Computational Mechanics/Fluids/Stokes

Stokes fluid

Strong form

In the strong form, the momentum and mass balance for an incompressible fluid reduce to:

$\begin{array}{ccccc}\text{(1)}& & -\mathrm{\nabla }\cdot \mathit{\sigma }& =\overrightarrow{f}& 𝗈𝗇\mathrm{\Omega }(t)\\ \text{(2)}& & \mathrm{\nabla }\cdot \overrightarrow{v}& =\overrightarrow{0}& 𝗈𝗇\mathrm{\Omega }(t)\end{array}$

In other words, this fluid can be described without taking inertia into account in the momentum balance ( $𝖱𝖾<<1$ ).

Weak Form

To obtain the weak form, we simply multiply the residuals of (1) and (2) with an appropriate test function and integrate over the whole domain:

$\begin{array}{ccc}\text{(3)}& & (\overrightarrow{w},-\mathrm{\nabla }\cdot \mathit{\sigma }-\overrightarrow{f}{)}_{\mathrm{\Omega }(t)}\\ \text{(4)}& & (\overrightarrow{w},\mathrm{\nabla }\cdot \overrightarrow{v}{)}_{\mathrm{\Omega }(t)}\end{array}$

Expanding the terms