Physlib.FluidDynamics.NavierStokes.Basic

The Navier-Stokes equations in $d$-dimensional space represent the system where both the classical continuity equation and the conservative momentum equation are satisfied. For a fluid with density $\rho(t, x)$, velocity field $\mathbf{u}(t, x)$, stress tensor field $\sigma(t, x)$, and external body force field $\mathbf{f}(t, x)$, the system is defined by the following equations: $$\begin{cases} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \\ \frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla \cdot \sigma + \rho \mathbf{f} \end{cases}$$ 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 divergence operator (vector divergence for the first equation and matrix divergence for the second). - $\sigma$ is the externally supplied stress tensor. - $\mathbf{f}$ is the body force per unit mass.

For a fluid in $d$-dimensional space described by the data $data$ (comprising density $\rho$, velocity $\mathbf{u}$, stress tensor $\sigma$, and body force $\mathbf{f}$), the convective Navier-Stokes equations are satisfied if both the classical continuity equation and the convective momentum equation hold: 1. **Continuity Equation**: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$ 2. **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 $\nabla \cdot (\rho \mathbf{u})$ is the divergence of the mass flux, and $\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u}$ represents the material derivative (convective acceleration) of the velocity field.

Consider a fluid in $d$-dimensional space with density $\rho(t, x)$, velocity field $\mathbf{u}(t, x)$, stress tensor $\sigma(t, x)$, and external body force $\mathbf{f}(t, x)$. Suppose that: 1. For every $x$, the functions $t \mapsto \rho(t, x)$ and $t \mapsto \mathbf{u}(t, x)$ are differentiable. 2. For every $t$, the velocity field $x \mapsto \mathbf{u}(t, x)$ and the momentum density field $x \mapsto \rho(t, x) \mathbf{u}(t, x)$ are differentiable. Then, the conservative form of the Navier-Stokes equations: $$\begin{cases} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \\ \frac{\partial (\rho \mathbf{u})}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = \nabla \cdot \sigma + \rho \mathbf{f} \end{cases}$$ holds if and only if the convective form of the Navier-Stokes equations: $$\begin{cases} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \\ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \nabla \cdot \sigma + \rho \mathbf{f} \end{cases}$$ holds. This equivalence relies on the fields being differentiable enough for the application of the product rules.