Tensors for Physics

7.4 Tensor Fields 97

7.4.3 Local Mass and Momentum Conservation,

Pressure Tensor

Letρ=ρ(r)andv=v(r)be the mass density and the local velocity field of a fluid.
Theconservation of massimplies thecontinuity equation

∂ρ

∂t

+∇ν(ρ vν)= 0. (7.49)

With the help of the substantial time derivative

d

dt

:=

∂

∂t

+vν∇ν, (7.50)

the continuity equation is equivalent to

dρ

dt

+ρ∇νvν= 0. (7.51)

The local conservation equation for the linear momentum densityρvμ, in the absence
of external forces, can be cast into the form

ρ

dvμ

dt

+∇νpνμ= 0. (7.52)

Herepνμis the pressure tensor. It characterizes the transport of momentum, which is
not of convective type. The convective transport is described by the termρvν∇νvμ,
which occurs in connection with the substantial derivative. The gradient∇νpνμ=kμ
describes an internal force density.
In thermal equilibrium, the pressure tensor of a fluid reduces to the isotropic
tensorPδμν, wherePis the hydrostatic pressure. In general, the pressure tensor can
be decomposed into its isotropic, its symmetric traceless and its antisymmetric parts,
cf. Chap. 6. Thus

pνμ=(P+ ̃p)δμν+pνμ+

1

2

ενμλpλ. (7.53)

In thermal equilibrium, the partp ̃of the scalar pressure is zero, just as pνμand the
axial vectorpλ=ελαβpαβwhich is associated with the antisymmetric part of the
pressure tensor.
The time change of the orbital angular momentumλ=ελκμrκvμcan be inferred
from the momentum conservation equation. More specifically, multiplication of
(7.52)byελκμrκand use ofrκ∇νpνμ=∇ν(rκpνμ)−pνμ∇νrκleads to

ρ

dλ

dt

+ελκμ∇ν(rκpνμ)=ελνμpνμ. (7.54)