Unsteady Stokes equations
1. Mathematical formulation
The unsteady Stokes problem is given by:
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:
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:
We denote:
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: