Tensors for Physics

320 16 Constitutive Relations

Erlangen, Germany and, at about the same time independently by F.R. McCourt in
his PhD work in Vancouver, Canada. Both advisers of the young scientists, viz. L.
Waldmann and R.F. Snider approved the results of the calculations but considered
them not worth being published since such effects cannot be measured. However,
less than two years later, experiments were performed successfully.

16.3.5 Angular Momentum Conservation, Antisymmetric

Pressure and Angular Velocity

Consider a fluid composed of particles with a rotational degree of freedom. Let the
fluid have an average rotational velocityw. The pertaining average internal angular
momentum is denoted byJ, andJ=θwis assumed, with an effective moment of
inertiaθ. In the absence of torques due to external fields, the total angular momentum,
i.e. the sum of the orbital angular momentumλ=mελνμrνvμ, associated with
average flow velocityvand the internal angular momentum obey a local conservation
equation. Heremis the mass of a particle,ρ/mis the number density. From the local
conservation of the linear momentum, cf. (7.54), follows

(ρ/m)

dλ

dt

+ελκμ∇ν(rκpνμ)=ελνμpνμ=pλ,

wherepλis the axial vector associated with the antisymmetric part of the pressure
tensor. The corresponding equation for the internal angular momentum reads

ρ

m

dJλ

dt

+...=−pλ, (16.62)

where the dots on the left hand side indicate gradient terms linked with the flux of
internal angular momentum. The opposite sign ofpλin the equations forλand
Jλguarantees the conservation of the total angular momentum. The change of the
rotational energyw·dJ/dt, taken into account in the energy balance, leads to an extra
termwλpλin the entropy production (16.42). Thus the part of the entropy production
involving axial vectors is

ρ

m

T

(

δs

δt

)axvec

irrev

=−pλ(ωλ−wλ). (16.63)

The ansatz
pλ=− 2 ηrot(ωλ−wλ), ηrot> 0 , (16.64)

is made where therotational viscosityηrotis a phenomenological coefficient. As
a consequence, for a spatially homogeneous system, the average internal angular
momentum obeys the relaxation equation