Eur. Phys. J. Plus (2017) 132: 142
https://doi.org/10.1140/epjp/i2017-11431-y

Regular Article

Unsteady translational motion of a slip sphere in a viscous fluid using the fractional Navier-Stokes equation

¹ Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, Beirut, Lebanon
² Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt

^* e-mail: emad.ashmawy@bau.edu.lb

Received: 12 December 2016
Accepted: 1 March 2017
Published online: 24 March 2017

Abstract

In this paper, we investigate the translational motion of a slip sphere with time-dependent velocity in an incompressible viscous fluid. The modified Navier-Stokes equation with fractional order time derivative is used. The linear slip boundary condition is applied on the spherical boundary. The integral Laplace transform technique is employed to solve the problem. The solution in the physical domain is obtained analytically by inverting the Laplace transform using the complex inversion formula together with contour integration. An exact formula for the drag force exerted by the fluid on the spherical object is deduced. This formula is applied to some flows, namely damping oscillation, sine oscillation and sudden motion. The numerical results showed that the order of the fractional derivative contributes considerably to the drag force. The increase in this parameter resulted in an increase in the drag force. In addition, the values of the drag force increased with the increase in the slip parameter.