Tensors for Physics

98 7 Fields, Spatial Differential Operators

A fluid composed of particles with an internal rotational degree of freedom, in gen-
eral, also possesses an internal angular momentum or spin density. The balance
equation for this quantity contains the axial vectorpλ=ελνμpνμin such a way, that
it cancels in the sum of the equations of change for the orbital and the internal angular
momenta. This is due to the conservation of the total angular momentum. As a con-
sequence, the antisymmetric part of the pressure tensor is identical to zero for fluids
composed of particles which have no rotational degree of freedom, like gaseous or
liquid Argon. In the hydrodynamic description of flow processes in molecular fluids,
and this includes water, the antisymmetric part of the pressure tensor relaxes to zero
on a time scale fast compared with typical hydrodynamical time changes, such that
the pressure tensor can be treated as being symmetric. Then constitutive laws are

needed forp ̃and pνμonly. In hydrodynamics, the relations

p ̃=−ηV∇λvλ, pνμ=− 2 η∇νvμ, (7.55)

are used. The non-negative coefficientsηandηVare theshear viscosityand the
volume viscosity, respectively. A justification of these constitutive laws and gener-
alizations thereof are treated in Sects.16.3,16.4,17.3, and17.4. Insertion of (7.55)
into the momentum balance yields a closed equation for the flow velocityv.For
ηV=0, this corresponds to theNavier-Stokes equations.

7.5 Maxwell Equations in Differential Form

7.5.1 Four-Field Formulation

The full Maxwell equations, in differential form, and in the conventionalfour-field
formulation,are

∇μDμ=ρ, εμνλ∇νHλ=jμ+

∂

∂t

Dμ, (7.56)

and

εμνλ∇νEλ=−

∂

∂t

Bμ, ∇μBμ= 0. (7.57)

The first pair of equations, referred to as theinhomogeneous Maxwell equations,
involve the densityρof electric charges and the electric current densityj. The second
pair are thehomogeneous Maxwell equations.HereEis the electric field,Dis the
electric displacement field. Frequently, bothHand themagnetic inductionBare
calledmagnetic field. For charges and currents in vacuum, one hasD=ε 0 EandB=
μ 0 H, whereε 0 andμ 0 are thedielectric permeabilityand themagnetic susceptibility
of the vacuum. When all charges and currents are represented byρandj,thetwo
fieldsEandBwould suffice for electrodynamics, and one could useε 0 =1,μ 0 =1,