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.
Highlights
Similar to the fractional telegraph equation, the equation governing the fluid flow popularly known as the Navier–Stokes equation has been solved for the fractional order time derivative [4]
This paper presents a semi analytical solution for the problem of unsteady translational motion of a porous sphere in an incompressible viscous fluid
The fluid flow exterior and interior of the porous sphere is governed by the modified Navier–Stokes equation with fractional time derivative and the modified Brinkman equation with fractional time derivative
Summary
Fractional differential equations are a type of differential equation where derivatives are not the traditional derivatives but are of fractional order. One way to include history in any mathematical model is to use fractional order derivative. History of any parameter can be taken into consideration using fractional order derivatives of that parameter. Fractional differential equations have been used in other different areas such as fractional telegraph equation. The telegraph equation, known as a damped wave equation is classified as a hyperbolic partial differential equation, which governs physically the voltage and current in an electrical transmission line with distance and time. Similar to the fractional telegraph equation, the equation governing the fluid flow popularly known as the Navier–Stokes equation has been solved for the fractional order time derivative [4]. Kumar et al [4] developed a Interfaces 2021, 5, 24. https://
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