Numerical simulations of unsteady flows in a rotating channel using a novel eigenfunction expansion method

Q Rubbab,A Ahmadian,S Salahshour,S Irshad,Y Mahsud,M Ferrara,M A Imran

doi:10.1063/5.0012874

Abstract

Starting flows of a viscous incompressible fluid, modeled by the time-fractional derivatives, within a rotating channel due to an impulsive pressure gradient are studied. Using the eigenfunction expansion, the analytic solutions in series form are obtained. The flow of the ordinary fluid is studied as a special case of the time-fractional problem. The convergence of series solutions is proved. In addition, using the classical analytical method, coupled with the Laplace transform and Stehfest’s algorithm, an approximate solution is found. The flow rates in x- and y-directions are determined. In the case of the ordinary fluid, the steady-state and transient components of velocities are obtained. The numerical calculations are carried out by using the Mathcad software. It is found that, for fractional fluids, the reversal flow is much attenuated if the values of the fractional parameter are less than 1.

Highlights

Fluid flows in rotating frames are important in many industrial applications
The flow of the ordinary fluid is studied as a special case of the time-fractional problem
The starting flow in a rotating parallel channel due to an impulsive pressure gradient was conducted by Wang,1 using the separation of variables method and the Laplace transformation scitation.org/journal/adv approach

Summary

INTRODUCTION

Fluid flows in rotating frames are important in many industrial applications. It is known that some natural phenomena, such as ocean circulations, hurricanes, tornadoes, and geophysical systems, imply rotating flows with heat and mass transfer. The purpose of the present research article is to study a case not considered before, namely, the starting flow of a viscous incompressible fluid, with time-fractional derivatives, within a rotating channel due to an impulsive pressure gradient. The Laplace transform of the complex velocity field is obtained by means of the classical analytical method in order to analyze the numerical results; the inverse Laplace transform may be obtained, its mathematical expression is quite complicated and makes it difficult to use for numerical calculations. For this reason, an approximate solution of the velocity field is obtained using Stehfest’s algorithm for the inverse Laplace transform. Graphs are used to illustrate the physical aspects of the fluid flow

PROBLEM STATEMENT

Eigenfunction expansion of the velocity field

Flow rates

NUMERICAL RESULTS AND DISCUSSION

CONCLUDING REMARKS