Transient Velocity Analysis of Three-Phase Immiscible Maxwell Fluids in Clear and Porous Cylindrical Media
Emeng Emeng Iyeme
*
Department of Mathematics, University of Cross River State, Calabar, Cross River State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
The transient flow behaviour of three immiscible fractional Maxwell fluids in a cylindrical domain was investigated. The domain comprises a central clear region and two surrounding porous layers. An incompressible and one-dimensional flow is assumed. This flow is driven by a quick, unexpected motion of the cylinder. A viscoelastic memory effect was captured by using fractional relations to formulate the governing equations. These equations are then non-dimensionalised to introduce parameters such as the Reynolds and Darcy numbers, porosity coefficient, and relaxation time. An analytical solution of the resulting system of equations using Olayiwola’s Generalised Polynomial Approximation Method (OGPAM) is presented. Velocity expressions are obtained for each region and used in the analysis of the effects of inertia, permeability, porosity, and viscoelastic memory on the fluid's motion. Graphical summaries using Maple 2018 reveal that velocity increases in the clear region and decreases in the porous layers due to the effect of resistance. Flow velocity is enhanced with the increase in Reynolds and Darcy numbers, while fluid motion in the porous region is aided by higher porosity. The importance of memory effects in viscoelasticity is emphasised as relaxation time significantly influences transient behaviour. The model is physically consistent, as continuity of velocity across interfaces is maintained. The study demonstrates the effectiveness of OGPAM in providing semi-analytical solutions and reducing computational complexity. The study can be applied to biomedical systems, enhanced oil recovery processes, and porous media flow.
Keywords: Transient, velocity, analysis, immiscible, clear, porous, cylindrical