Unsteady magnetohydrodynamic stagnation point flow-closed-form analytical solutions

doi:10.1007/s10483-019-2463-7

Abstract

Abstract:

This paper investigates the unsteady stagnation point flow and heat transfer of magnetohydrodynamic (MHD) fluids over a moving permeable flat surface. The unsteady Navier-Stokes (NS) equations are transformed into a similarity nonlinear ordinary differential equation, and a closed form solution is obtained for the unsteadiness parameter of 2. The boundary layer energy equation is transformed into a similarity equation, and is solved for a constant wall temperature and a time-dependent uniform wall heat flux case. The solution domain, velocity, and temperature profiles are calculated for different combinations of parameters including the Prandtl number, mass transfer parameter, wall moving parameter, and magnetic parameter. Two solution branches are obtained for certain combinations of the controlling parameters, and a stability analysis demonstrates that the lower solution branch is not stable. The present solutions provide an exact solution to the entire unsteady MHD NS equations, which can be used for validating the numerical code of computational fluid dynamics.

Key words: ordinary differential equations, singular perturbation, turning point problem, Navier-Stokes (NS) equation, magnetohydrodynamic (MHD), stretching/shrinking sheet, unsteady stagnation point flow

2010 MSC Number:

76W05

T. G. FANG, F. J. WANG, Bo GAO. Unsteady magnetohydrodynamic stagnation point flow-closed-form analytical solutions. Applied Mathematics and Mechanics (English Edition), 2019, 40(4): 449-464.

