Stokes

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The stationary Stokes model describes slow viscous flow of an incompressible Newtonian fluid, with inertia neglected. In a domain \(\Omega\), the velocity \(\boldsymbol{u}\) and pressure \(p\) satisfy momentum balance and incompressibility:

Here \(\nu>0\) is the constant viscosity coefficient and \(\boldsymbol{f}\) is the applied force. Pressure enforces the incompressibility constraint. With homogeneous velocity prescribed on \(\partial\Omega\), pressure is determined up to an additive constant; the normalization \(\int_\Omega p\,\mathrm dx=0\) fixes that constant. The mixed velocity–pressure discretization must preserve this coupling to give a stable approximation.

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