An implicit-Chebyshev pseudospectral method for the effect of radiation on power-law fluid past a vertical plate immersed in a porous medium

Abstract

An implicit-Chebyshev collocation spectral method is employed in this study. This method was used to compute the problem of unsteady free convection with heat transfer from an isothermal vertical flat plate to a non-Newtonian fluid saturated porous medium, which is modeled as a power-law fluid. Boundary layer and Boussinesq approximations have been incorporated. The Darcy–Brinkman–Forchheimer model is applied to describe the flow field, where the magnetic field and the radiation effects are taken into account. Because of the non-Newtonian rheology, these problems are non-linear and must be solved numerically. The domain of the problem is discretized according to the implicit-Chebyshev spectral collocation scheme. In this study, the spatial derivatives are computed with a differentiation matrix and the time derivatives are computed with Crank–Nicolson implicit finite-difference method. Numerical calculations are carried out for the various parameters entering into the problem. Velocity and temperature profiles are shown in tables and graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficients approach the steady state values.

Introduction

The study of the boundary layer behavior on continuous surfaces is an important field, because the analysis of such flows finds applications in different areas. Examples include the cooling of a metallic plate in a cooling bath, the boundary layer along material handling conveyers and the boundary layer along a liquid film in condensation processes. In the bulk of heat transfer over plates by either natural, forced or combined convection, many studies involving theoretical or experimental investigations have been published in the literature. Most of these studies are based upon the laminar boundary-layer approach. Also, the study of magneto-fluid flows in a slip-flow regime with heat transfer has important engineering applications, for example, in power generators, magneto-hydrodynamic (MHD) accelerators, refrigeration coils, transmission lines, electric transformers and heating elements. It serves as the basis for understanding some of the important phenomena occurring in heat exchangers. The uses of magnetic field to control the flow and heat transfer processes in fluids near different types of boundaries are now well known. This has led to considerable interest in the study of boundary layer flows subjected to an externally applied magnetic field. The effect of radiation on MHD flow and heat transfer problems has become more important industrially. Also, many processes in engineering occur at high temperatures and radiation heat transfer becomes very important for the design of pertinent equipment. Nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. At high operating temperature, radiation can be quite significant.

Pascal [1] presented similarity solutions for axisymmetric plane radial power law fluid flows through a porous medium. Darcy–Forchheimer natural, forced and mixed convection heat transfer in power-law fluid saturated porous media was studied by Shenoy [2]. The problem of forced convection heat transfer on a flat plate embedded in porous media for power-law fluids has been studied by Hady and Ibrahim [3]. Kinyanjui et al. [4] studied the MHD free convection heat and mass transfer of a heat generating fluid past an impulsively started infinite vertical porous plate with Hall current and radiation absorption. Anwar Hossain and Mike Wilson [5] discussed the Natural convection flow in a fluid-saturated porous medium enclosed by non-isothermal walls with heat generation. Numerical modeling of non-Newtonian fluid flow in a porous medium using a three-dimensional periodic array was presented by Inoue and Nakayama [6]. All these studies were concerned with steady flows. Pascal [7] presented similarity solutions to some unsteady flows of non-Newtonian fluids of power law behavior. Pascal and Pascal [8] studied the non-linear effects on some unsteady non-Darcian flows through porous media. Unsteady forced convection heat transfer on a flat plate embedded in the fluid-saturated porous medium with inertia effect and thermal dispersion is investigated by Cheng and Lin [9]. Israel-Cookey et al. [10] discussed the influence of viscous dissipation and radiation on unsteady MHD free-convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction. Recently, Chiem and Zhao [11] studied the problem of numerical study of steady/unsteady flow and heat transfer in porous media using a characteristics-based matrix-free implicit finite volume method on unstructured grids.

The non-Newtonian behavior of the fluid will render the heat transport problem much more difficult to analyze than that for a Newtonian fluid. This problem is non-linear, and therefore, unlike the Newtonian case. Then the heat transport velocity cannot be found simply by the first time-averaging second-order differential equation, which instead must be solved in full before the steady component can be separated from the complete solution. To solve numerically a partial differential equation, the finite-difference or finite-element methods are often used to compute the spatial derivatives. But these methods will typically require a large number of nodal points in order to yield satisfactory results. As a promising alternative, the spectral and pseudospectral methods have been greatly advanced in recent years. The spectral method distinguishes itself from the finite-difference and finite-element methods by the fact that global information is incorporated in computing a spatial derivative. The spectral method can yield greater accuracy for a smooth solution with far fewer nodes and therefore less computational time than the finite-difference and finite-element schemes. A wide variety of spectral schemes applied to fluid dynamics exist in the literature and have been reviewed by Canuto et al. [12] and Peyret [13]. Panchang and Kopriva [14] presented a numerical study of the Euler equations for water wave propagation over complicated bathymetry and gave a comparison of performance by the Chebyshev spectral and finite-difference methods for the one-dimensional case. They showed that, the Chebyshev spectral collocation method is remarkably superior in terms of accuracy and the number of nodes required.

Hence, the objective of this paper is to investigate the radiation effect in the presence of magnetic field on unsteady free convection with heat transfer from an isothermal vertical flat plate to a non-Newtonian power-law fluid saturated porous medium. The Darcy–Brinkman–Forchheimer model, which includes the effects of boundary and inertia forces is employed. The dimensionless non-linear partial differential equations are solved numerically by using an implicit-Chebyshev pseudospectral scheme. The values of friction factor and heat transfer coefficient are determined for steady and unsteady free convection. Also, the effects of radiation and magnetic field parameters on the flow and heat transfer have been shown in tables and graphically. In this work the maximum errors between the exact solution and numerical solutions for Burgers equation have been shown in table where we show that, the implicit-Chebyshev pseudospectral method is more accurate as compared with the finite-difference method (Crank–Nicolson).