Momentum Equation
\[\rho \frac{D\mathbf{U}}{Dt} = -\nabla p + \mathbf{J}\times\mathbf{B} + \mu \nabla^2\mathbf{U}\]
Source lecture(s): pc368_lec05_mhd
Physical Meaning
The momentum equation balances the rate of change of fluid momentum with forces: pressure gradients, electromagnetic Lorentz force, and viscous stress.
Variable Definitions
| Symbol | Definition | SI Units |
|---|---|---|
| \(\rho\) | Fluid mass density | kg m\(^{-3}\) |
| \(\mathbf{U}\) | Fluid velocity | m s\(^{-1}\) |
| \(p\) | Fluid pressure | Pa |
| \(\mathbf{J}\) | Current density | A m\(^{-2}\) |
| \(\mathbf{B}\) | Magnetic field | T |
| \(\mu\) | Dynamic viscosity | Pa·s |
Assumptions
- Single MHD fluid.
- Scalar, isotropic pressure.
- Ohmic dissipation via \(\mathbf{E} + \mathbf{U}\times\mathbf{B} = \eta\mathbf{J}\).
Derivation
Take the first velocity moment of the Boltzmann equation with collision operator that conserves momentum. The Lorentz force appears as \(\int (q/m)\mathbf{E} f d^3v + \int (q/m)\mathbf{v}\times\mathbf{B} f d^3v = \rho_c \mathbf{E} + \mathbf{J}\times\mathbf{B}\).
Applications
- Z-pinch equilibrium: \(dp/dr = B_\theta^2/\mu_0\).
- Tokamak force balance: Toroidal/poloidal field stresses.
- Alfvén waves: Inertial term \(\rho \partial \mathbf{U}/\partial t\) vs. magnetic tension.
Connections to Other Equations
- MHD: Part of the MHD system.
- Alfvén Speed: Derived from momentum + induction.