# See the MRF development

The section describes the development for the incompressible Navier-Stokes formulation in the rotating frame.

## Contents

## 1 Accelerations

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:

**Eqn [1]**

The incompressible Navier-Stokes equations in the inertial frame with constant molecular viscosity are:

**Eqn [2]**

**Eqn [3]**

Let's look at the left-hand side of the momentum equation of Eqn [2], by taking into account Eqn [1] for the acceleration term:

**Eqn [4]**

since

Also, it can be noted that

Eqn [3] can be written as

**Eqn [5]**

Eqn [5] represents the incompressible Navier-Stokes equations in the rotating frame, in terms of rotating velocities (convection velocity and convected velocity).

Eqn [5] 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 [5] can be written as

Eqn [5] can be written in terms of the absolute velocity:

**Eqn [6]**

## 5 Summary

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