# Unsteady Boundary Layer Flow over a Vertical Surface due to Impulsive and Buoyancy in the Presence of Thermal-Diffusion and Diffusion-Thermo using Bivariate Spectral Relaxation Method

Authors

1 School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa.

2 Department of Mathematical Sciences, Federal University of Technology, Akure, Ondo State Nigeria.

Abstract

In this article, unsteady boundary layer ﬂow formed over a vertical surface due to impulsive motion and buoyancy is investigated. The mathematical model which properly accounts for space and temperature-dependent internal heat source in a ﬂowing ﬂuid is incorporated into the energy equation. This model is presented in this study as a term which accounts for two different forms of internal heat generation during the short time period and long time period. Due to the ﬂuid ﬂow under consideration, the inﬂuence of thermal-diffusion and diffusion-thermo are incorporated into the governing equation since it may not be realistic to assume that both effects are of smaller order of magnitude than the effects described by Fourier’s or Fick’s law. The corresponding effect of internal heat source on viscosity is considered; the viscosity is assumed to vary as a linear function of temperature. The ﬂow model is described in terms of a highly coupled and nonlinear system of partial differential equations. The governing equations are non-dimensionalized by using suitable similarity transformation which unraveled the behavior of the ﬂuid ﬂow at short time and long time periods. The dimensionless system of non-linear coupled partial differential equations (PDEs) is solved using Bivariate Spectral Relaxation Method (BSRM). A parametric study of selected parameters is conducted and results of the surface shear stress, heat transfer and mass transfer at the wall are illustrated graphically and physical aspects of the problem are discussed.

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