MIXED CONVECTION BOUNDARY-LAYER FLOW OF A MICRO POLAR FLUID TOWARDS A HEATED SHRINKING SHEET BY HOMOTOPY ANALYSIS METHOD

Abstract

A comprehensive study of two dimensional stagnation flow of an incompressible micro polar fluid with heat transfer characteristics towards a heated shrinking sheet is analyzed analytically. The main goal of this paper is to find the analytic solutions using a powerful technique namely the Homotopy Analysis Method (HAM) for the velocity and the temperature distributions and to study the steady mixed convection in two-dimensional stagnation flows of a micro polar fluid around a vertical shrinking sheet. The governing equations of motion together with the associated boundary conditions are first reduced to a set of self-similar nonlinear ordinary differential equations using a similarity transformation and are then solved by the HAM. Some important features of the flow and heat transfer for the different values of the governing parameters are analyzed, discussed and presented through tables and graphs. The heat transfer from the sheet to the fluid decreases with an increase in the shrinking rate. Micro polar fluids exhibit a reduction in shear stresses and heat transfer rate as compared to Newtonian fluids, which may be beneficial in flow and thermal control of polymeric processing.

Dates

  • Submission Date2013-02-12
  • Revision Date2013-06-23
  • Acceptance Date2013-06-23
  • Online Date2013-07-06

