Control of Fluids and Fluid-Structure Interactions

Some Fluid Models

We consider a fluid flow occupying a bounded domain $\Omega _{F} \subset$ ${\mathbb{R}}^{N}$, with N = 2 or N = 3, at the initial time t = 0 and a domain $\Omega _{F}$(t) at time t > 0. Let us denote by ρ $(x,t) \in {\mathbb{R}}^{+}$ the density of the fluid at time t at the point $x\ \in \Omega _{F}$(t) and by u(x, t) ∈ ${\mathbb{R}}^{N}$ its velocity. The fluid flow equations are derived by writing the mass conservation

$$\frac{\partial \rho } {\partial t} + \mbox{ div}(\rho u) = 0\quad \mbox{ in}\;\,\Omega _{F}(t),\quad \mbox{ for}\;\,t > 0,$$

(1)

and the balance of momentum

$$\begin{array}{l} \rho \left (\frac{\partial u} {\partial t} + (u \cdot \nabla )u\right ) = \mbox{ div}\,\sigma \,\mbox{ +}\,\rho \,f\mbox{ } \\ \mbox{ in}\,\;\Omega _{F}(t),\quad \mbox{ for}\,t > \end{array}$$

(2)

where σ is the so-called constraint tensor and frepresents a volumic force. For an isothermal fluid, there is no need to complete the system by the balance of energy....