Stokes equation in a semi-infinite region: generalization of Lamb solution and applications to Marangoni flows

TL;DR

This paper extends Lamb's solution of the Stokes equation to semi-infinite hemispherical regions and applies it to analyze Marangoni flows caused by localized surface sources.

Contribution

The authors derive a complete general solution for Stokes flow in hemispherical geometries, expanding Lamb's classical solution for semi-infinite spaces.

Findings

Derived explicit general solution for Stokes flow in hemispherical regions.

Applied the solution to model Marangoni flows at liquid-air interfaces.

Provided insights into flow behavior near localized surface sources.

Abstract

We consider the creeping flow of a Newtonian fluid in a hemispherical region. In a domain with spherical, or nearly spherical, geometry, the solution of Stokes equation can be expressed as a series of spherical harmonics. However, the original Lamb solution is not complete when the flow is restricted to a semi-infinite space. The general solution in hemispherical geometry is then constructed explicitly. As an application, we discuss the solutions of Marangoni flows due to a local source at the liquid-air interface.