422 CHAPTER 22. CALCULUS OF VECTOR FIELDS

22.4.8 Bernoulli’s PrincipleConsider a possibly moving fluid with constant density ρ and let P denote the pressurein 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 mat-ters so for example there is no viscosity in the fluid. Thus the Cauchy stress in rectangularcoordinates should be

T =

 −P 0 00 −P 00 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 v̇ =−ρgk−∇P(x) . (22.18)

Now in all this, the coordinates are the spacial coordinates, and it is assumed they arerectangular. Thus x = (x,y,z)T and v is the velocity while v̇ is the total derivative ofv = (v1,v2,v3)

T given by vt + viv,i. Take the dot product of both sides of 22.18 with v.This yields

(ρ/2)ddt|v|2 =−ρg

dzdt− d

dtP(x) .

Therefore,ddt

(ρ |v|2

2+ρgz+P(x)

)= 0,

so there is a constant C′ such that

ρ |v|2

2+ρgz+P(x) =C′

For convenience define γ to be the weight density of this fluid. Thus γ = ρg. Divide by γ .Then

|v|2

2g+ z+

P(x)

γ=C.

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 whichcreates lift. It is also used to explain the concept of a venturi tube in which the air losespressure due to being pinched which causes it to flow faster. In many of these applica-tions, the assumptions used in which ρ is constant, and there is no other contribution to thetraction 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 smallenough that the conclusions are still roughly true. You can see how using balance of mo-mentum can be used to consider more difficult situations. For example, you might have abody force which is more involved than gravity.

2There were many Bernoullis. This is Daniel Bernoulli. He seems to have been nicer than some of the others.Daniel was actually a doctor who was interested in mathematics.He lived from 1700-1782.