Unsteady Dean flow formation in an annulus with partial slippage: A riemann-sum approximation approach

Abstract

This paper investigates time-dependent Dean Flow of an incompressible fluid under homogeneous slip, non-homogeneous slip, and no-slip boundary conditions. The fluid flow is due to the sudden application of an azimuthal pressure gradient. The general solutions of the governing momentum equations are obtained using a two-step approach called Laplace transformation and Riemann-sum approximation method of Laplace inversion. The velocity and skin friction are determined exactly in the Laplace domain and inverted back to time domain using a numerical approach known as Riemann-sum approximation. The steady-state solutions for the velocity and the skin friction are obtained for the validation of the method employed. Graphs are plotted for analysis and numerical values are tabulated for comparison of the Riemann-sum approximation and the exact solution at large values of time. From the analysis, it is observed that the velocity profile of the fluid is higher at the wall with the highest slip coefficients. Finally, the influence of the dimensionless time (T) and the slip coefficients is also discussed with the aid of graphical illustrations. Keywords: Unsteady, Dean flow, Riemann-sum approximation, Annulus, Azimuthal pressure gradient, Slip coefficients