One of the basic theorems of flows of an ideal fluid is Bernoulli’s theorem. Suppose that the fluid’s entropy is uniform, the entropy $s$ per unit mass of fluid is constant everywhere, and assuming the external force is conservative and the external force has a potential $\chi$ represented by $\mathbf{f} = -\nabla \chi$. Then, Euler’s equation of motion $\boxed{\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho} \nabla p + \mathbf{f}}$ reduces to $$ \frac{\partial \mathbf{u}}{\partial t} + (\nabla \times \mathbf{u}) \times \mathbf{u} = -\nabla \left(\frac{|\mathbf{u}|^2}{2} + h + \chi \right) $$ where $h$ is the enthalpy $(h = e + p/\rho)$ and $e$ the internal energy. This is valid for compressible flows.

$ \because \boxed{\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho} \nabla p + \mathbf{f}}, \quad \mathbf{f} = -\nabla \chi \quad \text{(external force is conservative)} $
$ \therefore \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho} \nabla p - \nabla \chi $
$ \because (\mathbf{u} \cdot \nabla)\mathbf{u} = \nabla \left(\frac{|\mathbf{u}|^2}{2}\right) - \mathbf{u} \times (\nabla \times \mathbf{u}), \quad \boldsymbol{\omega} = \nabla \times \mathbf{u} $
$ \therefore (\mathbf{u} \cdot \nabla)\mathbf{u} = \nabla \left(\frac{|\mathbf{u}|^2}{2}\right) - \mathbf{u} \times \boldsymbol{\omega} \Rightarrow \frac{\partial \mathbf{u}}{\partial t} + \nabla \left(\frac{|\mathbf{u}|^2}{2}\right) - \mathbf{u} \times \boldsymbol{\omega} = -\frac{1}{\rho} \nabla p - \nabla \chi $
$ \because s = \text{constant} \Rightarrow \nabla s = 0, \quad \frac{1}{\rho} \nabla p = \nabla h \quad \text{(for isentropic flow with uniform entropy)} $
$ \therefore \frac{\partial \mathbf{u}}{\partial t} + \nabla \left(\frac{|\mathbf{u}|^2}{2}\right) - \mathbf{u} \times \boldsymbol{\omega} = -\nabla h - \nabla \chi \Rightarrow \frac{\partial \mathbf{u}}{\partial t} - \mathbf{u} \times \boldsymbol{\omega} = -\nabla \left(\frac{|\mathbf{u}|^2}{2} + h + \chi\right) $
$ \because -\mathbf{u} \times \boldsymbol{\omega} = \boldsymbol{\omega} \times \mathbf{u} $
$ \therefore \boxed{ \frac{\partial \mathbf{u}}{\partial t} + (\nabla \times \mathbf{u}) \times \mathbf{u} = -\nabla \left(\frac{|\mathbf{u}|^2}{2} + h + \chi \right) } \text{where \(h = e + \frac{p}{\rho}$ with $e$ the internal energy} \)

¹ Tsutomu Kambe. (2007). Elementary Fluid Mechanics. World Scientific, Singapore.