Multivariable CalculusMultivariable Calculus (18).pdf

Chapter een
Some Physics
Fluid Mechanics
Suppose v(x, y, z, t) is the velocity at r = (x, y, z) = xi + yj + zk of a fluid flowing
smoothly through a region in space, and suppose ρ(x, y, z,t) is the density at r at time t. If
S is an oriented surface, it is not hard to convince yourself that the flux integral
òò ρv × dr
S
is the rate at which mass flows through the surface S. Now, if S is a closed surface, then
the mass in the region B bounded by S is, of course
òòò ρdV .
B
The rate at which this mass is changing is simply
¶ ¶ρ
òòò ρ= òòò
¶ dV ¶ dV .
t B B t
This is the same as the rate at which mass is flowing across S into B: ­ òò ρv × dr , where S
S
is given the outward pointing orientation. Thus,
¶ρ
òòò = ­òò ρ×
¶ dV v dr .
B t S
We now apply Gauss’s Theorem and get
¶ρ
òòò = ­òò ρ× = òòò ­ Ñ × ρ
¶ dV v dr ( v)dV.
B t S B
Thus,
æ ¶ρö
òòò ç + Ñ × ρ÷
è ¶ ( v)ødV .
B t
Meditate on this result. The region B is any region, and so it must be true that the
integrand itself is everywhere 0:

¶ρ
+ Ñ × (ρv) = 0 .
¶t
This is one of the fundamental equations of fluid dynamics. It is called the equation of
continuity.
In case the fluid is pressible, the continuity equation es quite simple.
¶ρ
pressible means simply that the density ρ is constant. Thus = 0 and so we have
¶t
¶ρ
+ Ñ × (ρv) = Ñ