Computational Mechanics/Fluids/Stokes

Strong form
In the strong form, the momentum and mass balance for an incompressible fluid reduce to: $$ \begin{align} \text{(1)}& & -\nabla\cdot\boldsymbol{\sigma} &= \vec{f} &~\mathsf{on}\Omega(t)\\ \text{(2)}& & \nabla\cdot\vec{v} &= \vec{0} &~\mathsf{on}\Omega(t) \end{align} $$ 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: $$ \begin{align} \text{(3)}& & (\vec{w},-\nabla\cdot\boldsymbol{\sigma}-\vec{f})_{\Omega(t)} \\ \text{(4)}& & (\vec{w},\nabla\cdot\vec{v})_{\Omega(t)} \end{align} $$

Expanding the terms