Scaling group transformation for MHD boundary layer flow over permeable stretching sheet in presence of slip flow with Newtonian heating effects

doi:10.1007/s10483-014-1873-7

Abstract

Abstract:

Taking into account the slip flow effects, Newtonian heating, and thermal radiation, two-dimensional magnetohydrodynamic (MHD) flows and heat transfer past a permeable stretching sheet are investigated numerically. We use one parameter group transformation to develop similarity transformation. By using the similarity transformation, we transform the governing boundary layer equations along with the boundary conditions into ordinary differential equations with relevant boundary conditions. The obtained ordinary differential equations are solved with the fourth-fifth order Runge-Kutta- Fehlberg method using MAPLE 13. The present paper is compared with a published one. Good agreement is obtained. Numerical results for dimensionless velocity, temperature distributions, skin friction factor, and heat transfer rates are discussed for various values of controlling parameters.

Key words: Newtonian heating, boundary layer, magnetohydrodynamic (MHD), Lie group analysis, heat transfer, slip flow

2010 MSC Number:

O357.4

A. A. AFIFY;M. J. UDDIN;M. FERDOWS. Scaling group transformation for MHD boundary layer flow over permeable stretching sheet in presence of slip flow with Newtonian heating effects. Applied Mathematics and Mechanics (English Edition), 2014, 35(11): 1375-1386.

