Physlib.FluidDynamics.CauchyFlow.Momentum

A $d$-dimensional Cauchy flow satisfies the **Cauchy momentum equation** in conservative form if, for every time $t$ and position $\mathbf{x}$, the following balance holds: $$\frac{\partial}{\partial t} (\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla \cdot \sigma + \rho \mathbf{f}$$ where: - $\rho$ is the mass density and $\mathbf{u}$ is the velocity field, such that $\rho \mathbf{u}$ is the momentum density. - $\rho \mathbf{u} \otimes \mathbf{u}$ is the convective momentum flux tensor, where $(\mathbf{u} \otimes \mathbf{u})_{ij} = u_i u_j$. - $\sigma$ is the stress tensor field. - $\mathbf{f}$ is the specific body-force field (force per unit mass). - $\nabla \cdot$ (or `matrixDiv`) denotes the matrix divergence, defined as the vector whose $i$-th component is the divergence of the $i$-th row of the matrix field: $(\nabla \cdot T)_i = \sum_j \frac{\partial T_{ij}}{\partial x_j}$.

For a Cauchy flow in $d$ dimensions, this proposition states that the convective Cauchy momentum equation holds at all times $t$ and positions $\mathbf{x}$: $$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \text{div} \sigma + \rho \mathbf{f}$$ where: - $\rho(t, \mathbf{x})$ is the mass density field. - $\mathbf{u}(t, \mathbf{x})$ is the velocity field. - $\sigma(t, \mathbf{x})$ is the stress tensor field. - $\mathbf{f}(t, \mathbf{x})$ is the specific body force field. - $\text{div} \sigma$ is the divergence of the stress tensor (the vector whose components are the divergences of the rows of $\sigma$). - $\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}$ is the material acceleration of the fluid.

For a $d$-dimensional Cauchy flow, suppose the classical continuity equation $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$ holds. Assuming that the density $\rho$ and velocity $\mathbf{u}$ are differentiable with respect to time, and that the momentum density $\rho \mathbf{u}$ and the velocity $\mathbf{u}$ are spatially differentiable, then the Cauchy momentum equation in conservative form $$\frac{\partial}{\partial t} (\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla \cdot \sigma + \rho \mathbf{f}$$ is equivalent to the Cauchy momentum equation in convective form $$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \nabla \cdot \sigma + \rho \mathbf{f}$$ where $\rho$ is the mass density, $\mathbf{u}$ is the velocity field, $\sigma$ is the stress tensor field, and $\mathbf{f}$ is the specific body-force field.