Stokes flow of incompressible liquid through a conical diffuser with partial slip boundary condition

TL;DR

This paper derives a new, practical solution for the flow of incompressible liquid through a conical diffuser with partial slip boundary conditions, extending classical no-slip solutions and analyzing flow vorticity effects.

Contribution

It introduces an alternative, more applicable form of the solution to the Navier-Stokes equations for this problem, incorporating partial slip boundary conditions and providing recurrent relations for flow variables.

Findings

Flow sliding induces vorticity in the diffuser.

Zero slip length recovers classical no-slip solution.

Solution applicable for small slip lengths.

Abstract

An alternative form of the general solution of the linearized stationary Navier-Stokes equations for an incompressible fluid in spherical coordinates is obtained by the vector potential method. A previously published solution to this problem, dating back to the paper by Sampson, is given in terms of a stream function, which leads to formulas that are difficult to apply in practice. The presented form of solution is applied to the problem of liquid flowing through a conical diffuser under a partial slip boundary condition for a certain slip length lambda. Recurrent relations are obtained that allow us to determine the velocity, pressure and stream function. The solution is analyzed in the first order of decomposition with respect to a small dimensionless parameter (lambda divided by r). It is shown that the sliding of the liquid over the surface of the cone leads to a vorticity of the flow. At zero slip length, we obtain the well-known solution to the problem of a diffuser with no-slip boundary condition corresponding to strictly radial streamlines.