ANALYTICAL SOLUTION OF UNSTEADY STOKES FLOW OF A MAXWELL FLUID WITH TIME-DEPENDENT SLIP IN A POROUS CHANNEL

Abstract

The present article shows an analytical closed form solution for the unsteady Stokes flow of an incompressible Maxwell fluid in a porous parallel plates channel with time dependent Navier slip boundary conditions. The governing partial differential equation is obtained by combining the Maxwell constitutive relation and a Darcy resistance term. The problem is reduced to a second order ordinary differential equation using a Laplace transform approach, and the solution is given in terms of hyperbolic functions. The slip length is defined as  which decays exponentially from an elevated initial value to a constant steady state, modeling progressive surface relaxation. The velocity field is divided into two components: steady-state and exponentially decaying transients. A limiting case analysis aligns with existing Newtonian and no slip solutions. The Stehfest numerical inversion method is verified for accuracy by comparison with finite-difference results. Theoretical analyses indicate that: (i) increasing the dimensionless relaxation time Λ slows down the flow development and lowers the maximum velocity; (ii) greater slip leads to a more uniform, plug-like velocity profile; and (iii) reducing the Darcy number diminishes the bulk flow speed. The analytical framework is directly applicable to microfluidic devices, biomedical transport, polymer filtration, and enhanced oil recovery.