The equations of motion given in Conservation of Momentum, Constitutive Equation for a Newtonian Fluid, and Navier-Stokes Momentum Equation are valid in an inertial frame of reference, one that is stationary or that is moving at a constant speed with respect to a stationary frame of reference.
Conservation of Momentum:
\[ \frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x},t) \mathbf{u}(\mathbf{x},t) dV + \int_{A^*(t)} \rho(\mathbf{x},t) \mathbf{u}(\mathbf{x},t) (\mathbf{u}(\mathbf{x},t) - \mathbf{b}) \cdot \mathbf{n} dA = \int_{V^*(t)} \rho(\mathbf{x},t) \mathbf{g} dV + \int_{A^*(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) dA \]Constitutive Equation for a Newtonian Fluid (Tensor form):
\[ T_{ij} = -p \delta_{ij} + \tau_{ij} = -p \delta_{ij} + 2\mu \left( S_{ij} - \frac{1}{3} S_{mm} \delta_{ij} \right) + \mu_v S_{mm} \delta_{ij} \]Navier-Stokes Momentum Equation (Tensor form):
\[ \rho \left( \frac{\partial u_j}{\partial t} + u_i \frac{\partial u_j}{\partial x_i} \right) = -\frac{\partial p}{\partial x_j} + \rho g_j + \frac{\partial}{\partial x_i} \left[ \mu \left( \frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) + \left( \mu_v - \frac{2}{3} \mu \right) \frac{\partial u_m}{\partial x_m} \delta_{ij} \right] \]