We denote \(\Omega \subseteq \mathbb{R}^d\), \(d = 2\)
or \(d=3\), the fluid domain and \(\partial \Omega = \Gamma_D \cup \Gamma_N\) its boundary. On \(\Gamma_D\) the Dirichlet boundary conditions are imposed and on \(\Gamma_N\) the
Neumann boundary conditions are imposed. The fluid density is given by \(\rho_f\) and its viscosity by \(\mu\). The fluid velocity \(u(x,t) : \Omega \times\) ]\(0,T\)] \(\mapsto \mathbb{R}^{d}\) and pressure \(u(x,t) : \Omega \times \)]\(0,T\)] \(\mapsto \mathbb{R}\) satisfy the Navier-Stokes equations:
\[\begin{equation*}
\left\{\begin{array}{rcl}
\rho_f \frac{\partial u}{\partial t} + \rho_f (u \cdot \nabla)u - \mu \Delta u + \nabla p &=& f \mbox{,} \quad \mbox{in } \Omega \texttt{,} \\
\nabla \cdot u &=& 0 \mbox{,} \quad \mbox{in } \Omega \texttt{,}\\
u(x,t) &=& h(x,t) \;, \quad \mbox{ on } \Gamma_D \mbox{,} \\
\sigma(x)\vec{n} &=& (-p I_d + 2 \mu D(u))\vec{n} = g(x,t) \;, \quad \mbox{ on } \Gamma_N \mbox{.}
\end{array}\right.\;
\end{equation*}\]
where \(f(x,t): \Omega \times\) ]\(0,T\)] \(\mapsto \mathbb{R}^{d}\) represents the external forces.
We use the following functional spaces as test and trial spaces :
\[X = H^1(\Omega)^{d} \mbox{,} \quad M = L^2_0(\Omega) \mbox{.}\]
The variational formulation of the Navier-Stokes problem is given by :
Find \((u,p) \in X \times M\) such that:
\[\int_{\Omega} \rho_f \frac{\partial u}{\partial t} \cdot v + \int_{\Omega} \rho_f (u \cdot \nabla)u \cdot v + 2 \mu \int_{\Omega} D(u):D(v) - \int_{\Omega} p \nabla \cdot v = \int_{\Gamma_N} g \cdot v + \int_{\Omega} f \cdot v \;, \quad \forall v \in X \mbox{,}\]
\[\int_{\Omega} q \nabla \cdot u = 0 \;, \quad \forall q \in M \mbox{,}\]
where \(D(u) = \frac{1}{2}(\nabla u + \nabla u^T)\), the deformation tensor.
\[\begin{align*}
a(u,v) &= \int_{\Omega} \rho_f \frac{\partial u}{\partial t} \cdot v + \int_{\Omega} \rho_f (u \cdot \nabla)u \cdot v + 2 \mu \int_{\Omega} D(u):D(v) \mbox{,} \\
b(v,p) &= - \int_{\Omega} p \nabla \cdot v \mbox{,} \\
F(v) &= \int_{\Omega} f \cdot v \mbox{,} \\
G(v) &= \int_{\Gamma_N} g \cdot v \mbox{.} \\
\end{align*}\]
Thus, the variational formulation can be rewritten as:
Find \((u,p) \in X \times M \) such that:
\[\begin{align*}
a(u,v) + b(v,p) &= G(v) + F(v) \;, \quad \forall v \in X \mbox{,} \\
b(u,q) &= 0 \;, \quad \forall q \in M \mbox{.}
\end{align*}\]
The integration in time is solved by the Euler-Backward method.
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