Starting flow in channels with boundary slip

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 start-up flows examined in this work include: (1) plane Couette flow between two plates, (2) rotary Couette flow between two coaxial cylinders, (3) Poiseuille flow through a parallel-plate channel, (4) Poiseuille flow through a rectangular channel, (5) Poiseuille flow through a circular channel, and (6) 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.