How to verify creeping flow past a liquid sphere (Hadamard-Rybczynski and Satapathy-Smith)

This article describes a possible test case for multiphase fluid-flow. Please consider it an example and use meaningful test cases for your use case.

There are several analytical solutions for the rise or decent of particles in Stokes flow (valid for very small Reynolds numbers). The setting is typically a spherical bubble/drop/particle rising or falling in a much larger spherical domain. The outer boundary is important to keep in mind because of the strong wall effects in creeping (highly viscous) flow.
For example, if a flow solver, say OpenFOAM, is to be tested against one of the analytical solutions, there will we be an unavoidable difference in the assumed shape (sphere) and the simulation boundary (cube/cuboid). Even if the outer domain is about ten times larger than the particle, the deviation in the local velocity field close to the interface may be up to 10% due to wall effects. This observation is not only due to the shape of the boundary but also due to its distance. Even solvers able to handle spherical domains will be effected, but there are ways to circumvent this issue. Note that without a special treatment at the outer boundary, the simulation domain should be at leat 100 to 1000 times larger than the particle! This ratio comes from observations in semi-analytical mass transfer computations (analytical velocity field + numerical solution of species transport). The wall effect in the concentration gradient at the particle surface only vanished (it wasn’t visible in plots, roughly speaking less than 1% deviation) if the outer domain was about 1000 times larger than the particle. A more convenient way to get around this problem is to include the wall effect in the solution, and this is exactly the assumption in which the derivations of Hadamard-Rybczynski and Satapathy-Smith differ:

  • Hadamard-Rybczynski: the outer domain boundary is assumed to be at infinity.
  • Satapathy-Smith: the outer domain boundary is assumed to be at a finite distance. This distance is part of the analytical solution.

The analytical solution of creeping flow past a spherical particle is extremely valuable because it might be used to

  • validate the hydrodynamic solution of a flow solver locally (velocity and pressure fields) and globally (rise velocity)
  • test interpolation methods or discretization schemes
  • compute semi-analytical solutions (numerical solutions based on the analytical velocity profile) to avoid error propagation.

Useful resources attached or linked

  • The book Analytical Solutions for Transport Processes contains an explicit expression for the velocity and pressure fields on page 168 (179 in the PDF).
  • The slides in the attached files below contain a nice overview of historical notes and related solutions.
  • The coefficients in the original article “The motion of single immiscible drops through a liquid” by Satapathy and Smith are incorrect.
    Paul Weber computed new coefficients (see the attached files) using Mathematica.
  • The utility satapathyVelocityField in the attached files below may be used to compute the analytical velocity field on OpenFOAM meshes.

See also