Lecture 2 Resistive-Viscous MHD Equations

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Lecture 2
Resistive–Viscous MHD Equations

Overview

This lecture turns the schematic MHD system of Lecture 1 into an honest transport model.
1.
It separates body forces from stresses transmitted across surfaces.
2.
It makes clear where viscosity and resistivity actually enter the equations.
3.
It identifies the two main channels of dissipation: viscous heating and Ohmic heating.

The equations in (1.7)–(1.13) look deceptively compact. Their real content depends on what is meant by stress, conductivity, and heat transport. In liquid metals the closures can be very close to the ordinary Navier–Stokes ones; in plasmas they are often anisotropic and collisionality-dependent. This lecture keeps the algebra at the single-fluid level but prepares the ground for that later distinction.

Historical Perspective

One of the oldest conceptual moves in continuum mechanics is the distinction between forces that act throughout a volume and forces transmitted through its boundary. In MHD this distinction becomes especially useful because the Lorentz force behaves as a body force in the momentum equation, while viscosity and pressure naturally enter through a stress tensor.

2.1 Body and Surface Forces

Let \(V\) be a material volume bounded by a surface \(S\) with outward area element \[ d\vect {S} = \hat {\vect {n}}\, dS. \] We write the total force density as the sum of a body-force term and a surface-force term.

Body forces. Typical examples are

\vect {f}_{\mathrm {body}} = \rho \vect {g} + \J \times \B + \rho _e \E . \tag{2.1}

Under the quasi-neutral approximation, \(\rho _e \approx 0\), so the dominant electromagnetic body force is the Lorentz force \(\J \times \B \) already used in (1.8).

Surface forces. Introduce a pressure-stress tensor \(\tens {P}\) so that the force exerted on the fluid by its surroundings is

d\vect {F}_{\mathrm {surf}} = - \tens {P}\cdot d\vect {S}. \tag{2.2}

The total surface force on the volume is then

\vect {F}_{\mathrm {surf}} = - \oint _S \tens {P}\cdot d\vect {S} = - \int _V \divergence \tens {P}\, dV. \tag{2.3}

Therefore the local momentum equation is

\rho \frac {D\uvec }{Dt} = -\divergence \tens {P} + \vect {f}_{\mathrm {body}}. \tag{2.4}

2.2 Symmetry and Angular Momentum

Angular momentum conservation requires the stress tensor to be symmetric. To see this, define

\vect {L} = \int _V \rho \, \vect {r}\times \uvec \, dV. \tag{2.5}

Using (2.4) and integrating by parts gives a surface torque plus the volume term

\int _V \epsilon _{ijk} P_{kj}\, dV. \tag{2.6}

This vanishes for arbitrary material volumes only if

\boxed { P_{ij}=P_{ji}. } \tag{2.7}

That result is worth remembering: the symmetry of the stress tensor is not a decorative assumption but the local statement of angular-momentum conservation.

2.3 Pressure and Viscous Stress

We decompose the pressure-stress tensor as

\boxed { \tens {P} = p\,\tens {I} - \tens {\Pi }, } \tag{2.8}

where \(p\) is the isotropic pressure and \(\tens {\Pi }\) is the viscous stress tensor.

Substituting (2.8) into (2.4) gives

\boxed { \rho \frac {D\uvec }{Dt} = -\grad p + \divergence \tens {\Pi } + \J \times \B + \rho \vect {g}. } \tag{2.9}

Equation (2.9) is the more explicit version of (1.8): the pressure term and the viscous term are simply the divergence of the surface stress written in split form.

For an isotropic Newtonian fluid,

\boxed { \tens {\Pi } = \mu \left [ \grad \uvec + (\grad \uvec )^T - \frac {2}{3} (\divergence \uvec )\tens {I} \right ] + \zeta (\divergence \uvec )\tens {I}, } \tag{2.10}

where \(\mu \) and \(\zeta \) are the shear and bulk viscosities. If the flow is incompressible and \(\mu \) is uniform, then

\divergence \tens {\Pi } = \mu \lap \uvec . \tag{2.11}

Caution

The isotropic Newtonian closure is not the last word in plasma MHD. It is a very useful model for liquid metals and for introductory derivations, but in a magnetized plasma the true viscous stress is generally anisotropic. That is exactly the issue revisited in Lecture 5.

2.4 Resistive Induction

The magnetic field evolves according to the induction equation (1.13). For uniform resistivity, using Ampère’s law (1.11) and \(\divergence \B =0\) from (1.12), one finds

\boxed { \pp {\B }{t} = \curl (\uvec \times \B ) + \eta _{\mathrm {m}} \lap \B , \qquad \eta _{\mathrm {m}} \equiv \frac {\eta }{\muo }. } \tag{2.12}

The first term on the right describes advection and stretching of magnetic field, while the second describes diffusion of magnetic structure. In later lectures this competition will reappear as the magnetic Reynolds number and the frozen-in limit.

2.5 Dissipation and Heating

Two positive-definite sinks of ordered mechanical or magnetic energy appear naturally:

\boxed { Q_{\mathrm {visc}} = \tens {\Pi }:\grad \uvec , \qquad Q_{\mathrm {ohm}} = \eta J^2. } \tag{2.13}

These are the terms that feed the internal-energy equation (1.14). Viscosity damps velocity gradients; resistivity damps current gradients. Both convert organized motion or field structure into heat.

2.6 Experimental Perspective

This lecture is one of the places where theory first starts to feel like hardware. In liquid-metal MHD, (2.9) and (2.12) can be almost embarrassingly literal: pressure gradients drive the flow, the Lorentz force brakes or redirects it, and the diffusive coefficients can often be measured independently. In laboratory plasmas the same equations still organize the dynamics, but the closure hidden in \(\tens {\Pi }\) and in the conductivity is much more delicate. That is why the question “what is the right MHD model?” is often really a question about transport.

Takeaways

Pressure and viscosity are best understood as parts of a stress tensor acting across surfaces.

The explicit resistive–viscous momentum equation is (2.9); the corresponding induction equation is (2.12).

Viscous and Ohmic dissipation, (2.13), are not afterthoughts. They are the channels through which ordered flow and magnetic structure are irreversibly degraded.