Deriving boundary condition on the free surface of a pool of water

Question

I would like to know how one could derive the Bernoulli’s equation (and hence the boundary condition on the free surface of a pool of water), assuming that the fluid is inviscid and irrotational, and starting with the equations of the conservation of mass and the conservation of momentum (the “Euler” equations).

Answer 1

The derivation can be found in numerous textbooks in fluid dynamics/continuum mechanics (e.g. "An Introduction to Fluid Dynamics" by G. K. Batchelor and "A First Course in Continuum Mechanics" by O. Gonzalez and A. M. Stuart). Due to the abundance of literature on the subject, I omit certain details related to notation which can normally be inferred from textbooks or online resources.

Assume that $B$ is a reference configuration, $\varphi : B \times [0,\infty) \longrightarrow E^3$ is a motion (assumes certain smoothness conditions), $v$ represents the velocity field, $\rho_0\in\mathbb{R}$ the density and $p:B_t \times [0,\infty) \longrightarrow \mathbb{R}$ the pressure, such that the following conditions are satisfied for all $t \geq 0$ and $x \in B_t$:

• (1) $\nabla \cdot v = 0$
• (2) $\rho_0\frac{\partial}{\partial t} v + \rho_0 (\nabla v) v = - \nabla p$
• (3) $\nabla \times v = 0$

Furthermore, assume that $B_t$ is open and simply connected for all $t\geq 0$. Then there exists $\phi:B_t \times [0,\infty) \longrightarrow \mathbb{R}$ such that $v=\nabla\phi$ for all $x$ and $t$ as above (see velocity potential for irrotational vector fields).

From (3), $\nabla v = \nabla v^T$. Therefore, $(\nabla v)v = \frac{1}{2} \nabla(v \cdot v)$ (this can be derived using the well known identity $\nabla (v \cdot w) = (\nabla v)^T w + (\nabla w)^T v)$. Substituting into (2) yields $\rho_0\frac{\partial}{\partial t} v + \rho_0 (\frac{1}{2} \nabla (v \cdot v)) = - \nabla p$. Selectively substituting $v=\nabla\phi$ and rearranging (taking into account that $\nabla$ and $\frac{\partial}{\partial t}$ commute) yields $\nabla (\frac{\partial}{\partial t} \phi + \frac{1}{2} (v \cdot v) + \rho_0^{-1} p) = 0$. Thence, $\frac{\partial}{\partial t} \phi + \frac{1}{2} (v \cdot v) + \rho_0^{-1} p = f$ for some function $f:[0, \infty) \longrightarrow \mathbb{R}$. For a steady field this simplifies further to $\frac{1}{2} (v \cdot v) + \rho_0^{-1} p = const$, which is known as the Bernoulli equation. With regard to its connection to the boundary condition on the free surface, I believe that this is off-topic here (most likely, this would be more suitable for Physics Stack Exchange), unless further background information is provided allowing to state it as a precise mathematical problem.