Consider a possibly moving fluid with constant density ρ and let P denote the pressure in
this fluid. If B is a part of this fluid the force exerted on B by the rest of the fluid is
∫∂B− PndA where n is the outer normal from B. Assume this is the only force which
matters so for example there is no viscosity in the fluid. Thus the Cauchy stress in
rectangular coordinates should be
( )
− P 0 0
T = ( 0 − P 0 ) .
0 0 − P
Then divT = −∇P. Also suppose the only body force is from gravity, a force of the form
−ρgk, so from the balance of momentum
ρ˙v = − ρgk − ∇P (x). (27.15)
(27.15)
Now in all this, the coordinates are the spacial coordinates, and it is assumed they are
rectangular. Thus x =
(x,y,z)
T and v is the velocity while
˙v
is the total derivative of
v =
(v1,v2,v3)
T given by vt + viv,i. Take the dot product of both sides of (27.15) with
v. This yields
(ρ∕2) d-|v|2 = − ρgdz −-dP (x).
dt dt dt
Therefore,
( )
d- ρ|v|2
dt 2 + ρgz + P (x) = 0,
so there is a constant C′ such that
ρ|v|2
-----+ ρgz + P (x ) = C′
2
For convenience define γ to be the weight density of this fluid. Thus γ = ρg. Divide by γ.
Then
2
|v|-+ z + P-(x) = C.
2g γ
This is Bernoulli’s2
principle. Note how, if you keep the height the same, then if you raise
|v|
, it follows the
pressure drops.
This is often used to explain the lift of an airplane wing. The top surface is curved,
which forces the air to go faster over the top of the wing, causing a drop in pressure
which creates lift. It is also used to explain the concept of a venturi tube in which the air
loses pressure due to being pinched which causes it to flow faster. In many of these
applications, the assumptions used in which ρ is constant, and there is no other
contribution to the traction force on ∂B than pressure, so in particular, there is no
viscosity, are not correct. However, it is hoped that the effects of these deviations from
the ideal situation are small enough that the conclusions are still roughly true. You can
see how using balance of momentum can be used to consider more difficult
situations. For example, you might have a body force which is more involved than
gravity.