Physlib.FluidDynamics.NavierStokes.Momentum

For a given dimension $d \in \mathbb{N}$ and a fluid state `fluid`, the momentum density is the vector field that assigns to each time $t$ and position $x \in \mathbb{R}^d$ the product of the fluid's mass density $\rho(t, x)$ and its velocity $\mathbf{u}(t, x)$. Mathematically, the momentum density field is given by the expression $\rho \mathbf{u}$.

Given a fluid state in $d$-dimensional space with density $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, the convective momentum flux is the $d \times d$ matrix field defined by $\rho (\mathbf{u} \otimes \mathbf{u})$. At any time $t$ and position $x$, its components are given by $(\rho \mathbf{u} \otimes \mathbf{u})_{ij} = \rho(t, x) u_i(t, x) u_j(t, x)$, where $\otimes$ denotes the outer product of the velocity vector with itself.

For a fluid in $d$-dimensional space with mass density $\rho(t, x)$, velocity field $\mathbf{u}(t, x)$, stress tensor field $\sigma(t, x)$, and body force field $\mathbf{f}(t, x)$, the momentum equation in conservative form is satisfied if for every time $t$ and position $x$: $$\frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla \cdot \sigma + \rho \mathbf{f}$$ where: - $\rho \mathbf{u}$ is the momentum density. - $\rho \mathbf{u} \otimes \mathbf{u}$ is the convective momentum flux matrix, with components $(\rho \mathbf{u} \otimes \mathbf{u})_{ij} = \rho u_i u_j$. - $\nabla \cdot$ denotes the matrix divergence operator, which maps a matrix field to a vector field by taking the divergence of its rows. - $\frac{\partial}{\partial t}$ denotes the partial derivative with respect to time. - $\sigma$ is the stress tensor field (not yet specialized to a Newtonian fluid). - $\mathbf{f}$ is the external body force per unit mass.

For a fluid in $d$-dimensional space with velocity field $\mathbf{u}(t, x)$, the convective term (also known as the nonlinear transport term) is a vector field $(\mathbf{u} \cdot \nabla) \mathbf{u}$ defined at each time $t$ and position $x$ by the sum: $$\sum_{j=0}^{d-1} u_j(t, x) \frac{\partial \mathbf{u}}{\partial x_j}(t, x)$$ where $u_j(t, x)$ 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.

For a fluid in $d$-dimensional space with velocity field $\mathbf{u}(t, x)$, the material acceleration is a vector field defined at each time $t$ and position $x$ as the sum of the partial time derivative of the velocity and the convective term: $$\frac{\partial \mathbf{u}}{\partial t}(t, x) + (\mathbf{u}(t, x) \cdot \nabla) \mathbf{u}(t, x)$$ where $\frac{\partial \mathbf{u}}{\partial t}$ denotes the local rate of change of velocity with respect to time, and $(\mathbf{u} \cdot \nabla) \mathbf{u}$ is the convective acceleration term.

For a fluid in $d$-dimensional space, the convective momentum equation is the proposition that for all times $t$ and positions $x$, the product of the fluid density $\rho(t, x)$ and the material acceleration equals the sum of the divergence of the stress tensor $\sigma(t, x)$ and the external body force. The equation is given by: $$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \text{div } \mathbf{\sigma} + \rho \mathbf{f}$$ where $\mathbf{u}$ is the velocity field, $\mathbf{\sigma}$ is the stress tensor field, $\mathbf{f}$ is the body force per unit mass, and $\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}$ is the material acceleration.

For a fluid in $d$-dimensional space with density $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, the left-hand side of the conservative momentum equation is a vector field defined at time $t$ and position $x$ as: $$\frac{\partial (\rho \mathbf{u})}{\partial t}(t, x) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u})(t, x)$$ where $\rho \mathbf{u}$ is the momentum density, $\rho \mathbf{u} \otimes \mathbf{u}$ is the convective momentum flux, and $\nabla \cdot$ denotes the matrix divergence operator.

