What is the equivalent form of the Bernoulli equation for constant-density inviscid flow that appears steady when viewed in a frame of reference that rotates at a constant rate about the z-axis, i.e., when \( \boldsymbol{\Omega} = (0, 0, \Omega_z) \) with \( \Omega_z \) constant?
[Sol] \(\because\) Momentum equation in a non-inertial frame of reference \(\rho \frac{D' \mathbf{u}'}{Dt} = -\nabla' p + \rho \mathbf{g} + \mu \nabla'^2 \mathbf{u}' - \rho \left[ \frac{d \mathbf{U}}{dt} + \frac{d \boldsymbol{\Omega}}{dt} \times \mathbf{x}' + 2 \boldsymbol{\Omega} \times \mathbf{u}' + \boldsymbol{\Omega} \times ( \boldsymbol{\Omega} \times \mathbf{x}' ) \right]\)
For steady (\( \partial / \partial t = 0 \)) inviscid (\( \mu = 0 \)) flow in a steadily rotating (\( d \boldsymbol{\Omega} / dt = 0 \)) coordinate system that does not accelerate (\( d \mathbf{U} / dt = 0 \)): \( \rho (\mathbf{u}' \cdot \nabla') \mathbf{u}' = -\nabla' p + \rho \mathbf{g} + \mu \nabla'^2 \mathbf{u}' - \rho \left[ 2 \boldsymbol{\Omega} \times \mathbf{u}' + \boldsymbol{\Omega} \times ( \boldsymbol{\Omega} \times \mathbf{x}' ) \right] \)
\(\therefore\) The body force term can be replaced with the gradient of the body-force potential \( \mathbf{g} = -\nabla \Phi \), and the \(\left[ 2 \boldsymbol{\Omega} \times \mathbf{u}' + \boldsymbol{\Omega} \times ( \boldsymbol{\Omega} \times \mathbf{x}' ) \right]\) can be evaluated when \( \boldsymbol{\Omega} = (0, 0, \Omega_z) \): \( \rho \nabla' \left( \frac{1}{2} | \mathbf{u}' |^2 \right) + \rho ( \nabla' \times \mathbf{u}' ) \times \mathbf{u}' = -\nabla' p - \rho \nabla' \Phi - \rho \left[ 2 \boldsymbol{\Omega} \times \mathbf{u}' - \Omega_z^2 ( x' \mathbf{e}_x' + y' \mathbf{e}_y' ) \right] \) \(\Rightarrow -\Omega_z^2 ( x' \mathbf{e}_x' + y' \mathbf{e}_y' ) = -\nabla' \left( \frac{1}{2} \Omega_z^2 (x'^2 + y'^2) \right)\)
\(\therefore\) \(\nabla' \left( \frac{\rho}{2} | \mathbf{u}' |^2 + p + \rho \Phi - \frac{1}{2} \rho \Omega_z^2 (x'^2 + y'^2) \right) = -2 \rho \boldsymbol{\Omega} \times \mathbf{u}' - \rho ( \nabla' \times \mathbf{u}' ) \times \mathbf{u}'\)
\(\therefore\) \(\mathbf{u}' \cdot \nabla' \left( \frac{\rho}{2} | \mathbf{u}' |^2 + p + \rho \Phi - \frac{1}{2} \rho \Omega_z^2 (x'^2 + y'^2) \right) = 0\) \(\Rightarrow \boxed{\frac{\rho}{2} | \mathbf{u}' |^2 + p + \rho \Phi - \frac{1}{2} \rho \Omega_z^2 (x'^2 + y'^2) = \text{constant along a streamline in the rotating frame}}\)
\(\frac{\partial p}{\partial r'} = + \frac{\rho}{2} \Omega_z^2 r'\) where \(r' = \sqrt{x'^2 + y'^2} \)