Physlib.FluidDynamics.Euler.Basic

For a fluid flow in $d$-dimensional space (represented by a `CauchyFlow`) with density $\rho$, velocity field $\mathbf{u}$, and Cauchy stress tensor $\sigma$, and a pressure field $p$, the property `Euler` holds if the following three conditions are satisfied: 1. The **classical continuity equation** holds: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$. 2. The **Cauchy momentum equation** holds, which describes the conservation of momentum in terms of the stress tensor $\sigma$. 3. The flow is **inviscid**, meaning the Cauchy stress tensor is determined by the pressure field as $\sigma = -p \mathbb{I}$, where $\mathbb{I}$ is the identity tensor.

For a fluid flow in $d$-dimensional space characterized by a density field $\rho$, a velocity field $\mathbf{u}$, and a Cauchy stress tensor $\boldsymbol{\sigma}$, and given a pressure field $p$, the convective Euler equations are satisfied if the following three conditions hold: 1. The classical continuity equation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$. 2. The convective Cauchy momentum equation: $\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \nabla \cdot \boldsymbol{\sigma} + \mathbf{f}$, where $\mathbf{f}$ represents the external body force. 3. The inviscid stress relation: The Cauchy stress tensor is related to the pressure by $\boldsymbol{\sigma} = -p \mathbf{I}$, where $\mathbf{I}$ is the identity tensor.

Let $\rho$, $\mathbf{u}$, $\boldsymbol{\sigma}$, and $p$ represent the density, velocity, Cauchy stress tensor, and pressure of a $d$-dimensional fluid flow, respectively. Assume the following differentiability conditions hold: 1. The density $\rho$ is differentiable with respect to time $t$. 2. The velocity $\mathbf{u}$ is differentiable with respect to time $t$. 3. The momentum density $\rho \mathbf{u}$ is differentiable with respect to space. 4. The velocity $\mathbf{u}$ is differentiable with respect to space. Then the conservative Euler equations are satisfied if and only if the convective Euler equations are satisfied. The Euler equations (in both forms) consist of: - The **continuity equation**: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$. - An **inviscid stress relation**: $\boldsymbol{\sigma} = -p \mathbf{I}$, where $\mathbf{I}$ is the identity tensor. - A **momentum equation**: In the conservative form, this is the Cauchy momentum equation $\frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla \cdot \boldsymbol{\sigma} + \mathbf{f}$ (where $\mathbf{f}$ is the body force); in the convective form, this is $\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \nabla \cdot \boldsymbol{\sigma} + \mathbf{f}$.