Physlib.FluidDynamics.FluidFlow.Momentum

For a fluid flow in $d$ dimensions, the momentum density is the vector field defined at each time $t$ and position $x$ by the product $\rho(t, x) \mathbf{u}(t, x)$, where $\rho$ is the mass density and $\mathbf{u}$ is the velocity of the fluid.

Given a fluid flow in $d$ dimensions with mass density field $\rho(t, \mathbf{x})$ and velocity field $\mathbf{u}(t, \mathbf{x})$, the convective momentum flux is the matrix-valued field defined by the tensor product $\rho (\mathbf{u} \otimes \mathbf{u})$. At a given time $t$ and position $\mathbf{x}$, the $(i, j)$-th entry of this matrix is $\rho(t, \mathbf{x}) u_i(t, \mathbf{x}) u_j(t, \mathbf{x})$ for $i, j \in \{0, \dots, d-1\}$.

For a fluid flow in $d$ dimensions with velocity field $\mathbf{u}(t, \mathbf{x})$, the convective term is the vector field defined at each time $t$ and position $\mathbf{x}$ by $$((\mathbf{u} \cdot \nabla) \mathbf{u})(t, \mathbf{x}) = \sum_{j=0}^{d-1} u_j(t, \mathbf{x}) \frac{\partial \mathbf{u}}{\partial x_j}(t, \mathbf{x})$$ where $u_j(t, \mathbf{x})$ is the $j$-th component of the velocity vector and $\frac{\partial \mathbf{u}}{\partial x_j}$ is the partial derivative of the velocity field with respect to the $j$-th spatial coordinate. This term represents the nonlinear transport of velocity by the flow itself.

For a fluid flow in $d$ dimensions with velocity field $\mathbf{u}(t, \mathbf{x})$, the material acceleration is the vector field defined at each time $t$ and position $\mathbf{x}$ by the sum of the partial time derivative of the velocity and the convective term: $$\frac{\partial \mathbf{u}}{\partial t}(t, \mathbf{x}) + ((\mathbf{u} \cdot \nabla) \mathbf{u})(t, \mathbf{x})$$ where $\frac{\partial \mathbf{u}}{\partial t}$ denotes the derivative of the velocity field with respect to time at a fixed position, and $(\mathbf{u} \cdot \nabla) \mathbf{u}$ is the convective term representing the nonlinear transport of velocity by the flow.

Let $d$ be a natural number representing the spatial dimension. Let $\rho: \text{Time} \to \mathbb{R}$ be a scalar field and $\mathbf{u}: \text{Time} \to \mathbb{R}^d$ be a velocity field, both evaluated at a fixed position. If $\rho$ and $\mathbf{u}$ are differentiable at time $t$, then the time derivative of their product $\rho \mathbf{u}$ at $t$ follows the product rule: $$\partial_t (\rho(t) \mathbf{u}(t)) = \rho(t) \partial_t \mathbf{u}(t) + (\partial_t \rho(t)) \mathbf{u}(t)$$ where $\partial_t$ denotes the derivative with respect to time.

For a fluid flow in $d$ dimensions with mass density $\rho(t, \mathbf{x})$ and velocity field $\mathbf{u}(t, \mathbf{x})$, the momentum density is defined as $(\rho \mathbf{u})(t, \mathbf{x})$. If the density and velocity are differentiable with respect to time at $(t, \mathbf{x})$, then the time derivative of the momentum density at that point satisfies the product rule: $$\frac{\partial}{\partial t} (\rho \mathbf{u})(t, \mathbf{x}) = \rho(t, \mathbf{x}) \frac{\partial \mathbf{u}}{\partial t}(t, \mathbf{x}) + \frac{\partial \rho}{\partial t}(t, \mathbf{x}) \mathbf{u}(t, \mathbf{x})$$ where $\partial_t$ denotes the partial derivative with respect to time.

For a fluid flow in $d$ dimensions, let $\rho$ denote the mass density field and $\mathbf{u}$ denote the velocity field. The momentum density is the vector field $\mathbf{p} = \rho \mathbf{u}$, and the convective momentum flux is the matrix field $\boldsymbol{\Pi} = \rho \mathbf{u} \otimes \mathbf{u}$, where the $(i, j)$-th component is $\Pi_{ij} = \rho u_i u_j$. For any time $t$, position $\mathbf{x}$, and indices $i, j \in \{0, \dots, d-1\}$, if the momentum density and velocity fields are spatially differentiable, then the partial derivative of the $(i, j)$-th component of the momentum flux with respect to the $j$-th spatial coordinate satisfies: $$\partial_j (\rho u_i u_j) = u_i \partial_j (\rho u_j) + (\partial_j u_i) \rho u_j$$ where $\partial_j$ denotes the partial derivative $\frac{\partial}{\partial x_j}$.

For a fluid flow in $d$ dimensions with mass density $\rho(t, \mathbf{x})$ and velocity field $\mathbf{u}(t, \mathbf{x})$, let the momentum density be $\rho \mathbf{u}$ and the convective momentum flux be the tensor field $\rho \mathbf{u} \otimes \mathbf{u}$. If the momentum density and the velocity field are spatially differentiable at time $t$ and position $\mathbf{x}$, then the matrix divergence of the convective momentum flux decomposes into continuity and convective parts as follows: $$\nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = (\nabla \cdot (\rho \mathbf{u})) \mathbf{u} + \rho (\mathbf{u} \cdot \nabla) \mathbf{u}$$ where $\nabla \cdot$ denotes the divergence operator and $(\mathbf{u} \cdot \nabla) \mathbf{u}$ is the convective term.

For a fluid flow in $d$ dimensions, let $\rho(t, \mathbf{x})$ denote the mass density field and $\mathbf{u}(t, \mathbf{x})$ denote the velocity field. Suppose that $\rho$ and $\mathbf{u}$ are differentiable with respect to time at $(t, \mathbf{x})$, and that the momentum density $\rho \mathbf{u}$ and the velocity $\mathbf{u}$ are spatially differentiable. Then the conservative momentum transport terms satisfy the following identity: $$\frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) + \left( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) \right) \mathbf{u}$$ where $\rho \mathbf{u} \otimes \mathbf{u}$ is the convective momentum flux, $\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}$ is the material acceleration, and $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u})$ is the continuity residual.