Physlib.FluidDynamics.ThermodynamicCauchyFlow.Bernoulli

Given a dimension $d$, a thermodynamic Cauchy flow, and an external potential $\Phi: \text{Space } d \to \mathbb{R}$, the Bernoulli function is the scalar field defined at each time $t$ and position $x$ as: $$B(t, x) = \frac{1}{2} |\mathbf{u}(t, x)|^2 + h(t, x) + \Phi(x)$$ where $\frac{1}{2} |\mathbf{u}|^2$ is the specific kinetic energy (with $\mathbf{u}$ being the velocity field) and $h$ is the specific enthalpy of the flow.

For a given dimension $d$, a thermodynamic Cauchy flow with velocity field $\mathbf{u}$ and specific enthalpy $h$, and an external potential $\Phi: \text{Space } d \to \mathbb{R}$, the local Bernoulli law is the property that the spatial gradient of the Bernoulli function $B(t, x) = \frac{1}{2} |\mathbf{u}(t, x)|^2 + h(t, x) + \Phi(x)$ vanishes at every time $t$ and position $x$: $$\nabla B(t, x) = 0$$

For a given dimension $d$, a thermodynamic Cauchy flow with velocity field $\mathbf{u}$ and specific enthalpy $h$, and an external potential $\Phi: \text{Space } d \to \mathbb{R}$, the Bernoulli Law is the predicate stating that the Bernoulli function $$B(t, x) = \frac{1}{2} |\mathbf{u}(t, x)|^2 + h(t, x) + \Phi(x)$$ is spatially constant at each time $t$. That is, for every time $t$ and any two positions $x, y$, the condition $B(t, x) = B(t, y)$ holds.