colloids and interfaces Article Axisymmetric Slow Motion of a Porous Spherical Particle in a Viscous Fluid Using Time Fractional Navier–Stokes Equation Jai Prakash * and Chirala Satyanarayana   Citation: Prakash, J.; Satyanarayana, C. Axisymmetric Slow Motion of a Porous Spherical Particle in a Viscous Fluid Using Time Fractional Navier–Stokes Equation. Colloids Interfaces 2021, 5, 24. https:// doi.org/10.3390/colloids5020024 Academic Editor: Huan J. Keh Received: 5 February 2021 Accepted: 8 April 2021 Published: 13 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Mathematics, École Centrale School of Engineering, Mahindra University, Hyderabad 500043, India; chiralasatya@gmail.com * Correspondence: jai.prakash@mahindrauniversity.edu.in 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. Keywords: fractional order; Stokes equation; Brinkman equation; stream function; porous particle 1. Introduction Fractional differential equations are a type of differential equation where deriva- tives are not the traditional derivatives but are of fractional order. Fractional differential equations have several applications in various branches of science and engineering. For example, in a porous media, when there is a fluid flow through it, there is a change in both solid and fluid properties of the porous media due to chemical reactions, mineral precipitation, etc., and, this results in a change in permeability of the porous media and viscosity of the fluid flowing through it over time. The phenomenon that solid and fluid properties change over time is represented by the term ‘memory’. To quantify the effect of history, ‘memory’ is incorporated in the mathematical model. Two types of memory, time memory and space memory, are found in literature. Space memory considers the previous space that the fluids have passed through [1]. 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. To consider time memory, fractional order derivatives in time are used, and to consider space memory, fractional order derivatives in space are used. Therefore, fractional differential equations are used to model mass transport from fractures in porous media to a porous rock matrix [2]. Fractional differential equations have been used in other different areas such as fractional telegraph equation. The telegraph equation, also 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. The fractional telegraph equation has been solved using a combination of homotopy analysis and Laplace transform methods [3]. 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 Colloids Interfaces 2021, 5, 24. https://doi.org/10.3390/colloids5020024 https://www.mdpi.com/journal/colloids