Physlib.FluidDynamics.CauchyFlow.Inviscid

A $d$-dimensional Cauchy flow is **inviscid** with respect to a scalar pressure field $p$ if, at every time $t$ and position $x$, the Cauchy stress tensor $\boldsymbol{\sigma}(t, x)$ is equal to the isotropic pressure stress $-p(t, x) \mathbb{I}$, where $\mathbb{I}$ is the $d \times d$ identity matrix.

Let $d$ be a natural number representing the spatial dimension. For a scalar pressure field $p: \text{Space } d \to \mathbb{R}$ at a given time, the matrix divergence of the inviscid pressure stress field defined by $x \mapsto -p(x) \mathbb{I}$ (where $\mathbb{I}$ is the $d \times d$ identity matrix) is equal to the negative gradient of the pressure field: $$\text{matrixDiv}(-p \mathbb{I}) = -\nabla p$$

For a $d$-dimensional Cauchy flow with stress tensor $\boldsymbol{\sigma}$ and a scalar pressure field $p$, if the flow is inviscid (meaning $\boldsymbol{\sigma}(t, x) = -p(t, x) \mathbb{I}$ for all time $t$ and position $x$, where $\mathbb{I}$ is the identity matrix), then for every time $t$, the matrix divergence of the stress tensor is equal to the negative gradient of the pressure: $$\text{matrixDiv } \boldsymbol{\sigma}(t) = -\nabla p(t)$$ where $\nabla p$ denotes the spatial gradient of the pressure field.