DOI Reference

10.2298/TSCI130212096R

References

  1. Rashidi, M.M., Erfani, E., A new analytical study of MHD stagnation-point flow in porous media with heat transfer, Computers and Fluids, 40 (2011), 1, pp. 172-178
  2. Hiemenz, K., Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder, Weber, Berlin, Germany, 1911
  3. Chamkha, A.J., Hydromagnetic plane and axisymmetric flow near a stagnation point with heat generation, International Communications in Heat and Mass Transfer, 25 (1998), 2, pp. 269- 278
  4. Chamkha, A.J., Issa, C., Effects of heat generation/ absorption and thermophoresis on hydromagnetic flow with heat and mass transfer over a flat surface, International Journal of Numerical Methods for Heat & Fluid Flow, 10 (2000), 4, pp. 432-449
  5. Mahapatra, T.R, Gupta, A.S., Magnetohydrodynamic stagnation-point flow towards a stretching sheet, Acta Mechanica, 152 (2001), 1-4, pp. 191-196
  6. Nazar, R., Amin, N., Filip, D., Pop, I., Unsteady boundary layer flow in the region of the stagnation point on a stretching sheet, International Journal of Engineering Science, 42 (2004), 11-12, pp. 1241-1253
  7. Abdelkhalek, M.M., The skin friction in the MHD mixed convection stagnation point with mass transfer, International Communications in Heat and Mass Transfer, 33 (2006), 2, pp. 249-258
  8. Layek, G.C., Mukhopadhyay, S., Samad, Sk.A., Heat and mass transfer analysis for boundary-layer stagnation point flow towards a heated porous stretching sheet with heat absorption/generation and suction/ blowing, International Communications in Heat and Mass Transfer, 34 (2007), 3, pp. 347-356
  9. Paullet, J., Weidman, P., Analysis of stagnation point flow toward a stretching sheet, International Journal of Non-Linear Mechanics, 42 (2007), 9, pp. 1084-1091
  10. Yian, L.Y., Amin, N., Pop, I., Mixed convection flow near a non orthogonal stagnation point towards a stretching vertical plate. International Journal of Heat and Mass Transfer, 50 (2007), 23-24, pp. 4855-4863
  11. Sharma, P.R., Singh, G., Effects of variable thermal conductivity and heat source/ sink on MHDflow near a stagnation point on a linearly stretching sheet, Journal of Applied Fluid Mechanics, 2 (2009), 1, pp. 13-21
  12. Wang, C.Y., Stagnation flow towards a shrinking sheet, International Journal of Non-Linear Mechanics, 43 (2008), 5, pp. 377-382
  13. Hayat, T., Abbas, Z., Sajid, M., MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface, Chaos, Solitons & Fractals, 39 (2009), 2, pp. 840-848
  14. Takhar, H.S., Kumari, M., Nath, G.G., Unsteady free convection flow under the influence of a magnetic field, Archive of Applied Mechanics, 63 (1993), 4-5, pp. 313-321
  15. Kumaran, V., Banerjee, A.K., Vanav Kumar, A., Vajravelu, K., MHD flow past a stretching permeable sheet, Applied Mathematics and Computation, 210 (2009), 1, pp. 26-32
  16. Hassanien, I.A., Al-arabi, T.H., Non Darcy unsteady mixed convection flow near the stagnation point on a heated vertical surface embedded in a porous medium with thermal radiation and variable viscosity, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 4, pp. 1366-1376
  17. Hayat, T., Abbas, Z., Javed, T., MHD stagnation point flow and heat transfer over a permeable surface through a porous space, Journal of Porous Media, 12 (2009), 2, pp. 183-195
  18. Hoyt, J.W., Fabula, A.G., The effect of additives on fluid friction, U.S. Naval Ordinance Test Station Report, 1964
  19. Eringen, A.C., Theory of Micro polar Continua, In Proceedings of the Ninth Midwestern Conference, Wisconsin, (Wiley) 16-18 August, New York, 1965, pp. 23
  20. Eringen, A.C., Simple Micro fluids, International Journal Engineering and Science, 2 (1964), 2, pp. 205-217
  21. Eringen, A.C., Theory of Micro polar Fluids, Journal of Mathematics and Mechanics, 16 (1966), pp. 1-18
  22. Ariman, T., Turk, M.A., Sylvester, N.D., Microcontinuum fluid mechanics-A review, International Journal of Engineering Science, 11 (1973), 8, pp. 905-930
  23. Ariman, T., Turk, M.A., Sylvester, N.D., Application of microcontinuum fluid mechanics, International Journal of Engineering Science, 12 (1974), 4, pp. 273-293
  24. Nazar, R., Amin, N., Filip, D., Pop, I., Stagnation point flow of a micropolar fluid towards a stretching sheet, International Journal of Non-Linear Mechanics, 39 (2004), 7, pp. 1227-1235
  25. Chang, C.-L., Numerical simulation of micropolar fluid flow along a flat plate with wall conduction and boundary effects, Journal of Physics D: Applied Physics, 39 (2006), 6, pp. 1132-1140
  26. Kumari, M., Pop I., Nath, G., Unsteady MHD boundary-layer flow and heat transfer of a non Newtonian fluid in the stagnation region of a two dimensional body, Magnetohydrodynamics, 43 (2007), 3, pp. 301-314
  27. Lok, Y.Y., Pop, I., Chamkha, A.J., Non-orthogonal stagnation point flow of a micropolar fluid, International Journal of Engineering Science, 45 (2007), 1, pp. 173-184
  28. Ishak, A., Nazar, R., Pop, I., Magnetohydrodynamic (MHD) flow of a micro polar fluid towards a stagnation point on a vertical surface, Computers & Mathematics with Applications, 56 (2008), 12, pp. 3188-3194
  29. Abel M.S., Nandeppanavar, M.M., Heat transfer in MHD viscoelastic boundary-layer flow over a stretching sheet with non uniform heat source/sink, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 5, pp. 2120-2131
  30. Ashraf, M., Anwar Kamal, M., Syed, K.S., Numerical study of asymmetric laminar flow of micropolar fluids in a porous channel, Computers & Fluids, 38 (2009), 10, pp. 1895-1902
  31. Ashraf, M., Anwar Kamal M., Syed, K.S., Numerical simulation of a micro polar fluid between a porous disk and a non-porous disk, Applied Mathematical Modelling, 33 (2009), 4, pp. 1933-1943
  32. Ashraf, M., Anwar Kamal, M., Syed, K.S., Numerical investigations of asymmetric flow of a micropolar fluid between two porous disks, Acta Mechanica Sinica, 25 (2009), 6, pp. 787-794
  33. Lyapunov, A.M., The general problem of the stability of motion, International Journal of Control, 55 (1992), 3, pp. 531-534
  34. Karmishin, A.V., Zhukov, A.I., Kolosov, V.G., Methods of Dynamics Calculation and Testing for Thin-Walled Structures, Mashinostroyenie, Moscow, Russia, 1990
  35. Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston and London, 1994
  36. Liao, S.J., Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, 2004
  37. Liao, S.J., On the analytic solution of magneto hydrodynamic flows of non-Newtonian fluids over a stretching sheet, Journal of Fluid Mechanics, 488 (2003), pp. 189-212
  38. Abbasbandy, S., Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chemical Engineering Journal, 136 (2008), 2-3, pp. 144-150
  39. Allan, F.M., Construction of analytic solution to chaotic dynamical systems using the homotopy analysis method, Chaos, Solitons & Fractals, 39 (2009), 4, pp. 1744-1752
  40. Sajid, M., Siddiqui, A.M., Hayat, T., Wire coating analysis using MHD Oldroyd 8-constant fluid, International Journal of Engineering Science, 45 (2007), 2-8, pp. 381-392
  41. Rashidi, M.M., Siddiqui, A.M., Asadi, M., Application of homotopy analysis method to the unsteady squeezing flow of a second grade fluid between circular plates, Mathematical Problems in Engineering, 2010 (2010), doi:10.1155/2010/706840
  42. Rashidi, M.M., Domairry,G., Dinarvand, S., The Homotopy Analysis Method for Explicit Analytical Solutions of Jaulent-Miodek Equations, Numerical Methods for Partial Differential Equations 25 (2) (2009), pp. 430-439.
  43. Rashidi, M.M., Mohimanian Pour, S.A., Analytic Approximate Solutions for Unsteady Boundary-Layer Flow and Heat Transfer due to a Stretching Sheet by Homotopy Analysis Method, Nonlinear Analysis: Modelling and Control 15 (1) (2010), pp. 83-95.
  44. Ziabakhsh, Z., Domairry, G., Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation,14 (2009), 4, pp. 1284-1294
  45. Rashidi, M.M., Momoniat, E., Rostami, B., Analytic approximate solutions for MHD boundary-layer viscoelastic fluid flow over continuously moving Stretching surface by Homotopy Analysis Method with two auxiliary parameters, Journal of Applied Mathematics, 2012 (2012), doi:10.1155/2012/780415
  46. Bergman, T.L., Lavine, A.S., Incropera, F.P., Dewitt, D.P., Introduction to heat transfer, John Wiley and Sons Inc., New York, USA, Sixth edition, 2011
  47. Guram, G.S., Anwar, M., Micropolar flow due to a rotating disc with suction and injection, ZAMM-Journal of Applied Mathematics and Mechanics, 61 (1981), 11, pp. 589-595
  48. Takhar, H.S., Bhargava, R., Agraval, R.S., Balaji, A.V.S., Finite element solution of micropolar fluid flow and heat transfer between two porous discs, International Journal of Engineering Science, 38 (2000), 17, pp. 1907-1922