Stokes Problem

The velocity field $ v$ and pressure $ p$ of an incompressible fluid is given as the solution of the Stokes problem

$\displaystyle -\left(\eta(v\hackscore{i,j}+ v\hackscore{i,j})\right)\hackscore{,j}+p\hackscore{,i}=f\hackscore{i}-\sigma\hackscore{ij,j}$ (90)

where $ \eta$ is the viscosity, $ F\hackscore{i}$ defines an internal force and $ \sigma\hackscore{ij}$ is an intial stress . We assume an incompressible media:

$\displaystyle -v\hackscore{i,i}=0$ (91)

Natural boundary conditions are taken in the form

$\displaystyle \left(\eta(v\hackscore{i,j}+ v\hackscore{i,j})\right)n\hackscore{j}-n\hackscore{i}p=s\hackscore{i}+\sigma\hackscore{ij} n\hackscore{j}$ (92)

which can be overwritten by constraints of the form

$\displaystyle v\hackscore{i}(x)=v^D\hackscore{i}(x)$ (93)

at some locations $ x$ at the boundary of the domain. The index $ i$ may depend on the location $ x$ on the boundary. $ v^D$ is a given function on the domain.



Subsections
esys@esscc.uq.edu.au