available. Cortell [5] has solved a roblem on the effect of radiation on Blasius flow by using fourth order Runge-Kutta aroach. Later, Sajid and Hayat [6] considered the influence of thermal radiation on the boundary layer flow due to an exonentially stretching sheet by solving the roblem analytically via homotoy analysis method (HAM). Bidin and Nazar [7] studied the boundary layer flow over an exonential stretching sheet with thermal radiation, using Keller-box method. Bala Anki Reddy and Bhaskar Reddy [8] analyze the thermal radiation effects on hydro-magnetic flow due to an exonentially stretching sheet. Rafael [9] studied about viscoelastic fluid flow and heat transfer over a stretching sheet under the effects of a non uniform heat source, viscous dissiation and thermal radiation. Elbashbeshy and Bazid [30] studied the heat transfer over a stretching surface in a orous medium, with internal heat generation and suction or injection. Nagbhooshan [3] analyzes the flow and heat transfer over an exonential stretching sheet under the effects of a temerature gradient deendent heat sink and thermal radiation. Barik et al. [3] analyze the heat and mass transfer on MHD flow through a orous medium over a stretching surface with heat source. Recently, Sreenivasulu and Bhaskar Reddy [33] studied the thermal radiation and chemical reaction effects on MHD stagnation-oint flow of a nanofluid over a orous stretching sheet embedded in a orous medium with heat absortion/generation using Lie Grou Analysis However, the interaction of artial sli flow on heat and mass transfer ast a stretching sheet immersed in a fluid of variable viscosity, has received little attention. Hence, the resent study an attemt is made to analyze a steady magnetohydrodynamic (MHD) sli flow over a stretching sheet in the resence of thermal radiation, heat source/sink and mass transfer. The governing boundary layer equations have been transformed to a two-oint boundary value roblem in similarity variables and the resultant roblem is solved numerically using the fourth order Runge-Kutta method with shooting technique. The effects of various governing arameters on the fluid velocity, temerature, concentration, skin-friction coefficient, Nusselt number and Sherwood number are shown in figures and analyzed in detail.. MATHEMATICAL ANALYSIS A steady two-dimensional sli flow of a viscous incomressible electrically conducting and radiating fluid ast a linearly stretching sheet with temerature deendent viscosity is considered. The flow is assumed to be in the x-direction, which is chosen along the late in the uward direction and y-axis normal to late. A uniform magnetic field is alied in the direction erendicular to the late. The transverse alied magnetic field and magnetic Reynolds number are assumed to be very small, so that the induced magnetic field is negligible. Under these assumtions along with the Bossiness and boundary layer aroximations, the system of equations, governing the flow field are given by International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 9

3 Radiation Effect on MHD Sli Flow Past a Stretching Sheet with Variable Viscosity and Heat Source/Sink u x v y 0 () u u T u u u v u x y T y y y B0 T T k T qr q x y c y c y c u v ( T T ) C C C u v D x y y () (3) (4) The boundary conditions for the velocity, temerature and concentration fields are u u cx L, v 0, T Tw, C Cw at y 0 y u 0, T T, C C as y (5) where u and v are the velocity comonents along the x and y axes, resectively, T is the flow temerature within the boundary layer and C is the fluid concentration within the boundary layer, * / is the kinematic fluid viscosity, B0 is the magnetic field of constant strength, qr is the radiative heat flux, q is the heat source/sink coefficient, T w is the temerature of the sheet, C w is the concentration of the sheet,t is the fluid temerature in the free-stream,c is the fluid concentration in the free-stream, is the coefficient of fluid viscosity, c is the secific heat, D is the coefficient of mass diffusivity, c is the stretching constant with c 0, L is the denote the sli length and k are resectively the and thermal conductivity. By using the Rosseland aroximation, the radiative heat flux q r is given by q r 4 3k T y 4 where is the Stefan-Boltzmann constant and k - the mean absortion coefficient. It should be noted that by using the Rosseland aroximation, the resent analysis is limited to otically thick fluids. If temerature differences within the flow are sufficiently small, then the equation (7) can be 4 linearized by exanding T into the Taylor series aboutt, which after neglecting higher order terms takes the form (6) T 4T T 3T In view of the equations (7) and (8), the equation (4) reduces to (7) 3 T T k 6 T T q 3 u v ( T T ) x y c k k y c (8) The continuity equation () is satisfied by the Cauchy Riemann equations u y and v x (9) where ( xy, ) is the stream function. The temerature deendent viscosity of the fluid is of the form International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 0

