Page 658 - Bird R.B. Transport phenomena

638 Chapter 20 Concentration Distributions with More Than One Independent Variable

(ii) that the tangential fluid velocity relative to the interface is negligible within the
concentration boundary layer. (This approximation is satisfactory for
fluid-fluid systems free of surfactants, when the interfacial drag is not too
large.)
(iii) that the concentration boundary layer along each interface is thin relative to the
local radii of interfacial curvature.
(iv) that the concentration boundary layers on nonadjacent interfacial elements do
not overlap.

Each of these approximations is asymptotically valid for small ЯЬ in nonrecirculating
АВ
flows with nonrigid interfaces and nonzero Da) /Dt—that is, with time-dependent con-
A
centration as viewed by an observer moving with the fluid. The systems considered in
part (a) of §20.3 are thus included, because they are time-dependent for such an observer
(though steady for a stationary one).
Interfacially embedded coordinates are used in this discussion, with a piecewise
smooth interfacial grid as in Fig. 20.4-1. Each interfacial element in the system is perma-
r
nently labeled with surface coordinates (u, w), and its position vector is (u, w, t). Each
s
point in a boundary layer is identified by its distance у from the nearest interfacial point,
together with the surface coordinates {u, w) of that point. The instantaneous position
vector of each point (u, w, y) at time t is then
r(w, w, y, t) = {u, w, t) + yn(w, w, t) (20.4-2)
r
s
relative to a stationary origin, as illustrated in Fig. 20.4-2. The function (w, w, t) gives
r
s
the trajectory of each interfacial point (u, w, 0), and the associated function п(м, w, t) =
(d/dy)r gives the instantaneous normal vector from each surface element toward its posi-
tive side. These functions are computable from fluid mechanics for simple flows, and
provide a framework for analyzing experiments in complex flows.

Dropl
Composite
drop

14
15

Fig. 20.4-1. Schematic illustration of embedded coordinates in a simple coa-
lescence process. W.E. Stewart, J.B. Angelo, and E.N. Lightfoot, AIChE Jour-
nal, 14,458^67 (1968).