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Mixed convection in the stagnation-point flow of a Maxwell fluid towards a vertical stretching surface. (English) Zbl 1196.35160

Summary: In the present analysis, we study the steady mixed convection boundary layer flow of an incompressible Maxwell fluid near the two-dimensional stagnation-point flow over a vertical stretching surface. It is assumed that the stretching velocity and the surface temperature vary linearly with the distance from the stagnation-point. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. Analytical and numerical solutions of the derived system of equations are developed. The homotopy analysis method (HAM) and finite difference scheme are employed in constructing the analytical and numerical solutions, respectively. Comparison between the analytical and numerical solutions is given and found to be in excellent agreement. Both cases of assisting and opposing flows are considered. The influence of the various interesting parameters on the flow and heat transfer is analyzed and discussed through graphs in detail. The values of the local Nusselt number for different physical parameters are also tabulated. Comparison of the present results with known numerical results of viscous fluid is shown and a good agreement is observed.

MSC:

35Q35 PDEs in connection with fluid mechanics
76A10 Viscoelastic fluids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35A24 Methods of ordinary differential equations applied to PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
80A20 Heat and mass transfer, heat flow (MSC2010)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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[1] Zierep, J.; Fetecau, C., Energetic balance for the Rayleigh-Stokes problem of a Maxwell fluid, Int. J. Eng. Sci., 45, 617-627 (2007) · Zbl 1213.76026
[2] Fetecau, C.; Fetecau, C., Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder, Int. J. Eng. Sci., 44, 788-796 (2006) · Zbl 1213.76014
[3] Vieru, D.; Nazar, M.; Fetecau, C.; Fetecau, C., New exact solutions corresponding to the first problem of Stokes for Oldroyd-B fluids, Comput. Math. Appl., 55, 1644-1652 (2008) · Zbl 1135.76005
[4] Tan, W. C.; Masuoka, T., Stokes first problem for an Oldroyd-B fluid in a porous half space, Phys. Fluids, 17, 023101-023107 (2005) · Zbl 1187.76517
[5] Rajagopal, K. R., Flow of viscoelastic fluids between rotating disks, Theor. Comput. Fluid Dyn., 3, 185-206 (1992) · Zbl 0747.76018
[6] Dunn, J. E.; Rajagopal, K. R., Fluids of differential type-critical-review and thermodynamic analysis, Int. J. Eng. Sci., 33, 689-729 (1995) · Zbl 0899.76062
[7] Fosdick, R. L.; Rajagopal, K. R., Thermodynamics and stability of fluids of third grade, Proc. R. Soc. Lond. A, 339, 351-377 (1980) · Zbl 0441.76002
[8] Rajagopal, K. R.; Na, T. Y., On Stoke’s problem for a non-Newtonian fluid, Acta Mech., 48, 233-239 (1983) · Zbl 0528.76003
[9] Hayat, T.; Ali, N.; Asghar, S., Hall effects on peristaltic flow of a Maxwell fluid in a porous medium, Phys. Lett. A, 363, 397-403 (2007) · Zbl 1197.76126
[10] Hayat, T.; Khan, S. B.; Khan, M., The influence of Hall current on the rotating oscillating flows of an Oldroyd-B fluid in a porous medium, Non-Linear Dyn., 47, 353-362 (2007) · Zbl 1180.76071
[11] Hayat, T.; Abbas, Z.; Sajid, M.; Asghar, S., The influence of thermal radiation on MHD flow of a second grade fluid, Int. J. Heat Mass Transfer, 50, 931-941 (2007) · Zbl 1124.80325
[12] Rajagopal, K. R., On boundary conditions for fluids of the differential type, (Sequira, A., Navier-Stokes Equations and Related Non-linear Problems (1995), Plenum Press: Plenum Press New York), 273-278 · Zbl 0846.35107
[13] Rajagopal, K. R., Boundedness and uniqueness of fluids of the differential type, Acta Cienca Indica, 18, 1-11 (1982)
[14] Rajagopal, K. R.; Gupta, A. S., An exact solution for the flow of a non-Newtonian fluid past an infinite plate, Meccanica, 19, 158-160 (1984) · Zbl 0552.76008
[15] Rajagopal, K. R.; Kaloni, P. N., Some remarks on boundary conditions for fluids of the differential type, (Graham, G. A.C.; Malik, S. K., Continuum Mechanics and its Applications (1989), Hemisphere: Hemisphere New York), 935-942
[16] Rajagopal, K. R.; Szeri, A. Z.; Troy, W., An existence theorem for the flow of a non-Newtonian fluid past an infinite porous plate, Int. J. Non-Linear Mech., 21, 279-289 (1986) · Zbl 0599.76013
[17] Sakiadis, B. C., Boundary layer behavior on continuous solid surfaces, II. The boundary layer on a continuous flat surface, AIChE J., 7, 221-225 (1961)
