Governing equations for fully compressible reacting flow

$\textbf{1. Continuity equation:}$

$\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho \textbf{V}) = 0$

$\textbf{Or in tensor notation:}$

$\frac{\partial\rho}{\partial t} + \frac{\partial\rho u_i}{\partial x_i}= 0$

$\textbf{2. Momentum equation:}$

$\frac{\partial\rho\textbf{V}}{\partial t} + \nabla\cdot(\rho\textbf{V}\otimes\textbf{V}) = \rho g -\nabla p + \nabla\cdot\tau$

$\textbf{Or in tensor notation:}$

$\frac{\partial\rho u_i}{\partial t} + \frac{\partial\rho u_i u_j}{\partial x_j} = \rho g_i -\frac{\partial p}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j}$

$\textbf{3. Energy equation:}$

$\frac{\partial\rho e}{\partial t} + \nabla\cdot(\rho\textbf{V}e) = -p(\nabla\cdot\textbf{V}) + \tau : \nabla\textbf{V} - \nabla\cdot\textbf{q}$

$\textbf{Or in tensor notation:}$

$\frac{\partial\rho e}{\partial t} + \frac{\partial\rho u_j e}{\partial x_j} = -p\frac{\partial u_j}{\partial x_j} + \tau_{ij}\frac{\partial u_i}{\partial x_j} - \frac{\partial q_j}{\partial x_j}$