TrendMD

References

[1] Sakiadis, B. C. Boundary-layer behavior on continuous solid surfaces, part I: boundary-layerequations for two-dimensional and axisymmetric flow. AIChE Journal, 7, 26-28 (1961)
[2] Sakiadis, B. C. Boundary-layer behavior on continuous solid surfaces, part II: the boundary-layeron a continuous flat surface. AIChE Journal, 7, 221-225 (1961)
[3] Sakiadis, B. C. Boundary-layer behavior on continuous solid surfaces, part III: the boundary-layeron a continuous cylindrical surface. AIChE Journal, 7, 467-472 (1961)
[4] Crane, L. J. Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik, 21,645-647 (1970)
[5] Chakrabarti, A. and Gupta, A. S. Hydromagnetic flow and heat transfer over a stretching sheet.Quarterly of Applied Mathematics, 37, 73-78 (1977)
[6] Andersson, H. I. MHD flow of a viscous fluid past a stretching surface. Acta Mechanica, 95,227-230 (1992)
[7] Chiam, T. C. Magnetohydrodynamic boundary layer flow due to a continuously moving flat plate.Computers & Mathematics with Applications, 26, 1-7 (1993)
[8] Vajravelu, K. and Hadjinicolaou, A. Convective heat transfer in an electrically conducting fluidat a stretching surface with uniform free stream. International Journal of Engineering Science,35, 1237-1244 (1997)
[9] Pop, I. and Na, T. Y. A note on MHD flow over a stretching permeable surface. MechanicsResearch Communications, 25, 263-269 (1998)
[10] Kumari, M. and Nath, G. Flow and heat transfer in a stagnation point flow over a stretchingsheet with a magnetic field. Mechanics Research Communications, 26, 469-478 (1999)
[11] Sajid, M. and Hayat, T. Influence of thermal radiation on the boundary layer flow due to exponentiallystretching sheet. International Communications in Heat and Mass Transfer, 35, 347-356(2008)
[12] Abo-Eldahab, E. M. and El Aziz, M. A. Blowing/suction effect on hydromagnetic heat transferby mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. International Journal of Thermal Sciences, 43, 709-719 (2004)
[13] Sahoo, B. Flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partialslip. Communications in Nonlinear Science and Numerical Simulation, 15, 602-615 (2010)
[14] Datti, P. S., Prasad, K. V., Abel, M. S., and Joshi, A. MHD viscoelastic fluid flow over a nonisothermalstretching sheet. International Journal of Engineering Science, 42, 935-946 (2004)
[15] Elbashbeshy, E. M. A. Heat transfer over a stretching surface with variable surface heat flux.Journal of Physics D: Applied Physics, 31, 1951-1954 (1998)
[16] Seddeek, M. A. Heat and mass transfer on a stretching sheet with a magnetic field in a visco-elasticfluid flow through a porous medium with heat source or sink. Computational Materials Science,38, 781-787 (2007)
[17] Shit, G. C. and Haldar, R. Effects of thermal radiation on MHD viscous fluid flow and heattransfer over nonlinear shrinking porous sheet. Appl. Math. Mech. -Engl. Ed., 32, 677-688 (2011)DOI 10.1007/s10483-011-1448-6
[18] Vajravelu, K., Prasad, K. V., Sujatha A., and Ng, C. O. MHD flow and mass transfer of chemicallyreactive upper convected Maxwell fluid past porous surface. Appl. Math. Mech. -Engl. Ed., 33,899-910 (2012) DOI 10.1007/s10483-012-1593-8
[19] Merkin, J. H. Natural convection boundary-layer flow on a vertical surface with Newtonian heating.International Journal of Heat and Fluid Flow, 15, 392-398 (1994)
[20] Lesnic, D., Ingham, D. B., and Pop, I. Free convection boundary layer flow along a vertical surfacein a porous medium with Newtonian heating. International Journal of Heat and Mass Transfer,42, 2621-2627 (1999)
[21] Lesnic, D., Ingham, D. B., and Pop, I. Free convection from a horizontal surface in a porousmedium with Newtonian heating. Journal of Porous Media, 3, 227-235 (2000)
[22] Lesnic, D., Ingham, D. B., Pop, I., and Storr, C. Free convection boundary layer flow above anearly horizontal surface in a porous medium with Newtonian heating. Heat and Mass Transfer,40, 665-672 (2004)
[23] Pop, I., Lesnic, D., and Ingham, D. B. Asymptotic solutions for the free convection boundarylayerflow along a vertical surface in a porous medium with Newtonian heating. Hybrid Methodsin Engineering, 2, 31-40 (2000)
[24] Salleh, M. Z., Nazar, R., and Pop, I. Forced convection boundary layer flow at a forward stagnationpoint with Newtonian heating. Chemical Engineering Communications, 196, 987-996 (2009)
[25] Salleh, M. Z., Nazar, R., and Pop, I. Mixed convection boundary layer flow about a solid spherewith Newtonian heating. Archives of Mechanics, 62, 283-303 (2010)
[26] Salleh, M. Z., Nazar, R., and Pop, I. Modeling of free convection boundary layer flow on a spherewith Newtonian heating. Acta Applicandae Mathematicae, 112, 263-274 (2010)
[27] Salleh, M. Z., Nazar, R., and Pop, I. Modelling of boundary layer flow and heat transfer over astretching sheet with Newtonian heating. Journal of the Taiwan Institute of Chemical Engineers,41, 651-655 (2010)
[28] Salleh, M. Z., Nazar, R., Arifin, N. M., Merkin, J. H., and Pop, I. Forced convection heat transferover a horizontal circular cylinder with Newtonian heating. Journal of Engineering Mathematics,69, 101-110 (2011)
[29] Merkin, J. H., Nazar, R., and Pop, I. The development of forced convection heat transfer neara forward stagnation point with Newtonian heating. Journal of Engineering Mathematics, 74,53-60 (2012)
