Physlib.FluidDynamics.NavierStokes.Continuity

For a fluid state in $d$-dimensional space with density $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, the classical continuity equation is satisfied if, for every time $t$ and spatial point $x$ where the density is differentiable with respect to time and the mass flux $\rho \mathbf{u}$ is spatially differentiable, the following relation holds: $$\frac{\partial \rho}{\partial t}(t, x) + \nabla \cdot (\rho \mathbf{u})(t, x) = 0$$ where $\frac{\partial \rho}{\partial t}$ is the partial derivative of density with respect to time, and $\nabla \cdot (\rho \mathbf{u})$ is the divergence of the momentum density (mass flux) with respect to the spatial coordinates.

For a fluid in $d$-dimensional space with a state defined by density $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, the continuity residual at time $t$ and position $x$ is the scalar value defined by: $$\frac{\partial \rho}{\partial t}(t, x) + \nabla \cdot (\rho(t, x) \mathbf{u}(t, x))$$ where $\frac{\partial}{\partial t}$ denotes the partial derivative with respect to time and $\nabla \cdot$ denotes the spatial divergence operator. This expression represents the local rate of change of mass density plus the divergence of the mass flux.

For a fluid in $d$-dimensional space with density $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, the smooth continuity equation is a property that holds if the following conditions are met: 1. For every spatial position $x \in \text{Space } d$, the density function $t \mapsto \rho(t, x)$ is differentiable with respect to time. 2. For every time $t \in \text{Time}$, the mass flux function $x \mapsto \rho(t, x) \mathbf{u}(t, x)$ is differentiable with respect to space. 3. The continuity residual vanishes for all $t$ and $x$: $$\frac{\partial \rho}{\partial t}(t, x) + \nabla \cdot (\rho(t, x) \mathbf{u}(t, x)) = 0$$ where $\frac{\partial}{\partial t}$ is the partial derivative with respect to time and $\nabla \cdot$ is the spatial divergence operator.

For a fluid state in $d$-dimensional space with density $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, if the smooth continuity equation holds—meaning $\rho$ is differentiable with respect to time $t$, the mass flux $\rho \mathbf{u}$ is differentiable with respect to space $x$, and the relation $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$ holds for all $t$ and $x$—then the classical continuity equation is also satisfied. That is, at any point $(t, x)$ where $\rho$ is differentiable in time and $\rho \mathbf{u}$ is differentiable in space, the relation $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$ holds.