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One way to account for the e ect of the small-scale motion is to express the small-scale

motion in the large-scale motion. It is not obvious that this is consistent with reality, but

8.6. VISCOSITY AND THE NAVIER-STOKES EQUATION 85

it appears to be the only way to avoid a complete description of the small-scale motion of

the uid (which would be impossible).

In order to do this, we assume there is some length scale that separates the small-

scale ow from the large scale ow, and we decompose the velocity in a long-wavelength

component vL and a short-wavelength component vS :

v = vL + vS : (8.48)

In addition, we will take spatial averages over a length scale that corresponds to the

length scale that distinguishes the large-scale ow from the small-scale ow. This average

D SE

The average of the small-scale ow is zero ( v = 0) while

is indicated by brackets:

D LE L

h i.

the average of the large-scale ow is equal to the large-scale ow ( v = v ) because

the large-scale ow by de nition does not vary over the averaging length. For simplicity

we will assume that the density does not vary.

Problem a: Show that the momentum equation for the large-scale ow is given by:

@( vL ) + ( vL vL ) + (D vS vS E) = F : (8.49)

@t r r

Show in particular why this expression contains a contribution that is quadratic in

the small-scale ow, but that the terms that are linear in vS do not contribute.

All terms in (8.49) are familiar, except the last term in the left hand side. This term

exempli es the e ect of the small-scale ow on the large-scale ow since it accounts for

the transport of momentum by the small-scale ow. It looks that at this point further

progress in impossible without knowing the small scale ow vS . One way to make further

D S SE

progress is to express the small-scale momentum current v v in the large scale ow.

vL

JS

Figure 8.3: he direction of momentum transport within a large-scale ow by small-scale

motions.

Consider the large-scale ow shown in gure (8.3). Whatever the small-scale motions

are, in general they will have the character of mixing. In the example of the gure, the

momentum is large at the top of the gure and the momentum is smaller at the bottom. As

CHAPTER 8. CONSERVATION LAWS

86

a rst approximation one may assume that the small-scale motions transport momentum

in the direction opposite to the momentum gradient of the large-scale ow. By analogy

with (8.25) we can approximate the momentum transport by the small-scale ow by:

D E

JS vS vS L (8.50)

; rv

where plays the role of a di usion constant.

Problem b: Insert this relation in (8.49), drop the superscript L of vL to show that

large-scale ow satis es:

@( v) + ( vv) = 2v + F : (8.51)

@t r r

This equation is called the Navier-Stokes equation. The rst term on the right hand side

accounts for the momentum transport by small-scale motions. E ectively this leads to

viscosity of the uid.

Problem c: Viscosity tends to damp motions at smaller length-scales more than motion

at larger length scales. Show that the term 2v indeed a ects shorter length scales

r

more than larger length scales.

Problem d: Do you think this treatment of the momentum ux due to small-scale mo-

tions is realistic? Can you think of an alternative?

Despite reservations that you may (or may not) have against the treatment of viscosity

in this section, you should realize that the Navier-Stokes equation (8.51) is widely used in

uid mechanics.

8.7 Quantum mechanics = hydrodynamics

As we have seen in section (6.4) the behavior of microscopic particles is described by

Schrodinger's equation

h @ (r t) = h2 2 (r t) + V (r) (r t) (6:13) again

i @t 2m

; ; r

rather than Newton's law. In this section we reformulate the linear wave equation (6.13)

as the laws of conservation of mass and momentum for a normal uid. In order to do this

write the wave function as

i

exp h ' :

= (8.52)

p

This equation is simply the decomposition of a complex function in its absolute value

and its phase, hence and ' are real functions. The factor h is added for notational

convenience.

8.7. QUANTUM MECHANICS = HYDRODYNAMICS 87

Problem a: Insert the decomposition (8.52) in Schrodinger's equation (6.13), divide by

i

exp h ' and separate the result in real and imaginary parts to show that and

p

' satisfy the following di erential equations:

1

@+ =0 (8.53)

t m

r r'

21

2+ h

1 2 22 =

@t ' + 2m : (8.54)

8m 2

jr'j jr j ; r ;V

The problem is that at this point we do not have a velocity yet. Let us de ne the

following velocity vector:

1

vm : (8.55)

r'

Problem b: Show that this de nition of the velocity is identical to the velocity obtained

in equation (6.19) of section (6.4).

Problem c: Show that with this de nition of the velocity, expression (8.53) is identical

to the continuity equation:

@+ ( v) =0 : (8:6) again

@t r

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