Maxwell fluid suction flow in a channel

Abstract

Two-dimensional, steady and incompressible suction flow of the upper-convected Maxwell fluid in a porous surface channel has been studied. The combined effects of viscoelasticity and inertia are considered. A similarity solution is assumed, resulting in a nonlinear system of ODEs that describes the relations between the two velocity components, the three deviatoric stresses and the pressure gradient. This system is solved using two methods: an analytical solution, based on a power series method in terms of the transverse coordinate across the channel, and a fourth-order Runge–Kutta numerical integration scheme. We first find the existing Newtonian flow solutions for suction and injection. For the Maxwell fluid, the solutions of the power series and the numerical integration are in complete agreement in the range of Reynolds and Deborah numbers 0 ≤ Re ≤ 30 and 0 ≤ De ≤ 0.3. They show that the suction flow exhibits a flattening of the longitudinal velocity profile near the centerline and the establishment of boundary layers near the porous surfaces as Reynolds number increases. It is also observed that when Deborah number increases, with a fixed Reynolds number, viscoelasticity affects the velocity profiles in the same way as inertia in a Newtonian fluid. The application of the self-similar solution to the injection flow of the Maxwell fluid is also discussed.