Axisymmetric Slow Motion of a Porous Spherical Particle in a Viscous Fluid Using Time Fractional Navier–Stokes Equation

Abstract

In this paper, we present the unsteady translational motion of a porous spherical particle in an incompressible viscous fluid. In this case, the modified Navier–Stokes equation with fractional order time derivative is used for conservation of momentum external to the particle whereas modified Brinkman equation with fractional order time derivative is used internal to the particle to govern the fluid flow. Stress jump condition for the tangential stress along with continuity of normal stress and continuity of velocity vectors is used at the porous–liquid interface. The integral Laplace transform technique is employed to solve the governing equations in fluid and porous regions. Numerical inversion code in MATLAB is used to obtain the solution of the problem in the physical domain. Drag force experienced by the particle is obtained. The numerical results have been discussed with the aid of graphs for some specific flows, namely damping oscillation, sine oscillation and sudden motion. Our result shows a significant contribution of the jump coefficient and the fractional order parameter to the drag force.