Equations of Motion

In many laboratory situations, the relevant distances and time scales are short enough so that a frame of reference attached to the earth (sometimes referred to as the laboratory frame of reference) is a suitable inertial frame of reference.
In atmospheric, oceanic, or geophysical studies where time and length scales are much larger, the earth’s rotation may play an important role, so an earth-fixed frame of reference must often be treated as a noninertial frame of reference.

In a noninertial frame of reference, the continuity equation is unchanged:

\[ \frac{\partial \rho (\mathbf{x},t)}{\partial t} + \nabla \cdot \left( \rho (\mathbf{x},t) \mathbf{u} (\mathbf{x},t) \right) = 0 \quad \text{or,} \] \[ \text{in index notation:} \quad \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i} (\rho u_i) = 0 \]

But the momentum equation must be modified:

\[ \rho \left( \frac{\partial u_j}{\partial t} + u_i \frac{\partial u_j}{\partial x_i} \right) = -\frac{\partial p}{\partial x_j} + \rho g_j + \frac{\partial}{\partial x_i} \left[ \mu \left( \frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) + \left( \mu_v - \frac{2}{3} \mu \right) \frac{\partial u_m}{\partial x_m} \delta_{ij} \right] \]