An ODE-Based Analytical Approach to Rayleigh Flow of Kelvin–Voigt Fluid

Abstract

We present some results from a project carried out in the Under- graduate Research in Mathematics (MATH 4380) course at TAMIU. The main goal of this study is to demonstrate that analytical solutions for a system of partial differential equations (PDEs) can be obtained using methods typically taught in an undergraduate ordinary differential equations (ODE) course, such as power series, undetermined coefficients, and Laplace transforms. In this study, we provide semi-analytical solutions for Rayleigh flow type problem utilizing the Kelvin-Voigt fluid model, which describes the behavior of vis- coelastic materials by incorporating both elastic and viscous properties. Heat transfer analysis is also considered under the assumption of local thermal equilibrium. The mathematical formulation is established based on funda- mental conservation laws, leading to a system of partial differential equations. To obtain approximate analytical solutions for the governing equations, we apply the Laplace transform and regular perturbation techniques. SAE 5W-30 Castrol Edge Engine Oil is considered as the working fluid for the simulations. We analyze how the Prandtl, Grashof, and Kelvin-Voigt numbers affect the flow pattern features. Our findings indicate that increasing the dimension- less Grashof number enhances the velocity fields, while the dimensionless temperature field decreases to zero more rapidly for larger Prandtl number