The vorticity equation \(\frac{D \boldsymbol{\omega}}{D t} = (\boldsymbol{\omega} \cdot \nabla) \mathbf{u} + \nu \nabla^2 \boldsymbol{\omega}\) is valid for a fluid of uniform density and viscosity observed from an inertial frame of reference. Here, this equation is generalized to a steadily rotating frame of reference and a variable density fluid.
The flow will be assumed incompressible (or nearly so).
The resulting vorticity equation is applicable to flows in rotating machinery and to oceanic and atmospheric fluid flows of sufficient size and duration to necessitate inclusion of the earth's rotation rate when conserving momentum.
The continuity and momentum equations for flow of a variable-density incompressible fluid observed in a steadily rotating frame of reference are
\[
\frac{\partial u_i}{\partial x_i} = 0
\quad \text{and} \quad
\frac{\partial u_i}{\partial t}
+ u_j \frac{\partial u_i}{\partial x_j}
+ \underbrace{2 \epsilon_{ijk} \Omega_j u_k}_{\substack{\Omega \text{ is the angular velocity} \\ \text{of the coordinate system}}}
= -\frac{1}{\rho} \frac{\partial p}{\partial x_i}
+ \underbrace{g_i}_{\substack{\text{effective gravity} \\ \text{(including centrifugal acceleration)}}}
+ \nu \frac{\partial^2 u_i}{\partial x_j^2}
\]
The advective acceleration and viscous diffusion terms can be rewritten to \(u_j \frac{\partial u_i}{\partial x_j}
= -\epsilon_{ijk} u_j \omega_k
+ \frac{\partial}{\partial x_i} \left( \frac{1}{2} u_j^2 \right)\) and \(\nu \frac{\partial^2 u_i}{\partial x_j^2}
= -\nu \epsilon_{ijk} \frac{\partial \omega_k}{\partial x_j}\), the the Coriolis acceleration term can be rewritten to\(2 \epsilon_{ijk} \Omega_j u_k
= -2 \epsilon_{ijk} u_j \Omega_k\), and conservative body forces \(g_i = - \frac{\partial \Phi}{\partial x_i}\) where \(\Phi\) is called the force potential
\[
\frac{\partial u_i}{\partial t}
+ \frac{\partial}{\partial x_i} \left( \frac{1}{2} u_j^2 + \Phi \right)
- \epsilon_{ijk} u_j \left( \omega_k + 2 \Omega_k \right)
= -\frac{1}{\rho} \frac{\partial p}{\partial x_i}
- \nu \epsilon_{ijk} \frac{\partial \omega_k}{\partial x_j}
\]
It is another form of the Navier-Stokes momentum equation, the rotating-frame-of-reference vorticity equation is obtained by taking its curl
Applying \(\epsilon_{nqi} \frac{\partial}{\partial x_q}\) on the left side of each term
\[
\epsilon_{nqi} \frac{\partial}{\partial x_q} \left( \frac{\partial u_i}{\partial t} \right)
+ 0
- \epsilon_{nqi} \epsilon_{ijk} \frac{\partial}{\partial x_q} \left[ u_j (\omega_k + 2\Omega_k) \right]
= -\epsilon_{nqi} \frac{\partial}{\partial x_q} \left( \frac{1}{\rho} \frac{\partial p}{\partial x_i} \right)
- \nu \epsilon_{nqi} \epsilon_{ijk} \frac{\partial^2 \omega_k}{\partial x_q \partial x_j}
\]