Equations of Motion

\[ \mathbf{a} = \frac{d \mathbf{u}}{dt} = \frac{d}{dt} \left( \mathbf{U} + \mathbf{u}' + \boldsymbol{\Omega} \times \mathbf{x}' \right) = \frac{d \mathbf{U}}{dt} + \mathbf{a}' + 2 \boldsymbol{\Omega} \times \mathbf{u}' + \frac{d \boldsymbol{\Omega}}{dt} \times \mathbf{x}' + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{x}') \]

In fluid mechanics, the acceleration \(\mathbf{a}\) of fluid particles is denoted \(\frac{D\mathbf{u}}{Dt}\) \[ \underbrace{\left( \frac{D\mathbf{u}}{Dt} \right)_{O123}}_{\text{Acceleration in inertial frame}} = \underbrace{\left( \frac{D'\mathbf{u}'}{Dt} \right)_{O'1'2'3'}}_{\text{Acceleration in noninertial frame}} + \underbrace{\frac{d\mathbf{U}}{dt}}_{\text{Acceleration of the noninertial frame}} + \underbrace{2\boldsymbol{\Omega} \times \mathbf{u}'}_{\text{Coriolis acceleration}} + \]\[ \underbrace{\frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{x}'}_{\text{Apparent acceleration from angular acceleration}} + \underbrace{\boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{x}')}_{\text{Centripetal acceleration}} \]

Substituting \(\left( \frac{D\mathbf{u}}{Dt} \right)_{O123} = \left( \frac{D'\mathbf{u}'}{Dt} \right)_{O'1'2'3'} + \frac{d\mathbf{U}}{dt} + 2\boldsymbol{\Omega} \times \mathbf{u}' + \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{x}' + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{x}')\) into \(\boxed{\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{u} \quad \text{(incompressible)}}\) produces \[ \rho \left( \frac{D'\mathbf{u}'}{Dt} \right)_{O'1'2'3'} = -\nabla' p + \rho \left[ \mathbf{g} - \frac{d\mathbf{U}}{dt} - 2\boldsymbol{\Omega} \times \mathbf{u}' - \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{x}' - \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{x}') \right] + \mu \nabla'^2 \mathbf{u}' \]

The equation above makes it clear that the primary effect of a noninertial frame is the addition of extra body force terms that arise from the motion of the noninertial frame. The terms \(\boxed{\mathbf{g} - \frac{d\mathbf{U}}{dt} - 2\boldsymbol{\Omega} \times \mathbf{u}' - \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{x}' - \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{x}')}\) reduce to \(\mathbf{g}\) alone when \(O'1'2'3'\) is an inertial frame \(\mathbf{U}\) = constant and \(\boldsymbol{\Omega} = 0\))