# Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow
## 2 Compressible Flow
### Continuous Flow Problem  
We consider compressible flow in a bounded domain \(\Omega_{t}\subset\mathbb{R}^{2}\) for \(t\in[0,T]\). The boundary of \(\Omega_{t}\) consists of four disjointed parts: \(\partial\Omega_{t}=\Gamma_{I}\cup\Gamma_{O}\cup\Gamma_{W}\cup\Gamma_{W_{t}}\), where \(\Gamma_{I}\) represents the inlet, \(\Gamma_{O}\) is the outlet and boundaries \(\Gamma_{W}\) and \(\Gamma_{W_{t}}\) denote impermeable fixed and moving walls, respectively. The time dependence of the domain \(\Omega_{t}\) is taken into account by using a regular one-to-one ALE mapping of the reference domain \(\Omega_{0}\) onto the current configuration \(\Omega_{t}:\mathcal{A}_{t}:\Omega_{0}\longrightarrow\bar{\Omega}_{t}\). Next, we define the domain velocity \(\bar{z}(\mathbf{X},t)=\frac{\partial}{\partial t}\mathcal{A}_{t}(\mathbf{X}),\quad t \in[0,T]\), \(\mathbf{X}\in\Omega_{0}\), \(\mathbf{z}(\mathbf{x},t)=\bar{z}(\mathcal{A}_{t}^{-1}(\mathbf{x}),t)\), \(t\in[0,T]\), \(\mathbf{x}\in\Omega_{t}\) and the ALE derivative of the vector function \(\mathbf{w}=\mathbf{w}(\mathbf{x},t)\), where \(\mathbf{x}\in\Omega_{t}\) and \(t\in[0,T]\): \(\frac{\mathcal{D}^{A}}{\partial t}\mathbf{w}(\mathbf{x},t)=\frac{\partial\mathbf{x}}{ \partial t}(\mathbf{X},t)\), where \(\mathbf{\tilde{w}}(\mathbf{X},t)=\mathbf{w}(\mathcal{A}_{t}(\mathbf{X}),t)\), \(\mathbf{X}\in\Omega_{0}\), \(\mathbf{x}=\mathcal{A}_{t}(\mathbf{X})\). Then the continuity equation, the N-S equations and the energy equation can be written in the ALE form  
\[\frac{\mathcal{D}^{A}\mathbf{w}}{\mathcal{D}t}+\sum\limits_{s=1}^{2}\frac{\partial \mathbf{g}_{s}(\mathbf{w})}{\partial x_{s}}+\mathbf{w}\mathrm{div}z=\sum\limits_{s=1}^{2} \frac{\partial\mathbf{R}_{s}(\mathbf{w},\
abla\mathbf{w})}{\partial x_{s}}, \tag{1}\]  
where \(\mathbf{w}=(\rho,\rho v_{1},\rho v_{2},E)^{T}\in\mathbb{R}^{4}\), \(\mathbf{g}_{s}(\mathbf{w})=f_{s}(\mathbf{w})-z_{s}\mathbf{w}\), \(f_{s}(\mathbf{w})=(\rho v_{s},\rho v_{1}v_{s}+\delta_{1s},\rho v_{2}v_{s}\\ +\delta_{2s},p,(E+p)v_{s})^{T},\mathbf{R}_{s}(\mathbf{w},\
abla\mathbf{w})=(0,\tau_{1}^{V}, \tau_{s2}^{V},\tau_{s1}^{V}v_{1}+\tau_{s2}^{V}v_{2}+k\frac{\partial\theta}{ \partial x_{s}})^{T}\), \(s=1,2\), \(\tau_{ij}^{V}=\lambda\delta_{ij}\mathrm{div}\mathbf{v}\), \(+2\mu d_{ij}(\mathbf{v})\), \(d_{ij}(\mathbf{v})=\frac{1}{2}\left(\frac{\partial v_{1}}{\partial x_{j}}+\frac{ \partial v_{j}}{\partial x_{i}}\right)\), \(i,j=1,2\). We have \(\mathbf{R}_{s}(\mathbf{w},\
abla\mathbf{w})=\sum\limits_{k=1}^{2}\mathbb{K}_{s,k}(\mathbf{w}) \frac{\partial\mathbf{w}}{\partial x_{k}}\), where \(\mathbb{K}_{s,k}(\mathbf{w})\) are \(4\times 4\) matrices depending on \(\mathbf{w}\), and \(f_{s}(\mathbf{w})=\mathbf{\Delta}(\mathbf{w})\mathbf{w}\) with \(\mathbf{\Delta}(\mathbf{w})=Df_{s}(\mathbf{w})/D\mathbf{w}\), see [11]. We use the following notation: \(p\)**--**pressure, \(\rho\)**--**fluid density, \(E\)**--**total energy, \(\mathbf{v}=(v_{1},v_{2})\)**--**velocity vector, \(\theta\)**--**absolute temperature, \(c_{v}>0\)**--**specific heat at constant volume, \(\gamma>1\)**--**Poisson adiabatic constant, \(\mu>0\)**--**dynamic viscosity, \(\lambda=-2\mu/3\)**--**second viscosity coefficients, \(k>0\)**--**heat conduction coefficient, \(\tau_{ij}^{V}\)**--**components of the viscous part of the stress tensor. Equation (1) is completed by the following thermodynamical relations for pressure and absolute temperature  
\[p=(\gamma-1)\left(E-\rho\frac{|\mathbf{v}|^{2}}{2}\right),\quad\theta=\frac{1}{c_ {v}}\left(\frac{E}{\rho}-\frac{|\mathbf{v}|^{2}}{2}\right) \tag{2}\]  
and equipped with initial and boundary conditions  
\[