Computational Mechanics/Fluids/Stokes

Stokes fluid

Strong form

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

${\begin{aligned}{\text{(1)}}&&-\nabla \cdot {\boldsymbol {\sigma }}&={\vec {f}}&~{\mathsf {on}}~~\Omega (t)\\{\text{(2)}}&&\nabla \cdot {\vec {v}}&={\vec {0}}&~{\mathsf {on}}~~\Omega (t)\end{aligned}}$

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

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:

${\begin{aligned}{\text{(3)}}&&({\vec {w}},-\nabla \cdot {\boldsymbol {\sigma }}-{\vec {f}})_{\Omega (t)}\\{\text{(4)}}&&({\vec {w}},\nabla \cdot {\vec {v}})_{\Omega (t)}\end{aligned}}$

Expanding the terms