See the MRF development
The section describes the development for the incompressible Navier-Stokes formulation in the rotating frame.
To start, we will look at the acceleration term for a rotating frame .
Notation: I: inertial, R: rotating
For a general vector:
For the position vector:
The acceleration is expressed as:
The incompressible Navier-Stokes equations in the inertial frame with constant molecular viscosity are:
Let's look at the left-hand side of the momentum equation of Eqn , by taking into account Eqn  for the acceleration term:
Also, it can be noted that
Eqn  can be written as
Eqn  represents the incompressible Navier-Stokes equations in the rotating frame, in terms of rotating velocities (convection velocity and convected velocity).
Eqn  can be further developed so the convected velocity is the velocity in the inertial frame.
The term can be developed as:
So, the steady term of left-hand side of Eqn  can be written as
Eqn  can be written in terms of the absolute velocity:
In summary, for multiple frames of reference, the incompressible Navier-Stokes equations for steady flow can be written
Frame Convected velocity Steady incompressible Navier-Stokes equations Inertial absolute velocity Rotating relative velocity Rotating absolute velocity