Transient Natural Convection Inside Porous Cavities: Hybrid Numerical-Analytical Solution and Mixed Symbolic-Numerical Computation

Abstract

A hybrid numerical-analy tical solution for two-dimensional transient-free convection is presented and applied to a vertical porous cavity, based on application of the ideas in the generalized integral transform technique ( GITT) . The integral transformation process reduces the original coupled partial differential equations ( PDEs) , for stream function and temperature, into an in nite system of nonlinear ordinary differential equations ( ODEs) for the transformed potentials, which is adaptively truncated and numerically solved through subroutine NDSolve from the Mathematica software system. All the analytical steps in the solution procedure are symbolically evaluated through the Mathematica package, mixing with the numerical computations and graphic representation. INTROD UCTIONTransport phenomena in porous media represent an important segment of the heat and mass transfer ¢eld and have a variety of applications in engineering. The effect of free convection as a result of the gravitational body force is of particular interest from both practical and theoretical points of view. Engineering applications include, among others, thermal insulations, radioactive waste disposal, solar energy collectors, and geothermal energy analysis. Other applications of the porous medium modeling are discussed by Kakac°, Kilkis, , Kulacki, and Arinc°[1] and more recently by Kaviany [2]. In addition to the interesting physics pertinent to this class of problems, the associated system of coupled PDEs for temperature and stream function provides an important test case in the development of new solution techniques for convection-diffusion problems.Within the last two decades, the ideas in the so-called GITT [3,4,5], progressively advanced toward the establishment of an alternative hybrid numerical-analytical approach based on the formal analytical principles in the classical integral transform method [6] for a priori nontransformable diffusion and convection-diffusion problems. This note is one more step in the effort to reach the full potential of this promising approach by testing its performance in the class of problems represented by transient-free convection within a two-dimensional