4 Incompressible Navier-Stokes
The incompressible Navier-Stokes equations, Euler equations (inviscid limit), vorticity dynamics, and the pressure Poisson equation.
4.1 Equations of Motion
Fluid constants: density \(\rho {\gt} 0\), dynamic viscosity \(\mu \ge 0\).
\(\rho \ne 0\).
\(\rho {\gt} 0\).
Kinematic viscosity \(\nu = \mu /\rho \).
\(\nu \ge 0\).
Incompressible Navier-Stokes: \(\nabla \! \cdot \! \mathbf{v} = 0\) and \(\rho \frac{D \mathbf{v}}{D t} = -\nabla p + \mu \nabla ^2\mathbf{v} + \mathbf{f}\).
\(\frac{D \mathbf{v}}{D t} = -\frac{1}{\rho }\nabla p + \nu \nabla ^2\mathbf{v} + \frac{1}{\rho }\mathbf{f}\).
\(\nabla \! \cdot \! (\rho \mathbf{v}) = 0\) since \(\rho \) is constant and \(\nabla \! \cdot \! \mathbf{v} = 0\).
4.2 Euler Equations
When \(\mu = 0\): \(\rho \frac{D \mathbf{v}}{D t} = -\nabla p + \mathbf{f}\).
When \(\mu = 0\): \(\frac{D \mathbf{v}}{D t} = -\frac{1}{\rho }\nabla p + \frac{1}{\rho }\mathbf{f}\).
4.3 Vorticity
Vorticity: \(\mathbf{\omega }(t,x) = \nabla \! \times \! \mathbf{v}(t,x)\).
\(\nabla \! \cdot \! \mathbf{\omega } = 0\): vorticity is solenoidal.
\(\nabla \! \times \! (\nabla p) = 0\).
Smoothness bundle for vorticity/pressure derivations: \(C^3\) velocity, curl-time commutativity, and divergence-time commutativity.
Partial derivatives of \(C^2\) vector fields are differentiable.
\(\nabla \! \times \! ((\mathbf{v}\cdot \nabla )\mathbf{v}) = (\mathbf{v}\cdot \nabla )\mathbf{\omega } - (\mathbf{\omega }\cdot \nabla )\mathbf{v}\) for divergence-free \(\mathbf{v}\). The cornerstone identity for vorticity dynamics.
\(\nabla \! \times \! (\nabla ^2\mathbf{v}) = \nabla ^2(\nabla \! \times \! \mathbf{v})\) for sufficiently smooth incompressible velocity fields.
The vortex stretching term \((\mathbf{\omega }\cdot \nabla )\mathbf{v}\) is unique to 3D.
4.4 Pressure Poisson Equation
The pressure is determined as a constraint enforcing incompressibility.