$$ \frac{\partial \mathbf{u}}{\partial t} + (\nabla \times \mathbf{u}) \times \mathbf{u} = -\nabla \left(\frac{|\mathbf{u}|^2}{2} + h + \chi \right) $$ From this equation, one can derive two important equations One is the case of irrotational flows for which \(\boldsymbol{\omega} = 0\) The other is the case of steady flows

In steady flows, the field variables like the velocity is independent of time \(t\), and their time derivatives vanish identically. In this case $$ \boldsymbol{\omega} \times \mathbf{u} = -\nabla \left( \frac{u^2}{2} + h + \chi \right), \quad u^2 = |\mathbf{u}|^2 $$

Since the vector product \(\boldsymbol{\omega} \times \mathbf{u}\) is perpendicular to the vector \(\mathbf{u}\) (also perpendicular to \(\boldsymbol{\omega}\)), the inner product of \(\mathbf{u}\) and \(\boldsymbol{\omega} \times \mathbf{u}\) vanishes. Taking the inner product of \(\mathbf{u}\) with \(\boxed{\boldsymbol{\omega} \times \mathbf{u} = -\nabla \left( \frac{u^2}{2} + h + \chi \right), \quad u^2 = |\mathbf{u}|^2 }\), we have $$ \mathbf{u} \cdot \nabla \left( \frac{u^2}{2} + h + \chi \right) = \frac{\partial}{\partial s} \left( \frac{u^2}{2} + h + \chi \right) = 0 $$ where \(u = |\mathbf{u}|\). The derivative \(\partial / \partial s\) in the middle is the differentiation along a stream-line parameterized with a variable \(s\).