The Stokes Problem

In this section we discuss how to solve the Stokes problem which is defined as follows:

We want to calculate the velocity field $v$ and pressure $p$ of an incompressible fluid . They are given as the solution of the Stokes problem

$$-\left(\eta(v\hackscore{i,j}+ v\hackscore{j,i})\right)\hackscore{,j}+p\hackscore{,i}=f\hackscore{i}-\sigma\hackscore{ij,j}$$ (101)

where $f\hackscore{i}$ defines an internal force and $\sigma\hackscore{ij}$ is an initial stress . The viscosity $\eta$ may weakly depend on pressure and velocity. If relevant we will use the notation $\eta(v,p)$ to express this dependency.

We assume an incompressible media:

$$-v\hackscore{i,i}=0$$ (102)

Natural boundary conditions are taken in the form

$$\left(\eta(v\hackscore{i,j}+ v\hackscore{j,i})\right)n\hackscore{... ...\hackscore{i} n\hackscore{j} v\hackscore{j}+\sigma\hackscore{ij} n\hackscore{j}$$ (103)

which can be overwritten by constraints of the form

$$v\hackscore{i}(x)=v^D\hackscore{i}(x)$$ (104)

at some locations $x$ at the boundary of the domain. $s\hackscore{i}$ defines a normal stress and $\alpha\ge 0$ the spring constant for restoring normal force. The index $i$ may depend on the location $x$ on the boundary. $v^D$ is a given function on the domain.