For a fluid in $d$-dimensional space with mass density $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, the left-hand side of the convective momentum equation is the vector field defined by the product of the density and the material acceleration: $$\rho(t, x) \left( \frac{\partial \mathbf{u}}{\partial t}(t, x) + (\mathbf{u}(t, x) \cdot \nabla) \mathbf{u}(t, x) \right)$$ where $\frac{\partial \mathbf{u}}{\partial t}$ is the local time derivative of the velocity and $(\mathbf{u} \cdot \nabla) \mathbf{u}$ is the convective acceleration term.

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

For a fluid in $d$ dimensions with mass density $\rho$ and velocity field $\mathbf{u}$, the time derivative of the momentum density $\rho \mathbf{u}$ at a fixed position $x$ and time $t$ satisfies the product rule: $$\partial_t (\rho(t, x) \mathbf{u}(t, x)) = \rho(t, x) \partial_t \mathbf{u}(t, x) + (\partial_t \rho(t, x)) \mathbf{u}(t, x)$$ provided that the functions $t \mapsto \rho(t, x)$ and $t \mapsto \mathbf{u}(t, x)$ are differentiable at $t$.

Given a fluid in $d$-dimensional space with density $\rho$ and velocity field $\mathbf{u}$, let the convective momentum flux be defined as $(\rho \mathbf{u} \otimes \mathbf{u})_{ij} = \rho u_i u_j$ and the momentum density as $(\rho \mathbf{u})_j = \rho u_j$. If the momentum density and the velocity field are spatially differentiable at time $t$, then the partial derivative of the $(i, j)$-th component of the momentum flux with respect to the $j$-th spatial coordinate satisfies the product rule: $$\frac{\partial}{\partial x_j} (\rho u_i u_j) = u_i \frac{\partial (\rho u_j)}{\partial x_j} + \frac{\partial u_i}{\partial x_j} (\rho u_j),$$ where $u_k$ denotes the $k$-th component of the velocity vector field $\mathbf{u}$.

For a fluid in $d$-dimensional space with density $\rho$ and velocity field $\mathbf{u}$, suppose that at time $t$ the momentum density $\rho \mathbf{u}$ and the velocity field $\mathbf{u}$ are spatially differentiable. Then the matrix divergence of the convective momentum flux $\rho \mathbf{u} \otimes \mathbf{u}$ at position $x$ is given by: $$\nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = (\nabla \cdot (\rho \mathbf{u})) \mathbf{u} + \rho (\mathbf{u} \cdot \nabla) \mathbf{u}$$ where $\nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u})$ denotes the matrix divergence of the tensor field, $\nabla \cdot (\rho \mathbf{u})$ is the divergence of the momentum density vector field, and $(\mathbf{u} \cdot \nabla) \mathbf{u}$ is the convective (nonlinear transport) term.

For a fluid in $d$-dimensional space with density $\rho(t, x)$ and velocity field $\mathbf{u}(t, x)$, suppose that at a given time $t$ and position $x$, the density and velocity are differentiable with respect to time, and both the momentum density $\rho \mathbf{u}$ and the velocity field $\mathbf{u}$ are spatially differentiable. Then the left-hand side of the conservative momentum equation is equal to the left-hand side of the convective momentum equation plus the product of the continuity residual and the velocity field: $$\frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) + \left( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) \right) \mathbf{u}$$ where $\otimes$ denotes the outer product, $\nabla \cdot$ denotes the divergence operator (matrix divergence for the tensor term and vector divergence for the momentum density), and $(\mathbf{u} \cdot \nabla) \mathbf{u}$ is the convective acceleration term.

For a fluid in $d$-dimensional space with density $\rho$, velocity field $\mathbf{u}$, stress tensor field $\sigma$, and body force field $\mathbf{f}$, suppose that the classical continuity equation holds: $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$ Additionally, assume that: 1. The density $\rho$ and velocity $\mathbf{u}$ are differentiable with respect to time $t$ for all positions $x$. 2. The momentum density $\rho \mathbf{u}$ and the velocity field $\mathbf{u}$ are spatially differentiable for all times $t$. Then, the conservative momentum equation: $$\frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla \cdot \sigma + \rho \mathbf{f}$$ is equivalent to the convective momentum equation: $$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \nabla \cdot \sigma + \rho \mathbf{f}$$ where $\otimes$ denotes the outer product and $\nabla \cdot$ denotes the divergence operator (matrix divergence for the tensor field and vector divergence for the vector field).