[30] Chaundary, R. C. and Preeti, J. Unsteady free convection boundary-layer flow past an impulsivelystarted vertical surface with Newtonian heating. Romanian Journal of Physics, 51, 911-925 (2006)
[31] Chaundary, R. C. and Preeti, J. An exact solution to the unsteady free convection boundary layerflow past an impulsively started vertical surface with Newtonian heating. Journal of EngineeringPhysics and Thermophysics, 80, 954-960 (2007)
[32] Beavers, G. S. and Joseph, D. D. Boundary conditions at a naturally permeable wall. Journal ofFluid Mechanics, 30, 197-207 (1967)
[33] Ariel, P. D. Two dimensional stagnation point flow of an elastico-viscous fluid with partial slip.Zeitschrift für angewandte Mathematik und Mechanik, 88, 320-324 (2008)
[34] Fang, T., Zhang, J., and Yao, S. Slip MHD viscous flow over a stretching sheet--an exact solution.Communications in Nonlinear Science and Numerical Simulation, 14, 3731-3737 (2009)
[35] Aman, F., Ishak, A., and Pop, I. Mixed convection boundary layer flow near stagnationpointon vertical surface with slip. Appl. Math. Mech. -Engl. Ed., 32, 1599-1606 (2011) DOI10.1007/s10483-011-1526-x
[36] Yazdi, M. H., Abdullah, S., Hashim, I., and Sopian, K. Slip MHD liquid flow and heat transferover non-linear permeable stretching surface with chemical reaction. International Journal of Heatand Mass transfer, 54, 3214-3225 (2011)
[37] Bhattacharyya, K., Mukhopadhyay, S., and Layek, G. C. Slip effects on boundary layer stagnationpointflow and heat transfer towards a shrinking sheet. International Journal of Heat and MassTransfer, 54, 308-313 (2011)
[38] Hansen, A.G. Similarity Analysis of Boundary Layer Problems in Engineering, Prentice Hall,Englewood Cliffs, U. S.A. (1964)
[39] Olver, P. J. Application of Lie Groups to Differential Equations, Springer-Verlag, New York (1986)
[40] Bluman, G. W. and Kumei, S. Symmetries and Differential Equations, Springer, New York (1989)
[41] Fan, J. R., Shi, J. M., and Xu, X. Z. Similarity solution of free convective boundary-layer behaviorat a stretching surface. Heat and Mass Transfer, 35, 191-196 (1999)
[42] Afify, A. A. Some new exact solutions for MHD aligned creeping flow and heat transfer in secondgrade fluids by using Lie group analysis. Nonlinear Analysis-Theory Methods & Applications, 70,3298-3306 (2009)
[43] Abdul-Kahar, R., Kandasamy, R., and Muhaimin, I. Scaling group transformation for boundarylayer flow of a nanofluid past a porous vertical stretching surface in the presence of chemicalreaction with heat radiation. Computers and Fluids, 52, 15-21 (2011)
[44] Afify, A. A. and Elgazery, N. S. Lie group analysis for the effects of chemical reaction on MHDstagnation-point flow of heat and mass transfer towards a heated porous stretching sheet withsuction or injection. Nonlinear Analysis: Modelling and Control, 17, 1-15 (2012)
[45] Hamad, M. A. A. and Ferdows, M. Similarity solution of boundary layer stagnation-point flow towardsa heated porous stretching sheet saturated with a nanofluid with heat absorption/generationand suction/blowing: a Lie group analysis. Communication in Nonlinear Science and NumericalSimulation, 17, 132-140 (2012)
[46] Rosmila, A. B., Kandasamy, R., and Muhaimin, I. Lie symmetry group transformation for MHDnatural convection flow of nanofluid over linearly porous stretching sheet in presence of thermalstratification. Appl. Math. Mech. -Engl. Ed., 33, 593-604 (2012) DOI 10.1007/s10483-012-1573-9
[47] Uddin, M. J., Khan, W. A., and Ismail, A. I. M. Free convection boundary layer flow from aheated upward facing horizontal flat plate embedded in a porous medium filled by a nanofluidwith convective boundary condition. Transport in Porous Media, 92, 867-881 (2012)
[48] Ferdows, M., Uddin, M. J., and Afify, A. A. Scaling group transformation for MHD boundarylayer free convective heat and mass transfer flow past a convectively heated nonlinear radiatingstretching sheet. International Journal of Heat and Mass Transfer, 56, 181-187 (2013)
[49] Sparrow, E. M. and Cess, R. D. Radiation Heat Transfer, Hemisphere, Washington, D. C. (1978)
[50] Helmy, K. A. MHD boundary layer equations for a power-law fluid with variable electric conductivity.Meccanica, 30, 187-200 (1995)
[51] Tapanidis, T., Tsagas, G., and Mazumdar, H. P. Application of scaling group of transformationsto viscoelastic second grade fluid flow. Nonlinear Functional Analysis and Applications, 8, 345-350(2003)
[52] Mukhopadhyay, S. and Layek, G. C. Effects of variable fluid viscosity on flow past a heatedstretching sheet embedded in a porous medium in presence of heat source/sink. Meccanica, 47,863-876 (2012)
[53] Cortell, R. Heat and fluid flow due to non-linearly stretching surfaces. Applied Mathematics andComputation, 217, 7564-7572 (2011)
[54] White, R. E. and Subramanian, V. R. Computational Methods in Chemical Engineering withMaple, Springer, New York (2010)
[55] Uddin, M. J., Yusoff, N. H. M., Bég, O. A., and Ismail, A. I. M. Lie group analysis and numericalsolutions of boundary layer flow of non-Newtonian nanofluids along a horizontal plate in porousmedium with internal heat generation. Physica Scripta, 87, 025401 (2013)