Before elaborating on the stress tensor, we will formulate the equations for

momentum conservation.

9.2.3 Conservation of linear momentum

The momentum density, or the amount of linear momentum per unit vol-

ume, de¬ned with respect to a ¬xed coordinate system, is given by ρu. This

is the same as the mass ¬‚ux density J (see (9.2)). Conservation of momen-

tum means that “ in the absence of external forces “ the amount of linear

momentum increases with time as a result of the net in¬‚ux of momentum,

or “ in other words “ that the time derivative of the momentum density

equals minus the divergence of the momentum ¬‚ux density. Since momen-

tum density is a vector, the momentum ¬‚ux density must be a tensor. We

call it Π.

The momentum conservation is expressed by

‚ ‚ ‚

(ρu) = ’∇ · Π, (ρu± ) = ’ Π±β . (9.9)

‚t ‚t ‚xβ

This expression can be proved to be valid (see below) when the following

de¬nition of the momentum ¬‚ux density tensor is adopted:

Π±β = ’σ±β + ρu± uβ . (9.10)

This de¬nition makes sense. There are two contributions: momentum can

change either because a force gives an acceleration, or because particles ¬‚ow

in or out of a region. The momentum ¬‚ux density tensor element Π±β is

1 For a more detailed discussion of the stress tensor and its relation to pressure, see Chapter 17,

Section 17.7

284 Coarse graining from particles to ¬‚uid dynamics

the ± component of the outward ¬‚ow of momentum through a unit area

perpendicular to the xβ axis.

Proof

‚ρu± ‚u± ‚ρ

=ρ + u±

‚t ‚t ‚t

‚ρuβ

Du± ‚u±

’ ρuβ ’ u±

=ρ

Dt ‚xβ ‚xβ

‚σ±β ‚ ‚

’ (ρu± uβ ) = ’

= Π±β .

‚xβ ‚xβ ‚xβ

In the ¬rst line we have used (9.1) and (9.3) and in the second line (9.8).

9.2.4 The stress tensor and the Navier“Stokes equation

The stress tensor σ is (in an isotropic ¬‚uid) composed of a diagonal pressure

tensor and a symmetric viscous stress tensor „ :

σ = ’p1 + „ , σ±β = ’p δ±β + „±β . (9.11)

In an isotropic Newtonian ¬‚uid where viscous forces are assumed to be

proportional to velocity gradients, the only possible form2 of the viscous

stress tensor is

‚u± ‚uβ 2

+ ζ ’ · δ±β ∇ · u.

„±β = · + (9.12)

‚xβ ‚x± 3

The tensor must be symmetric with ‚u± /‚xβ + ‚uβ /‚u± as o¬-diagonal ele-

ments, because these vanish for a uniform rotational motion without internal

friction, for which u = ω — r (ω being the angular velocity). We can split

the viscous stress tensor into a traceless, symmetric part and an isotropic

part:

⎛ ‚u ⎞

‚uy

2 ‚x ’ 2 ∇ · u ‚ux ‚ux

+ ‚uz

+ ‚x

x

3 ‚y ‚z ‚x

⎜ ‚uy ⎟

‚uy ‚uy

2 ‚y ’ 3 ∇ · u

‚ux ‚uz

2

„ = ·⎝ ⎠

‚x + ‚y ‚z + ‚y

‚uy

2 ‚uz ’ 2 ∇ · u

‚uz ‚ux ‚uz

+ +

‚x ‚z ‚y ‚z ‚z 3

⎛ ⎞

100

+ζ∇ · u ⎝ 0 1 0 ⎠ . (9.13)

001

2 For a detailed derivation see, e.g., Landau and Lifschitz (1987).

9.2 The macroscopic equations of ¬‚uid dynamics 285

There can be only two parameters: the shear viscosity coe¬cient · related

to shear stress and the bulk viscosity coe¬cient ζ related to isotropic (com-

pression) stress.

For incompressible ¬‚uids, with ∇·u = 0, the viscous stress tensor simpli¬es

to the following traceless tensor:

‚u± ‚uβ

„±β = · + (incompressible). (9.14)

‚xβ ‚x±

For incompressible ¬‚uids there is only one viscosity coe¬cient.

The divergence of the viscous stress tensor yields the viscous force. For

space-independent coe¬cients, the derivatives simplify considerably, and the

viscous force is then given by

1

f visc = ∇ · „ = ·∇2 u + ζ + · ∇(∇ · u),

3

‚ 2 uβ

‚„±β 1

visc 2

f± = = ·∇ u± + ζ + · . (9.15)

‚xβ 3 ‚x± ‚xβ

Combining (9.8) and (9.15) we obtain the Navier“Stokes equation (which is

therefore only valid for locally homogeneous Newtonian ¬‚uids with constant

viscosity coe¬cients):

Du ‚u

+ ρ(u · ∇)u = ∇ · σ + f ext

ρ =ρ

Dt ‚t

1

= ’∇p + ·∇2 u + ζ + · ∇(∇ · u) + f ext . (9.16)

3

Note that for incompressible ¬‚uids the equation simpli¬es to

‚u 1 ·

+ (u · ∇)u = ’ ∇p + ∇2 u + f ext (incompressible). (9.17)

‚t ρ ρ

The viscosity occurs in this equation only as the quotient ·/ρ, which is called

the kinematic viscosity and usually indicated by the symbol ν.

9.2.5 The equation of state

The Navier“Stokes equation (9.16) and the continuity equation (9.3) are

not su¬cient to solve, for example, the time dependence of the density and

velocity ¬elds for given boundary and initial conditions. What we need in

addition is the relation between pressure and density, or, rather, the pressure

changes that result from changes in density. Under the assumption of local

286 Coarse graining from particles to ¬‚uid dynamics

thermodynamic equilibrium, the equation of state (EOS) relates pressure,

density and temperature:

f (ρ, p, T ) = 0. (9.18)

We note that pressure does not depend on the ¬‚uid velocity or its gradient:

in the equation of motion (see (9.8) and (9.11)) the systematic pressure

force has already been separated from the velocity-dependent friction forces,

which are gradients of the viscous stress tensor „ .

The equation of state expresses a relation between three thermodynamic

variables, and not just pressure and density, and is therefore “ without

further restrictions “ not su¬cient to derive the pressure response to den-