Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

doi:10.1016/j.amc.2010.06.026

Applied Mathematics and Computation

Volume 217, Issue 2, 15 September 2010, Pages 837-842

https://doi.org/10.1016/j.amc.2010.06.026 Get rights and content

Abstract

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.

Introduction

The term “similarity solution” in fluid mechanics was first introduced by Blasius [1] when solving an application problem of Prandtl’s boundary layer theory. The idea is to simplify the governing equations by reducing the number of independent variables, by a coordinate transformation. Analogous to dimensional analysis, instead of parameters, like the Reynolds number, the coordinates themselves are collapsed into dimensionless groups that scale the velocities [2]. The terminology “similarity” is used because, despite the growth of the boundary layer with distance x from the leading edge, the velocity profile u/U_∞ remains geometrically similar. The same concept was then extended to the temperature profile. However, not all problems admit similarity solutions, since they depend on various factors, such as the surface geometries, boundary conditions, and the surface heating conditions.

The heat transfer part of the above problem was solved by Pohlhausen [3], by assuming uniform plate temperature. Recently, Aziz [4] and Magyari [5] studied the similar problem, but with convective boundary condition. Aziz demonstrated that similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x^−1/2. He reported the results for Prandtl number Pr = 0.1, 0.72 and 10. However, the numerical results reported in [4] for Pr = 0.1 are not enough accurate, owing to the small boundary layer thickness set in all the computations. It is well known that the Prandtl number Pr is a ratio of viscous to conduction effects, the lower the Prandtl number, the thicker the thermal boundary layer [6].

The objective of the present study is to extend the work of Aziz [4], by introducing the effects of suction and injection on the flat surface, besides giving accurate numerical results for Pr = 0.1. The process of suction and injection (blowing) has its importance in many engineering applications such as in the design of thrust bearing and radial diffusers, and thermal oil recovery. Suction is applied to chemical processes to remove reactants. Blowing is used to add reactants, cool the surfaces, prevent corrosion or scaling and reduce the drag (see Labropulu et al. [7]).

Section snippets

Problem formulation

Consider a steady two-dimensional laminar boundary layer flow over a static permeable flat plate immersed in a viscous fluid of temperature T_∞. It is assumed that the free stream moves on the top of the solid surface with a constant velocity U_∞. The boundary layer equations are [4], [6]: $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,$ $u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = ν \frac{\partial^{2} u}{\partial y^{2}},$ $u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = α \frac{\partial^{2} T}{\partial y^{2}},$ where u and v are the velocity components in the x and y directions, respectively, T is the fluid temperature in the boundary layer, ν is the kinematic

Results and discussion

The ordinary differential equations (8), (9) subject to the boundary conditions (12) were solved numerically using the symbolic algebra software Maple described in [4]. Table 1 presents the comparison for the values of −θ′(0) with those reported by Aziz [4], which shows an excellent agreement for Pr = 0.72 and Pr = 10. We note that the values of θ(0) as reported in [4] can be calculated from Eq. (12). For Pr = 0.1, the results reported in [4] are not enough accurate, owing to the small boundary layer

Conclusions

The problem of steady laminar boundary layer flow and heat transfer over a stationary permeable flat plate immersed in an uniform free stream with convective boundary condition was considered. Similarity solutions are possible if the convective heat transfer from the lower surface and the mass transpiration rate at the surface vary like x^−1/2, where x is the distance from the leading edge of the solid surface. It was found that suction increases the surface shear stress and as a consequence

Acknowledgements

The author would like to express his very sincere thanks to the editor and the anonymous referee for their valuable comments and suggestions. The financial supports received from Ministry of Science, Technology and Innovation, Malaysia (Project Code: 06-01-02-SF0610) and Universiti Kebangsaan Malaysia (Project Code: UKM-GGPM-NBT-080-2010) are gratefully acknowledged.

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Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

Abstract

Introduction

Section snippets

Problem formulation

Results and discussion

Conclusions

Acknowledgements

Commun. Nonlinear Sci. Numer. Simul.

Appl. Math. Comput.

Int. J. Heat Mass Transfer

Appl. Math. Comput.

Appl. Math. Model.

Grenzschichten in flüssigkeiten mit kleiner reibung

Z. Math. Phys.

Viscous Fluid Flow