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Viscosity

Submitted by pidoc on March 13, 2009

The Navier-Stokes equation is the equation which results when the Newtonian stress
tensor, (41), is inserted into the general equation of motion, (20):
r
Dvi
Dt
= -
∂
∂xi
p +
2
3
m - l
Ê
Ë
ˆ¯
—⋅ vv È
Î Í
˘
˚ ˙
+
∂
∂xj
m
∂vj
∂xi
+
∂vi
∂xj
Ê
Ë Á
ˆ
¯ ˜
È
Î Í
˘
˚ ˙
+ rGi (44)
For constant m and l, this equation can be written in vector notation as
22
r
Dvv
Dt
= -—p +
1
3
m + l
Ê
Ë
ˆ¯
—(— ⋅vv) + m—2 vv + r
vG
(45)
where
—2 =
∂
∂x2 +
∂
∂y2 +
∂
∂z2 (46)
is a scalar operator, operating in (45) on the vector vv , just like D/Dt on the left side is the
well-known scalar operator that operates on vv .
For incompressible flows with constant viscosity,
∂vj
∂xj
=—⋅ vv = 0 , (47)
and one obtains from (44) or (45)
r
Dvi
Dt
= -
∂p
∂xi
+ m
∂ 2vi
∂xj∂x j
+ rGi , (48)
or, in vector form,
r
Dvv
Dt
= -—p + m—2 vv + r
vG
. (49)
As mentioned above, (48) or (49) are in many cases a very good approximation even
when the flow is compressible. Written out fully in Cartesian coordinates, (48) reads
r
∂vx
∂t
+ vx
∂vx
∂x
+ vy
∂vx
∂y
+ vz
∂vx
∂z
Ê
Ë Á
ˆ
¯
= -
...

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