View source for Computational Mechanics/Fluids/Stokes

== Stokes fluid ==

===Strong form===
In the strong form, the momentum and mass balance for an incompressible fluid reduce to:
 <math>
  \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}
</math>
In other words, this fluid can be described without taking inertia into account in the momentum balance ( <math>\mathsf{Re} << 1</math> ).

===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:
 <math>
 \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}
</math>

Expanding the terms
[[Category:Computational mechanics]]