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Boundary Layer Flow Over a Moving Vertical Flat Plate with Convective Thermal Boundary Condition

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Abstract

This paper studies the steady boundary layer flow over an impermeable moving vertical flat plate with convective boundary condition at the left side of the flat plate. The governing partial differential equations are transformed into a system of ordinary (similarity) differential equations by using corresponding similarity variables. These equations were then solved numerically using the function bvp4c from Matlab for different values of the Rayleigh number Ra, the convective heat transfer parameter \(\gamma \), and the Prandtl number Pr. This paper demonstrates that a similarity solution is possible if the convective boundary condition heat transfer is associated with the hot or cooled fluid on the left side of the flat plate proportional to \(x^{-1/4}\). For the sake of comparison of the numerical results, the case of the static flat plate \((\sigma =0)\) has been also studied. For the case of a moving flat plate \((\sigma =1)\), it is shown that the solutions have two branches in a certain range of the positive (assisting flow) and negative (opposing flow) values of the Rayleigh number Ra. In order to test the physically available solutions, a stability analysis has been also performed. The effects of the governing parameters on the skin friction, heat transfer, wall temperature, velocity and temperature profiles, as well as on the streamlines and isotherms are investigated. Comparison with results from the open literature shows a very good agreement.

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Abbreviations

\(a,A,b,c_1,c_1\) :

Constants

\(C_\mathrm{f}\) :

Skin friction coefficient

g :

Acceleration due to gravity

\(Gr_x\) :

Local Grashof number

\(h_\mathrm{f}\) :

Heat transfer coefficient

k :

Thermal conductivity

L :

Characteristic length of the plate

\(Nu_x\) :

Local Nusselt number

Pr :

Prandtl number

Ra :

Rayleigh numbers

\(Re_x\) :

Local Reynolds number

t :

Time

T :

Fluid temperature

\(T_\mathrm{f}\) :

Temperature of the hot fluid

\(T_\infty \) :

Temperature of the ambient fluid

\(T_\mathrm{w}\) :

Temperature of the plate

u, v :

Velocity components along and normal to the plate

\(U_\mathrm{w}(x)\) :

Velocity of the moving plate

x, y :

Coordinates along and normal to the plate

\(\alpha \) :

Thermal diffusivity

\(\beta \) :

Coefficient of thermal expansion

\(\varepsilon \) :

Eigenvalue parameter

\(\gamma \) :

Convective heat transfer

\(\eta \) :

Similarity variable

\(\mu \) :

Dynamic viscosity

\(\nu \) :

Kinematic viscosity

\(\theta \) :

Dimensionless temperature

\(\rho \) :

Density

\(\sigma \) :

Moving parameter

\(\tau \) :

Dimensionless time

\(\psi \) :

Dimensionless stream function

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Acknowledgments

The first author (A.V. Roşca) wishes to thank Romanian National Authority for Scientific Research, CNCS—UEFISCDI, project number PN-II-RU-TE-2011-3-0013, while the second author (Md. J. Uddin) would like to express his thanks to USM for the financial support.

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Authors and Affiliations

  1. Faculty of Economics and Business Administration, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Alin V. Roşca

  2. School of Mathematical Sciences, Universiti of Sains Malaysia, 11800 USM, Penang, Malaysia

    Md. J. Uddin

  3. Department of Mathematics, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Ioan Pop

Authors
  1. Alin V. Roşca
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  2. Md. J. Uddin
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  3. Ioan Pop
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Correspondence to Ioan Pop.

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Communicated by Ahmad Izani MD Ismail.

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Roşca, A.V., Uddin, M.J. & Pop, I. Boundary Layer Flow Over a Moving Vertical Flat Plate with Convective Thermal Boundary Condition. Bull. Malays. Math. Sci. Soc. 39, 1287–1306 (2016). https://doi.org/10.1007/s40840-015-0275-1

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  • DOI: https://doi.org/10.1007/s40840-015-0275-1

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