Fundamental solutions in the Stokes flow

The aim of this project is the computation of fundamental solutions for the pressure associated to a Stokeslet in an axisymmetric domain.
Linearity of the Stokes flow allows to use the powerful technique of superposition of effects, in many cases in fact one can successfully describe a physical phenomenon with a superposition of solutions to the singularly forced governing equations, those are the so called fundamental solutions of the equation (Stokeslet for singularly forced Stokes equations). The Stokeslet describe the velocity field due to an applied point force, in the same way the Stresslet describe the stresses. Similarly one can find an expression for the pressure fundamental solution and its associated stress tensor. Starting from the well known fundamental solutions in 3D cartesian coordinates an integration in the azimuthal direction will be performed in order to find their axisymmetric formulation.
The knowledge of the fundamental solution allows for the implementation of a Boundary Integral Method in order to find the pressure in the all domain from specified Boundary Conditions; therefore a validation will be carried on against well known analytical results: Poiseuille flow in a channel, rising droplet due to buoyancy. Only basic knowledge of MATLAB is required, since most of the work deal with elegant analytical solutions, both in the fundamental solutions derivation as well as in the validation procedure.

Supervisor: Giacomo Gallino