[Ex] Pulsating flow in a tube, flow past undulating body (sperm, bacteria), or flow through a turbomachine. Momentum Eq. for an incompressible flow: $$ \rho \left( \frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \vec{\nabla}) \vec{u} \right) = - \vec{\nabla} p + \rho \vec{g} + \mu \, \vec{\nabla}^2 \vec{u} $$ Define dimensionless variables: $$ \vec{x}^{★} = \frac{\vec{x}}{L}, \quad t^{★} = \Omega t, \quad \vec{u}^{★} = \frac{\vec{u}}{U}, \quad \vec{g}^{★} = \frac{\vec{g}}{g}, \quad p^{★} = \frac{p - p_{\infty}}{\rho U^2} $$ $$ \rho \left(\frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \vec{\nabla}) \vec{u} \right) = - \vec{\nabla} p + \rho \vec{g} + \mu \, \vec{\nabla}^2 \vec{u} \Rightarrow \left[ \frac{\Omega L}{U} \right] \frac{\partial \vec{u}^{★}}{\partial t^{★}} + (\vec{u}^{★} \cdot \vec{\nabla}^{★}) \vec{u}^{★} = - \vec{\nabla}^{★} p^{★} + \left[ \frac{gL}{U^2} \right] \vec{g}^{★} + \left[ \frac{\mu}{\rho U L} \right] \vec{\nabla}^{★2} \vec{u}^{★} $$ where \(\vec{\nabla}^{★} = L \vec{\nabla}\)