# Sharp Interface Capturing in Compressible Multi-Material Flows with a Diffuse Interface Method
## 2 Physical Model and Governing Equations  
Considering a compressible multi-phase flow with \(N\) phases, where each phase is indexed by \(\phi\in\{1,...,N\}\), and is modelled as a compressible medium with a thermodynamic state defined by its:  
1. mass density \(\rho_{\phi}\);
2. specific internal energy \(e_{\phi}\);
3. temperature \(T_{\phi}\);
4. pressure \(p_{\phi}\).  
Further, each phase \(\phi\) obeys a coupled system of governing equations, composed of the balance equations of mass, momentum and energy for each phase. We consider here the situation where the pressures are at the instantaneous equilibrium everywhere, so that we have:  
\[p_{\phi}=p\qquad\forall\phi\in\{1,...,N\}. \tag{1}\]  
where \(p\) is an equilibrium pressure. We also consider that every control volume (in the terminology of the Eulerian framework) is filled by different phases, such that:  
\[\sum_{\phi=1}^{N}\alpha_{\phi}=1, \tag{2}\]  
with \(\alpha_{\phi}\in[0,1]\) is the volume fraction of any phase \(\phi\). Additionally, the system is completed with two equations of state (EOS), relating the four thermodynamic variables. The well-known complete set of volume-averaged equations for any phase \(\phi\) is written here in terms of the internal energies, as shown below:  
\[\frac{\partial(\rho_{\phi}\alpha_{\phi})}{\partial t}+\
abla\cdot(\rho_{\phi} \alpha_{\phi}\mathbf{u}_{\phi})=0, \tag{3}\]  
\[\frac{\partial(\rho_{\phi}\alpha_{\phi}\mathbf{u}_{\phi})}{\partial t}+\
abla \cdot(\rho_{\phi}\alpha_{\phi}\mathbf{u}_{\phi}\otimes\mathbf{u}_{\phi})=- \alpha_{\phi}\
abla(p_{\phi})+\mathbf{F}_{\phi}, \tag{4}\]  
\[\frac{\partial(\rho_{\phi}\alpha_{\phi}\epsilon_{\phi})}{\partial t}+\
abla \cdot(\rho_{\phi}\alpha_{\phi}\epsilon_{\phi}\mathbf{u}_{\phi})=-p_{\phi} \bigg{[}\frac{\partial(\alpha_{\phi})}{\partial t}+\
abla\cdot(\alpha_{\phi} \mathbf{u}_{\phi})\bigg{]}, \tag{5}\]  
\[p_{\phi}=p\qquad\forall\phi\in\{1,...,N\}, \tag{6}\]  
\[\sum_{\phi}\alpha_{\phi}=1, \tag{7}\]  
\[\text{EOS}_{1}(\rho_{\phi},\epsilon_{\phi},T_{\phi},p_{\phi})=0, \tag{8}\]  
\[\text{EOS}_{2}(\rho_{\phi},\epsilon_{\phi},T_{\phi},p_{\phi})=0\qquad\forall \phi\in\{1,...,N\}, \tag{9}\]  
The above system is identified by the unknowns \(\big{[}\mathbf{u}_{\phi},\rho_{\phi},\alpha_{\phi},\epsilon_{\phi},T_{\phi},p _{\phi}\big{]}\). \(\mathbf{u}_{\phi}\) is the phase velocity and we denote the components of the vector \(\mathbf{u}_{\phi}=(u_{\phi},v_{\phi})\) in 2D and \(\mathbf{u}_{\phi}=(u_{\phi},v_{\phi})\) in 3D. EOS\({}_{i}\) stands for the Equation Of State number \(i;i=1,2\). The form of EOS\({}_{i}\) mentioned above is generic. The specific definitions are given in Sections 6 and 7.  
Furthermore, in the present study, as evident from the above system of equations, we have separate momentum equations for each phase with independent velocities per phase. It is known that without the velocity relaxation terms, this system is conditionally hyperbolic with the loss of hyperbolicity under certain conditions [27; 28]. This issue can be fixed by using a large friction source term \(\mathbf{F}_{\phi}\) for balancing velocities between the phases.  
In this work we deal with the situation where there is a full mechanical equilibrium. This implies that the phases share a common velocity, which is achieved through the use of a term \(\mathbf{F}_{\phi}\). The form of which is:  
\[\mathbf{F}_{\phi}=D\bigg{[}\sum_{\phi^{*}\
eq\phi}^{N}\mathbf{u}_{\phi^{*}}-(N -1)\mathbf{u}_{\phi}\bigg{]}. \tag{10}\]  
where \(D\) is a friction constant which determines the rate at which velocities equilibrate, and \(\phi^{*}\) is a conjugate phase i.e., \(\phi^{*}\
eq\phi\).