4 R.L.V.Renuka Devi where * a b( Tw T ) (0) *is the constant value of the coefficient of viscosity in the free stream and a, b are constants with b(>0) having unit K. Here, we aly the viscosity temerature relation a* b* T which accords with the relation e at * when second and higher order terms are neglected from the exansion. The exression of kinematic viscosity becomes * a b( Tw T ), where * */ the constant value of the kinematic fluid viscosity. In view of the equations (9) and (0), the equations (), (4) and (8) reduce to 3 T B0 * b * a b( T ) w T 3 y x y x y y y y y 3 T T k 6 T T q 3 ( T T ) y x x y c k k y c C C C D y x x y y The corresonding boundary conditions are () () (3) cx L, 0, T T, w C C y y x w at y 0 y 0, T T, C C as y (4) Next, we introduce the dimensional variables for ψ, T and C as c * xf ( ) and ( ) T T C Tw T, ( ) Cw C C (5) where the similarity variable is defined as / yc / *. In view of (5), the equations (), () and (3) reduce to ( a A A ) f ff A ' f '' f ' Mf ' 0 (6) N " Pr f ' Pr Q 0 (7) " Scf ' 0 (8) where A b( Tw T ) is the viscosity arameter and Pr * c / - is the magnetic arameter, N arameter, Sc 6 T 3kk * - the Schmidt number. D The transformed boundary conditions are f 0, f ' f '',, at the radiation arameter, Q k is the Prandtl number, M B 0 q - the heat source/sink cc c International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page

5 Radiation Effect on MHD Sli Flow Past a Stretching Sheet with Variable Viscosity and Heat Source/Sink f ' 0as (9) where / Lc ( / *) is the sli arameter. The arameters of engineering interest for the resent roblem are the skin friction coefficient, local Nusselt number and the local Sherwood number which indicate hysically wall shear stress and rates of heat and mass transfer resectively. The skin-friction coefficient is given by The local Nusselt number may be written as The local Sherwood number may be written as Thus the values roortional to the skin-friction coefficient, Nusselt number and the Sherwood number are f ''(0), '(0) and '(0) resectively. 3. METHOD OF SOLUTION The set of couled non-linear governing boundary layer equations (6) - (8) together with the boundary conditions (9) are solved numerically by using Runge-Kutta fourth order technique along with shooting method. First of all, the higher order non-linear differential equations (6) - (8) are converted into simultaneous linear differential equations of first order and they are further transformed into an initial value roblem by alying the shooting technique (Jain et al.[34]). The resultant initial value roblem is solved by emloying Runge-Kutta fourth order technique. The ste size =0.05 is used to obtain the numerical solution with five decimal lace accuracy as the criterion of convergence. From the rocess of numerical comutation, the skin-friction coefficient, the Nusselt number and the Sherwood number, which are resectively roortional to f ''(0), '(0) and '(0), are also sorted out and their numerical values are resented in a tabular form. 4. RESULTS AND DISCUSSION Cf Nu Re Re In order to get a clear insight of the hysical roblem, the velocity, temerature and concentration have been discussed by assigning numerical values to the governing arameters encountered in the roblem. The effects of various arameters on the velocity are deicted in Figs. -5. The effects of various arameters on the temerature are deicted in Figs The effects of various arameters on the concentration are deicted in Figs Fig. shows the dimensionless velocity for different values of viscosity arameter (A). It is observed that the velocity increases with increasing values of viscosity arameter and the momentum boundary layer thickness increases with A. Fig. shows the dimensionless velocity rofiles for different values of magnetic arameter (M). It is seen that, as exected, the velocity decreases with an increase of magnetic arameter. The magnetic arameter is found to retard the velocity at all oints of the flow field. It is because that the alication of transverse magnetic field will result in a resistive tye force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity. Also, the boundary layer thickness decreases with an increase in the magnetic arameter. Fig.3 illustrates the effect of the sli arameter (δ) on the velocity field. The flow is decelerated due to the enhancement in the sli arameter. The effect of radiation arameter (N) on the velocity is illustrated in Fig.4. It is noticed that the velocity increases with increasing values of the radiation arameter. Fig.5 illustrates the effect of heat source/sink arameter (Q) on the velocity. It is noticed that as the heat source/sink arameter increases, the velocity increases. Fig. 6 shows the dimensionless temerature for different values of viscosity arameter. It is observed that the temerature decreases with increasing values of viscosity arameter and the thermal boundary layer thickness decreases with A. The effect of the magnetic arameter on the temerature is International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page x x Sh f ''(0) Re x '(0) '(0)