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References

[1] YANG, K. T. Unsteady laminar boundary layers in an incompressible stagnation flow. Transactions of ASME:Journal of Applied Mechanics, 25, 421-427(1958)
[2] WILLIAMS Ⅲ, J. C. Nonsteady stagnation-point flow. AIAA Journal, 6(12), 2419-2427(1968)
[3] JANKOWSKI, D. F. and GERSTING, J. M. Unsteady three-dimensional stagnation-point flow. AIAA Journal, 8(1), 187-188(1970)
[4] TEIPEL, I. Heat transfer in unsteady laminar boundary layers at an incompressible threedimensional stagnation flow. Mechanics Research Communications, 6(1), 27-32(1979)
[5] WANG, C. Y. The unsteady oblique stagnation point flow. Physics of Fluids, 28(7), 2046-2049(1985)
[6] RAJAPPA, N. R. Nonsteady plane stagnation point flow with hard blowing. Zeitschrift für angewandte Mathematik und Mechanik, 59, 471-473(1979)
[7] BURDE, G. I. Nonsteady stagnation-point flows over permeable surfaces:explicit solutions of the Navier-Stokes equations. Journal of Fluids Engineering-Transactions of the ASME, 117, 189-191(1995)
[8] LUDLOW, D. K., CLARKSON, P. A., and BASSOM, A. P. New similarity solutions of the unsteady incompressible boundary layer equations. The Quarterly Journal of Mechanics and Applied Mathematics, 53(2), 175-206(2000)
[9] MA, P. H. and HUI, W. H. Similarity solutions of the two dimensional unsteady boundary layer equations. Journal of Fluid Mechanics, 206, 537-559(1990)
[10] TAKHAR, H. S., CHAMKHA, A. J., and NATH, G. Unsteady axisymmetric stagnation-point flow of a viscous fluid on a cylinder. International Journal of Engineering Science, 37, 1943-1957(1999)
[11] ESWARA, A. T. and NATH, G. Effect of large injection rates on unsteady mixed convection flow at a three-dimensional stagnation point. International Journal of Non-Linear Mechanics, 34, 85-103(1999)
[12] KUMARI, M. and NATH, G. Unsteady flow and heat transfer of a viscous fluid in the stagnation region of a three-dimensional body with a magnetic field. International Journal of Engineering Science, 40, 411-432(2002)
[13] SESHADRI, R., SREESHYLAN, N., and NATH, G. Unsteady three-dimensional stagnation point flow of a viscoelastic fluid. International Journal of Engineering Science, 35, 445-454(1997)
[14] XU, H., LIAO, S. J., and POP, I. Series solution of unsteady boundary layer flows of nonNewtonian fluids near a forward stagnation point. Journal of Non-Newtonian Fluid Mechanics, 139, 31-43(2006)
[15] NAZAR, R., AMIN, N., FILIP, D., and POP, I. Unsteady boundary layer flow in the region of the stagnation point on a stretching sheet. International Journal of Engineering Science, 42, 1241-1253(2004)
[16] BARIS, S. and DOKUZ, M. S. Three-dimensional stagnation point flow of a second grade fluid towards a moving plate. International Journal of Engineering Science, 44, 49-58(2006)
[17] FANG, T. G., LEE, C. F., and ZHANG, J. The boundary layers of an unsteady incompressible stagnation-point flow with mass transfer. International Journal of Non-Linear Mechanics, 46, 942-948(2011)
[18] ZHONG, Y. and FANG, T. Unsteady stagnation-point flow over a plate moving along the direction of flow impingement. International Journal of Heat and Mass Transfer, 54, 3103-3108(2011)
[19] MAGYARI, E. and WEIDMAN, P. D. Comment on "Unsteady stagnation-point flow over a plate moving along the direction of flow impingement" by Y. Zhong and T. Fang. Internatioal Journal Heat Mass Transfer, 54, 3103-3108(2011)". International Journal of Heat and Mass Transfer, 55(4), 1425-1426(2012)
[20] FANG, T. and ZHONG, Y. Reply to "Comment on "Unsteady stagnation-point flow over a plate moving along the direction of flow impingement" by Y. Zhong and T. Fang. Internatioal Journal Heat Mass Transfer, 54, 3103-3108(2011)". International Journal of Heat and Mass Transfer, 55(4), 1425-1426(2012)
[21] SOID, S. K., ISHAK, A., and POP, I. MHD stagnation point flow over a stretching/shrinking sheet. 2015 International Symposium on Mathematical Sciences and Computing Research, 355-360(2015)
[22] CHEN, H., LIANG, H., WANG, F., and SHEN, M. Unsteady MHD stagnation point flow toward a shrinking sheet with thermal radiation and slip effects. Heat Transfer-Asian Research, 45(8), 730-745(2016)
[23] TURKYILMAZOGLU, M., NAGANTHRAN, K., and POP, I. Unsteady MHD rear stagnationpoint flow over off-centred deformable surfaces. International Journal of Numerical Methods for Heat and Fluid Flow, 27(7), 1554-1570(2017)
[24] ZAIB, A., BHATTACHARYYA, K., UROOJ, S. A., and SHAFIE, S. Dual solutions of an unsteady magnetohydrodynamic stagnation-point flow of a nanofluid with heat and mass transfer in the presence of thermophoresis. Proceedings of the Institution of Mechanical Engineers (Part E:Journal of Process Mechanical Engineering), 232(2), 155-164(2018)
[25] ZHU, J., ZHENG, L. C., and ZHANG, Z. G. Effects of slip condition on MHD stagnation-point flow over a power-law stretching sheet. Applied Mathematics and Mechanics (English Edition), 31(4), 439-448(2010) https://doi.org/10.1007/s10483-010-0404-z
[26] ZHENG, L., WANG, L., and ZHANG, X. Analytic solutions of unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform heat source/sink. Communications in Nonlinear Science and Numerical Simulation, 16(2), 731-740(2011)
[27] ZHANG, Y. and ZHENG, L. Analysis of MHD thermosolutal Marangoni convection with the heat generation and a first-order chemical reaction. Chemical Engineering Science, 69(1), 449-455(2012)
[28] ZHENG, L., ZHANG, C., ZHANG, X., and ZHANG, J. Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium. Journal of the Franklin Institute, 350(5), 990-1007(2013)
[29] LIN, Y., ZHENG, L., ZHANG, X., MA, L., and CHEN, G. MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generation. International Journal of Heat and Mass Transfer, 84, 903-911(2015)
[30] ZHANG, C., ZHENG, L., ZHANG, X., and CHEN, G. MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. Applied Mathematical Modelling, 39(1), 165-181(2015)
[31] FANG, T., ZHANG, J., and YAO, S. Slip MHD viscous flow over a stretching sheet-an exact solution. Communications in Nonlinear Science and Numerical Simulations, 14(11), 3731-3737(2009)
[32] FANG, T., YAO, S., and POP, I. Flow and heat transfer over a generalized stretching/shrinking wall problem-exact solutions of the Navier-Stokes equations. International Journal of NonLinear Mechanics, 46, 1116-1127(2011)
[33] FANG, T. and JING, W. Closed-form analytical solutions of flow and heat transfer for an unsteady rear stagnation-point flow. International Journal of Heat and Mass Transfer, 62, 55-62(2013)
[34] WOLFRAM, S. Mathematica -A System for Doing Mathematics by Computer, 2nd ed., AddisonWesley Publishing Company, New York (1993)
[35] MERKIN, J. On dual solutions occurring in mixed convection in a porous medium. Journal of Engineering Mathematics, 20, 171-179(1985)
[36] SOID, S. K., ISHAK, A., and POP, I. Unsteady MHD flow and heat transfer over a shrinking sheet with ohmic heating. Chinese Journal of Physics, 55(4), 1626-1636(2017)