Physlib.FluidDynamics.FluidFlow.Incompressible

For a given fluid flow in $d$ dimensions, the incompressibility residual is a scalar-valued function of time $t$ and position $x \in \text{Space } d$. It is defined as the divergence of the velocity field $\mathbf{v}$ at that time and position: $$(t, x) \mapsto (\nabla \cdot \mathbf{v}(t, \cdot))(x)$$ where $\mathbf{v}(t, \cdot)$ is the velocity vector field at time $t$, and $\nabla \cdot$ denotes the divergence operator.

A fluid flow in $d$ dimensions is defined as **classically incompressible** if, for every time $t$ and position $x \in \text{Space } d$, the differentiability of the velocity field $\mathbf{v}(t, \cdot)$ at $x$ implies that the incompressibility residual (the divergence of the velocity field) is zero: $$\nabla \cdot \mathbf{v}(t, x) = 0$$

A fluid flow in $d$ dimensions is defined to be smooth incompressible if for every time $t$, the velocity field $\mathbf{v}(t, \cdot): \text{Space } d \to \text{Space } d$ is differentiable, and the incompressibility residual (defined as the divergence of the velocity field) vanishes for all times $t$ and positions $x \in \text{Space } d$: $$\nabla \cdot \mathbf{v}(t, x) = 0$$

For a fluid flow in $d$ dimensions, if the flow is smooth incompressible, then it is also classically incompressible. Specifically, if for every time $t$ the velocity field $\mathbf{v}(t, \cdot)$ is globally differentiable and its divergence vanishes ($\nabla \cdot \mathbf{v}(t, x) = 0$ for all $x$), then it satisfies the classical incompressibility condition where $\nabla \cdot \mathbf{v}(t, x) = 0$ holds at any point $(t, x)$ where the velocity field is differentiable.