Chapter 1
Convection
Material volumes and transport theorem. Let ρ(x,t) and u(x, t) be
the density and velocity fields of a fluid. Suppose u(x, t) is given. In prin-
ciple, trajectories of all fluid particles can be determined, and the density
ρ(x,t) at any time t can be determined from an initial condition consisting
of given values of ρ(x, 0). To understand this determination, first look at
time sequences of regions in space corresponding to the same fluid parti-
cles, called material volumes. By definition, the total mass of fluid inside a
R(0)
x
X(x, t)
R(t)
Path of fluid particle
from time 0 to time t
Figure 1.1.
material volume R(t) is independent of t, so
(1.1)
d
dt
R(t)
ρ(x,t) dx = 0.
In general, given a vector field u(x, t) on
Rn,
one can construct a flow map
of
Rn
into itself. The image of x, denoted by X(x, t), satisfies the ODE
initial value problem
(1.2)
˙
X = u(x, t), all t,
X(x, 0) = x, given.
3
http://dx.doi.org/10.1090/gsm/109/01
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