Unsteady Stokes equations

Table of Contents

1. Mathematical formulation

The unsteady Stokes problem is given by:

\[\begin{equation*}\label{p:3} \left\{\begin{array}{rcl} \frac{\partial u}{\partial t} - \mu \Delta u + \nabla p &=& f \;, \quad \mbox{ in } \Omega \times [0,T[ \mbox{,} \\ \nabla \cdot u &=& 0 \;, \quad \mbox{ in } \Omega \times [0,T[ \mbox{,} \\ u(x,t) &=& h(x,t) \;, \quad \mbox{ on } \Gamma_D \times [0,T[ \mbox{,} \\ \sigma(x,t)\vec{n} &=& (-p I_d + 2 \mu D(u))\vec{n} = g(x,t) \;, \quad \mbox{ on } \Gamma_N \times [0,T[ \mbox{.} \end{array}\right.\; \end{equation*}\]

where \(\Omega \subseteq \mathbb{R}^d\), \(d = 2\) or \(d=3\), is the domain of the fluid. \(\partial \Omega\) its boundary, \(\partial \Omega = \Gamma_D \cup \Gamma_N\), where \(\Gamma_D\) denotes the portion of the boundary where the Dirichlet boundary conditions are imposed, and \(\Gamma_N\) the portion where Neumann boundary conditions are imposed. We have \(u(x,t) : \Omega \times [0,T[ \mapsto \mathbb{R}^d\) and \(p(x,t) : \Omega \times [0,T[ \mapsto \mathbb{R}\) respectively the velocity and the pressure of the fluid and \(\mu \in \mathbb{R}^{+}\) its dynamic viscosity. The external forces are described by \(f(x,t) : \Omega \times [0,T[ \mapsto \mathbb{R}^d\).

Variational formulation

Using the same notations as in the section Stationary Stokes equations, we find the following variational formulation:

Find \((u,p) \in X^t = \{u | u(\cdot,t) \in H^1(\Omega)^d \} \times M^t = \{ p | p(\cdot, t)\in L^2_0(\Omega)\}\) such that:

\[\begin{align*} \int_{\Omega}\frac{\partial u}{\partial t} v + a(u,v) + b(v,p) &= G(v) + F(v) \;, \quad \forall v \in X^t \mbox{,} \\ b(u,q) &= 0 \;, \quad \forall q \in M^t \mbox{.} \end{align*}\]

To numerically solve the integration in time, we use the Euler-Backward method. We divide the time interval [\(0,T\)[ into \(K\) sub-intervals and define \(\Delta t = \frac{T}{K}\), the time step.

The integral at time \(t_k = k* \Delta t\) is approximated by:

\[\begin{align*} \int_{\Omega} \frac{\partial u^k}{\partial t} v &\approx \int_{\Omega} \frac{u^k - u^{k-1}}{\Delta t} v \mbox{ , }\\ &\approx \int_{\Omega} \frac{u^k}{\Delta t} v - \int_{\Omega} \frac{u^{k-1}}{\Delta t} v \mbox{ . }\\ \end{align*}\]

We denote:

\[m(u^k,v) = \int_{\Omega} \frac{\partial u^k}{\partial t} v \quad \mbox{ and } \quad m(u^{k-1},v) = \int_{\Omega} \frac{\partial u^{k-1}}{\partial t} v\]

Putting this approximation into the variational formulation, it is expressed at time \(t_k\) by:

Find \((u^k,p^k) \in X^{t_k} = \{u | u(\cdot,t_k) \in H^1(\Omega)^d \} \times M^{t_k} = \{ p | p(\cdot, t_k)\in L^2_0(\Omega)\}\) such that:

\[\begin{align*} m(u^k,v) + a(u^k,v) + b(v,p^k) &= m(u^{k-1},v) + G(v) + F(v) \;, \quad \forall v \in X^{t_k} \mbox{,} \\ b(u^k,q) &= 0 \;, \quad \forall q \in M^{t_k} \mbox{.} \end{align*}\]