6 R.L.V.Renuka Devi illustrated in Fig.7. It is observed that as the magnetic arameter increases, the temerature increases. The effect of the sli arameter on the temerature is illustrated in Fig.8. It is seen that as the thermal buoyancy arameter increases, the temerature decreases. Fig. 9 deicts the variation of the thermal boundary-layer with the Prandtl number (Pr). It is noticed that the thermal boundary layer thickness decreases with an increase in the Prandtl number. Fig. 0 shows the variation of the thermal boundary-layer with the radiation arameter. It is observed that the thermal boundary layer thickness increases with an increase in the radiation arameter. The effect of heat source/sink arameter on the temerature is illustrated in Fig.. It is observed that as the heat source/sink arameter increases, the temerature increases. Fig. shows the dimensionless concentration for different values of viscosity arameter. It is observed that the concentration decreases with increasing values of viscosity arameter. The effect of magnetic arameter on the concentration field is illustrated Fig.3. As the magnetic arameter increases the concentration is found to be increasing. Fig. 4 illustrates the effect of Schmidt number on the concentration. As the Schmidt increases, a decreasing trend in the concentration field is noticed. Figs.5,6,7 and 8 show the variation of the skin friction and Nusselt number resectively. It is observed that the skin friction is found to increase with an increase in the heat source/sink arameter or radiation number. It is noticed that the Nusselt number decrease with an increase in the heat source/sink arameter or radiation number. Fig. 9 shows the variation Schmidt number on Sherwood number resectively. It is observed that the Sherwood number is found to decrease with an increase in the Schmidt number. In Table, the resent results are comared with those of Anderson [8] and Bhattacharya et al. [] and found that there is a erfect agreement. 5. CONCLUSIONS In the resent study, the steady boundary layer sli flow and heat transfer ast a stretching sheet with temerature deendent viscosity is considered in the resence of thermal radiation, heat source/sink and mass transfer. The governing equations are aroximated to a system of non-linear ordinary differential equations by similarity transformations. Numerical calculations are carried out for various values of the dimensionless arameters of the roblem. The resent solutions are validated by comaring with the existing solutions. Our results show a good agreement with the existing work in the literature. The results are summarized as follows The viscosity arameter enhances the velocity and reduces and the temerature concentration. Magnetic field elevates the temerature and concentration, and reduces the velocity. The radiation enhances the velocity and temerature. The heat source/sink enhances the velocity and temerature. The radiation arameter elevates the skin friction and reduces the heat transfer. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 3

7 Radiation Effect on MHD Sli Flow Past a Stretching Sheet with Variable Viscosity and Heat Source/Sink International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 4

8 R.L.V.Renuka Devi REFERENCES [] Crane, L. J., Flow ast a stretching late, J. Al. Math. Phys. (ZAMP),, , (970). [] Pavlov,K. B., Magnetohydrodynamic flow of an incomressible viscous fluid caused by the deformation of a lane surface, Magn. Gidrod., vol. 0, 46-48, (974). International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 5

