Starting flow in channels with boundary slip

Abstract

This paper contains a collection of problems and their analytical solutions for starting flows under the effect of boundary slip in channels of various geometries. The startup flows examined in this work include: (i) plane Couette flow between two plates, (ii) rotary Couette flow between two coaxial cylinders, (iii) Poiseuille flow through a parallel-plate channel, (iv) Poiseuille flow through a rectangular channel, (v) Poiseuille flow through a circular channel, and (iv) Poiseuille flow through an annulus. It is first shown that, using a depletion layer to model the effective slip, the slip length may attain its steady state much faster than the starting flow will do. This supports the use of a constant Navier slip length in the present problems. Transient solutions are derived in the form of eigenfunction expansions, where the eigenvalues are determined, by solving the characteristic equations numerically, as a function of the channel geometry and the slip lengths. From the leading eigenvalue, it is found that the boundary slip will in general lengthen the transient period of a starting flow.

Computing hydrodynamic eigenmodes of channel flow with slip -- A highly accurate algorithm

“…Due to this observation, a selection of coefficients for a wide range of values for S is given in Appendix A with 16 significant digits in a normalized floating point format. As reported in [14,2,17,1], the time scale required to reach a certain fraction of the stationary velocity field increases with increasing slip length on the channel walls. However, these time scales have not been quantified.…”

“…While the results coincide for the analytically available case S = 0, a significant difference for the two other cases S = 0.5, 2 can be observed. Even though the results provided in [16,17] agree with the results obtained here up to the last significant digit, the precision provided in [16,17] is not sufficient to accurately distinguish between different slip lengths. In this case, the first coefficient for S = 0.5 from [16] coincides with the results for S = 0.2, with a similar error for the results with S = 2 from [17].…”

“…Note that the solutions of Eq. ( 12) are also required for the analytic start up solution in a Couette flow setup [17]. We also give a factorization of the characteristic equation Eq.…”

“…9 also shows a comparison to coefficients from literature. In [16] five coefficients are given for S = 0 and S = 0.5, and in [17] five coefficients are provided for the case S = 2. However as A n = 0 if n is even, two of those coefficients are not required for the computation of the velocity field and hence have been omitted in Fig.…”

Homotopy analysis for the influence of Navier slip flow in a vertical channel with cross diffusion effects

“…Recently, Rundora and Makinde [7] analyzed the influence of Navier slip and variable viscosity through a porous medium with asymmetric convective boundary conditions. Most recently, Ng [8] investigated the nature of starting flow in channels with boundary slip.…”