[18] Phan-Thien, N., Plane and axisymmetric stagnation flow of a Maxwellian fluid, Rheol. Acta, 22, 127-130 (1983) · Zbl 0511.76011
[19] Zheng, R.; Phan-Thien, N.; Tanner, R. I., On the flow past a sphere in a cylindrical tube: Limiting Weissenberg number, J. Non-Newtnonian Fluid Mech., 36, 27-49 (1990)
[20] Nazar, R.; Amin, N.; Filip, D.; Pop, I., Unsteady boundary layer flow in the region of the stagnation point on the stretching sheet, Int. J. Eng. Sci., 42, 1241-1253 (2004) · Zbl 1211.76042
[21] Mahapatra, T. R.; Gupta, A. S., Heat transfer in stagnation-point flow towards a stretching sheet, Heat Mass Transfer, 38, 517-521 (2002)
[22] Mahapatra, T. R.; Gupta, A. S., Stagnation-point flow of a viscoelastic fluid towards a stretching surface, Int. J. Non-Linear Mech., 39, 811-820 (2004) · Zbl 1221.76035
[23] Ishak, A.; Nazar, R.; Pop, I., Mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet, Meccanica, 41, 509-518 (2006) · Zbl 1163.76412
[24] Hayat, T.; Abbas, Z.; Pop, I., Mixed convection in the stagnation point flow adjacent to a vertical surface in a viscoelastic fluid, Int. J. Heat and Mass Transfer, 51, 3200-3206 (2008) · Zbl 1144.80315
[25] Liao, S. J., Beyond Perturbation: Introduction to Homotopy Analysis Method (2003), Chapman & Hall/CRC Press: Chapman & Hall/CRC Press Boca Raton
[26] S.J. Liao, On the proposed homotopy analysis technique for non-linear problems and its applications, Ph. D. Dissertation, Shanghai Jiao Tong University 1992; S.J. Liao, On the proposed homotopy analysis technique for non-linear problems and its applications, Ph. D. Dissertation, Shanghai Jiao Tong University 1992
[27] Liao, S. J., A new branch of solutions of boundary-layer flows over a permeable stretching plate, Int. J. Non-Linear Mech., 42, 819-830 (2007) · Zbl 1200.76046
[28] Tan, Y.; Liao, S. J., Series solution of three-dimensional unsteady laminar viscous flow due to a stretching surface in a rotating fluid, ASME J. Appl. Mech., 74, 1011-1018 (2007)
[29] Xu, H.; Liao, S. J.; Wu, G. X., A family of new solutions on the wall jet, European J. Mech. B/Fluids, 27, 322-334 (2008) · Zbl 1154.76335
[30] Cheng, J.; Liao, S. J.; Mohapatra, R. N.; Vajravelu, K., Series solutions of nano boundary layer flows by means of the homotopy analysis method, J. Math. Anal. Appl., 343, 233-245 (2008) · Zbl 1135.76016
[31] Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A, 360, 109-113 (2006) · Zbl 1236.80010
[32] Tan, Y.; Abbasbandy, S., Homotopy analysis method for quadratic Recati differential equation, Commu. Non-linear Sci. Numer. Simul., 13, 539-546 (2008) · Zbl 1132.34305
[33] Abbasbandy, S.; Zakaria, F. S., Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear Dyn., 51, 83-87 (2008) · Zbl 1170.76317
[34] Inc, M., On numerical solution of Burger’s equation by homotopy analysis method, Phys. Lett. A, 372, 356-360 (2008) · Zbl 1217.76019
[35] Hayat, T.; Abbas, Z.; Sajid, M., On the analytic solution of MHD flow of a second grade fluid over a shrinking sheet, ASME J. Appl. Mech., 74, 1165-1171 (2007)
[36] Sajid, M.; Hayat, T.; Asgher, S., Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet, Int. J. Heat Mass Transfer, 50, 1723-1736 (2007) · Zbl 1140.76042
[37] Hayat, T.; Abbas, Z.; Javed, T., Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet, Phys. Lett. A, 372, 637-647 (2008) · Zbl 1217.76014
[38] Hayat, T.; Abbas, Z., Channel flow of a Maxwell fluid with chemical reaction, Z. Angew. Math. Phys., 59, 124-144 (2008) · Zbl 1133.76053
[39] Khan, M.; Abbas, Z.; Hayat, T., Analytic solution for flow of Sisko fluid through a porous medium, Trans. Porous Med., 71, 23-37 (2008)
[40] Abbas, Z.; Sajid, M.; Hayat, T., MHD boundary layer flow of an upper-convected Maxwell fluid in porous channel, Theor. Comput. Fluid Dyn., 20, 229-238 (2006) · Zbl 1109.76065
[41] Hayat, T.; Javed, T.; Sajid, M., Analytical solution for rotating flow and heat transfer analysis of a third-grade fluid, Acta Mech., 191, 219-229 (2007) · Zbl 1117.76069
[42] Abbas, Z.; Wang, Y.; Hayat, T.; Oberlack, M., Hydromagnetic flow in a viscoelastic fluid due to the oscillatory stretching surface, Int. J. Non-Linear Mech., 43, 783-793 (2008) · Zbl 1203.76169
[43] Liao, S., Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul., 14, 983-997 (2009) · Zbl 1221.65126
[44] Hayat, T.; Abbas, Z.; Sajid, M., Series solution for the upper-convected Maxwell fluid over a porous stretching plate, Phys. Lett. A, 358, 396-403 (2006) · Zbl 1142.76511
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