9 Radiation Effect on MHD Sli Flow Past a Stretching Sheet with Variable Viscosity and Heat Source/Sink [3] Po, I. and Na,T. Y., Unsteady flow ast a stretching sheet, Mech. Res. Commun., 3, 43-4, (996). [4] Andersson,H. I., Aarseth,J. B. and Dandaat, B. S., Heat transfer in a liquid film on an unsteady stretching surface, Int. J. Heat Mass Transfer, 43, 69-74, (000). [5] Fisher; E.G., Extrusion of Plastics, Wiley, New York, 344, (976). [6] Altan, T., Oh, S.,and Gegel, H., Metal Forming Fundamentals and Alications, American Society of Metals, Metals Park, OH, 353, (979). [7] Tadmor, Z., and Klein, I., Engineering rinciles of lasticating extrusion, Polymer Science and Engineering Series, Van Nostrand Reinhold, New York, (970). [8] Habibi Matin, M., Dehsara, M., Abbassi, A., Mixed convection MHD flow of nanofluid over a non-linear stretching sheet with effects of viscous dissiation and variable magnetic field, Mechanika. 8(4), 45-43, (0). [9] Hamad, M.A.A., Po, I., and Ismail, A.I.M., Magnetic field effects on free convection flow of a nanofluid ast a vertical semi-infinite flat late, Nonlinear Analysis: Real World Alications,, , (0). [0] Kandasamy, R., Loganathan, P., and Puvi Arasu, P., Scaling grou transformation for MHD boundary-layer flow of a nanofluid ast a vertical stretching surface in the resence of suction/injection, Nuclear Engineering and Design,.4, , (0). [] Lai,F. C. and Kulacki,F. A., The effect of variable viscosity on convective heat transfer along a vertical surface in a saturated orous medium, Int. J. Heat Mass Transfer, 33, 08-03, (990). [] Po,I., Gorla,R. S. R. and Rashidi,M., The effect of variable viscosity on flow and heat transfer to a continuous moving flat late, Int. J. Eng. Sci., 30, l-6, (99). [3] El-Aziz, M. A., Temerature deendent viscosity and thermal conductivity effects on combined heat and mass transfer in MHD three-dimensional flow over a stretching surface with Ohmic heating, Meccanica, 4, , (007). [4] Pantokratoras, A., Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity: A numerical reinvestigation, Int. J. Heat Mass Transfer, 5, 04-0, (008). [5] Mukhoadhyay, S., Unsteady boundary layer flow and heat transfer ast a orous stretching sheet in resence of variable viscosity and thermal diffusivity, Int. J. Heat Mass Transfer, 5, 53-57, (009). [6] Malarvizhi, G., Ramanaiah, G., and Po, I., Free and mixed convection about a vertical late with re-scribed temerature or heat flux, ZAMM, 74, 9-3, (994). [7] Beavers,G. S. and Joseh, D. D., Boundary condition at a naturally ermeable wall, J. Fluid Mech., 30, 97-07, (967). [8] Anderson,H. I., Sli flow ast a stretching surface, Acta Mech., 58, -5, (00). [9] Wang, C. Y., Flow due to a stretching boundary with artial sli - an exact solution of the Navier-Stokes equations, Chem. Eng. Sci., 57, , (00). [0] Ariel,P. D., Hayat,T. and Asghar,S., The flow of an elastico-viscous fluid ast a stretching sheet with artial sli, Acta Mech., 87, 9-35, (006). [] Ariel, P. D., Two dimensional stagnation oint flow of an elastico-viscous fluid with artial sli, Z. Angew. Math. Mech., 88, 30-34, (008). [] Bhattacharyya,K. and Layek, G. C., Chemically reactive solute distribution in MHD boundary layer flow over a ermeable stretching sheet with suction or blowing, Chem. Eng. Commun., 97, , (00). [3] Krishnendu Bhattacharyya, Layek, G. C., Rama Subba Reddy Gorla, Boundary Layer Sli Flow and Heat Transfer Past a Stretching Sheet with Temerature Deendent Viscosity. Thermal Energy and Power Engineering,, Issue, 38-43, (03). [4] Mukhoadhyay,S., Layek,G. C. and Samad,S. A., Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity, Int. J. Heat Mass Transfer, 48, , (005). International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 6

10 R.L.V.Renuka Devi [5] Cortell, R., Radiation effects in the Blasius flow, Alied Mathematics and Comutation 98, 33-33, (008). [6] Sajid, M. and Hayat, T., Influence of thermal radiation on the boundary layer ow due to an exonentially stretching sheet, International Communications in Heat and Mass Transfer, l.35, , (008). [7] Bidin, B., and Nazar, R., Numerical solution of the boundary layer flow over an exonentially stretching sheet with thermal radiation, Euroean journal of scientific research,.33, No.4,70-77., (009). [8] Bala Anki Reddy, P., and Bhaskar Reddy, N., Thermal radiation effects on hydro-magnetic flow due to an exonentially stretching sheet, International Journal of Alied Mathematics and Comutation, 3(4), , (0). [9] Rafael Cortell Bataller, Viscoelastic Fluid Flow and Heat Transfer over a Stretching Sheet under the effects of a non-uniform Heat Source, Viscous Dissiation and Thermal Radiation, Int. Journal of Heat and Mass Transfer, 50, 35-36, (007). [30] Elbashbeshy,E. M. A. and Bazid,M. A. A., Heat transfer in a orous medium over a stretching surface with internal heat generation and suction or injection, Al. Math. Comut., 58, , (004). [3] Nagbhooshan, J.K., Veena, P.H., Rajagoal, K., and Pravin, V.K., Flow and Heat Transfer over an Exonential Stretching Sheet under the Effects of a Temerature Gradient Deendent Heat Sink and Thermal Radiation, IOSR Journal of Mathematics (IOSRJM),, Issue 5, -9, (0). [3] Barik, R. N., Dash, G. C., and Rath, P. K., Heat and mass transfer on MHD flow through a orous medium over a stretching surface with heat source, Mathematical Theory and Modeling,,7, (0). [33] Bird, R. B., Sewart, W. E. and Lightfoot, E. N., Transort Phenomena, New York, John Wiley and Sons, (960). [34] Jain, M.K., Iyengar, S.R.K. and Jain, R.K., Numerical Methods for Scientific and Engineering Comutation, Wiley Eastern Ltd., New Delhi, India, (985). International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 7