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