Sains Malaysiana 43(8)(2014): 1239-1247

Note on Dual Solutions for the Mixed Convection Boundary Layer Flow Close to the Lower Stagnation Point of a Horizontal Circular Cylinder: Case of Constant Surface Heat Flux

(Nota Dua Penyelesaian bagi Aliran Lapisan Sempadan Perolakan Bercampur Hampir

dengan Rendah Silinder Bulat Mendatar: Kes Fluks Haba Permukaan Malar)

Alin V. Roşca¹, Natalia C. Roşca² & Ioan Pop²*

¹Faculty of Economics and Business Administration, Department of Statistics, Forecasts and Mathematics

Babeş-Bolyai University, R-400084 Cluj-Napoca, Romania

²Faculty of Mathematics and Computer Science, Department of Mathematics, Babeş-Bolyai University

R-400084 Cluj-Napoca, Romania

Received: 29 April 2013/Accepted: 4 December 2013

ABSTRACT

The paper reconsiders the problem of the mixed convection boundary layer flow near the lower stagnation point of a horizontal circular cylinder with a second order slip velocity model and a constant surface heat flux studied recently by Roşca et al. (2013). The ordinary (similarity) differential equations are solved numerically using the function bvp4c from Matlab for different values of the governing parameters. It is found that the similarity equations have two branches, upper and lower branch solutions, in a certain range of the mixed convection parameters. A stability analysis has been performed to show that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and therefore, not physically possible. This stability analysis is different by that presented by Roşca et al. (2013), who have presented a time-dependent analysis to determine the stability of the solution branches.

Keywords:  Dual solutions; mixed convection; numerical solution; second-order slip flow; similarity solution; stagnation point

ABSTRAK

Kertas ini mempertimbangkan semula masalah aliran lapisan sempadan perolakan bercampur berhampiran titik genang rendah silinder bulat mendatar dengan model halaju gelincir peringkat kedua dan fluks haba permukaan malar yang dikaji oleh Roşca et al. (2013) sebelum ini. Persamaan pembezaan biasa (keserupaan) diselesaikan secara berangka menggunakan bvp4c fungsi dari Matlab bagi nilai berbeza daripada parameter pengelasan. Adalah didapati bahawa keserupaan persamaan mempunyai dua cabang, penyelesaian cabang atas dan bawah dalam sesetengah julat parameter perolakan bercampur. Analisis kestabilan yang telah dijalankan menunjukkan bahawa penyelesaian cabang atas adalah stabil dan tersedia secara fizikal, manakala penyelesaian cabang bawah adalah tidak stabil dan oleh itu, tidak mungkin tersedia secara fizikal. Analisis kestabilan ini adalah berbeza daripada yang dikemukakan oleh Roşca et al. (2013) yang telah menyampaikan analisis bersandar- masa untuk menentukan kestabilan cabang penyelesaian.

Kata kunci: Dua penyelesaian; halaju gelincir peringkat kedua; penyelesaian berangka; penyelesaian persamaan; perolakan bercampur; titik stagnasi

REFERENCES

Amin, N. & Riley, N. 1995. Mixed convection at a stagnation point. Quart. J. Mech. Appl. Math. 48: 111-121.

Ariel, P.D. 1994. Stagnation point flow with suction: An approximate solution. ASME J. Appl. Mech. 61: 976-978.

Aziz, A. 2009. A similarity solution for laminar thermal boundary layer over flat plate with   convective surface boundary condition. Commun Non. Sci. Numer Simulat. 14: 1064-1068.

Bian, X. & Rangel, R. H. 1996. The viscous stagnation flow solidification problem. Int. J. Heat Mass Transfer 39: 3581-3594.

Fang, T. & Lee, C.F. 2005. A moving-wall boundary layer flow of a slightly rarefied gas free  stream over a moving flat plate. Appl. Math. Lett. 18: 487-495.

Fang, T. & Lee, C.F. 2006. Exact solutions of incompressible Couette flow with porous walls for slightly rarefied gases. Heat Mass Transfer 42: 255-262.

Fang, T., Yao, S., Zhang, J. & Aziz, A.  2010. Viscous flow over a shrinking sheet with a second order  slip flow model. Commun. Nonlinear Sci. Numer. Simulat. 15: 1831-1842.

Harris, S.D., Ingham, D.B. & Pop, I. 2009. Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip. Transport Porous Media77: 267-285.

Hiemenz, K. 1911. Die Grenzschicht an einem in den gleichformigen Flssigkeitsrom eingetauchten geraden Kreiszylinder. Dinglers Polytech. J. 326: 321-324.

Makinde, O.D. & Olanrewaju, P.O. 2010. Buoyancy effects on the thermal boundary layer over a vertical flat plate with a convective surface boundary conditions. ASME  Fluid Eng. 132: 044502-1 -044502-4.

McCroskey, W.J. 1977. Some current research in unsteady fluid dynamics - The 1976 Freeman scholarship lecture. ASME Journal of Fluid Engineering 99: 8-39.

Merkin, J.H. 1985.  On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math. 20: 171-179.

Merkin, J.H. & Pop, I. 2011. The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition. Commun. Nonlinear Sci. Numerical Simul. 16: 3602-3609.

Ramachandran, N., Chen, T.S. & Armaly, B.F. 1988. Mixed convection in stagnation flows adjacent to vertical surfaces. ASME J. Heat Transfer 110: 373-377.

Roşca, A.V. & Pop, I. 2013. Flow and heat transfer over a vertical permeable stretching/  shrinking sheet with a second order slip. Int. J. Heat Mass Transfer 60: 355-364.

Roşca, N.C., Roşca, A.V., Merkin, J.H. & Pop, I. 2013. Mixed convection boundary-layer flow near the lower stagnation point of a horizontal circular cylinder with a second-order wall velocity condition and a constant surface heat flux. The IMA Journal of Applied Mathematics. doi:10.1093/imamat/hxt045.

Seshadri, R., Sreeshylan, N. & Nath, G. 2002. Unsteady mixed convection flow in the stagnation region of a heated vertical plate due to impulsive motion. Int. J. Heat Mass Transfer 45: 1345-1352.

Wang, C.Y. 2003. Stagnation flows with slip: Exact solutions of the Navier-Stokes Equations. J. Appl. Math. Phys. (ZAMP) 54: 184-189.

Wang, C.Y. 2006. Stagnation slip flow and heat transfer on a moving plate. Chem. Engng. Sci. 61: 7668-7672.

Weidman, P.D., Kubitschek, D.G. & Davis, A.M.J. 2006. The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int. J. Engng. Sci. 44: 730-737.

Wu, L. 2008. A slip model for rarefied gas flows at arbitrary Knudsen number. Appl. Phys. Lett. 93: 253103.

*Corresponding author; email: popm.ioan@yahoo.co.uk