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) ˙ = u(x, t), all t, X(x, 0) = x, given. 3
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