where the integral is taken over a path C0i starting at an arbitrary reference

point r 0 and ending at r i .

The generalization to a coarse-grained quantity is straightforward: the

δ-function in (9.41) is replaced by the weight function w and the reference

point is chosen at the position r. Thus we de¬ne the averaged stress tensor

as

def

σ±β (r) = ’ w(r ’ r c ) dxcβ ,

int

Fi± (9.42)

Ci

i

where the integral is taken over a path Ci starting at r and ending at r i . It

is logical to choose straight lines for the paths. The divergence of this stress

tensor now yields the averaged internal force per unit volume, as de¬ned in

(9.38):

(∇ · σ)(r) = f int (r). (9.43)

Proof

‚

(∇ · σ)± = ’ w(r ’ r c )dxcβ

int

Fi±

‚xβ Ci

i

ri ‚

w(r ’ r c )dxcβ

int

= Fi±

‚xcβ

r

i

Fi± w(r ’ r i ).

int

= (9.44)

i

9.3.3 Conservation of mass

The mass conservation law of continuum mechanics (9.3):

‚ρ

+ ∇ · J = 0, (9.45)

‚t

9.3 Coarse graining in space 293

is valid and exact for the averaged quantities.

Proof Note that ρ (see (9.35)) is time dependent through the time depen-

dence of r i , and that the gradient of w with respect to r i equals minus the

gradient of w with respect to r:

‚ρ

=’ mi (∇w(r ’ r i )) · v i

‚t

i

= ’∇ · mi v i w(r ’ r i )

i

= ’∇ · J

9.3.4 Conservation of momentum

The momentum conservation law of continuum mechanics (9.9):

‚ ‚

(ρu± ) = ’ Π±β (9.46)

‚t ‚xβ

(valid in the absence of external forces) is valid and exact for the averaged

quantities.

Proof After applying (9.37) and (9.39) we must prove that, in the absence

of external forces,

‚σ±β

‚J± ‚

’ mi vi± viβ w(r ’ r i ).

= (9.47)

‚t ‚xβ ‚xβ

i

Filling in (9.36) on the l.h.s., we see that there are two time-dependent

terms, vi± and r i , that need to be di¬erentiated:

‚w(r ’ r i )

‚J±

mi vi± w(r ’ r i ) ’

= ™ mi vi± viβ

‚t ‚xβ

i i

‚

= f± ’ mi vi± viβ w(r ’ r i ).

int

‚xβ

i

Since the divergence of σ equals f int (r) (see (9.43)), we recover the r.h.s. of

(9.47).

294 Coarse graining from particles to ¬‚uid dynamics

9.3.5 The equation of motion

The equation of motion of continuum mechanics (9.6):

Du

ρ

= f (r) (9.48)

Dt

now has a slightly di¬erent form and contains an additional term. Working

out the l.h.s. we obtain

Du± ‚J± ‚J± ‚ρ ‚ρ

’ u±

ρ = + uβ + uβ , (9.49)

Dt ‚t ‚xβ ‚t ‚xβ

and carrying through the di¬erentiations, using (9.35) and (9.36), we ¬nd

‚w(r ’ r i )

Du±

= f± (r) ’ mi (vi± ’ u± )(viβ ’ uβ )

ρ

Dt ‚xβ

i

‚

σ±β ’ mi (vi± ’ u± )(viβ ’ uβ )w(r ’ r i ) . (9.50)

=

‚xβ

i

The step to the last equation follows since the terms with the partial deriva-

tives ‚u± /‚xβ and ‚uβ /‚xβ vanish. For example:

‚u± ‚u±

(viβ ’ uβ )w(r ’ r i ) = (Jβ ’ ρuβ ) = 0,

mi

‚xβ ‚xβ

i

because J = ρu.

It thus turns out that there is an extra term in the ¬‚uid force that is not

present in the equation of motion of continuum mechanics. It has the form

of minus the divergence of a tensor that represents the weighted particle

velocity deviation from the ¬‚uid velocity. This term is also exactly the

di¬erence between the particle-averaged momentum ¬‚ux density (9.39) and

the momentum ¬‚ux density (9.10) as de¬ned in ¬‚uid mechanics. Let us call

this term the excess momentum ¬‚ux density Πexc :

mi [vi± ’ u± (r)][viβ ’ uβ (r)]w(r ’ r i ).