Page 659 - Bird R.B. Transport phenomena

§20.4 Boundary Layer Mass Transport with Complex Interfacial Motion 639

Time t'

Boundary
layer adjacent
to interface
Stationary
origin of
coordinates

Fig. 20.4-2. Element dS (shaded) of a deforming interfacial area shown at two different times,
V and t, with the adjacent boundary layer. The vectors are (at time t):

OP = r (u, w, t) = position vector of a point on the interface
s
PQ = yn(w, w, t) = vector of length у normal to the interface locating a point in
the boundary layer
OQ = r(w, w, y, t) = position vector for a point in the boundary layer
The element of interfacial area consists of the same material particles as it moves through
space. The magnitude of the area changes with time and is given by dS = -r 1 du X —^ dw
Similarly, the magnitude of the volume of that part of the boundary layer between у and

The instantaneous volume of a spatial element du dw dy in the boundary layer (see
Fig. 20.4-2) is

dV = Vg(w, w, y, t) du dw dy (20.4-3)
in which Vg(w, w, y, t) is the following product of the local interfacial base vectors,
(d/du)r s and (d/dv)r , and the normal unit vector {d/dy)x s = n,
s
(20.4-4)

and is considered nonnegative in this discussion. The second equality follows because n
is collinear with the vector product of the local interfacial base vectors, which lie in the
plane of the interface. Correspondingly, the instantaneous area of the interfacial element
du dw in Fig. 20.4-2 is

dS = s{u, w, t)du dw (20.4-5)
in which s(u, w, t) is the following product of the interfacial basis vectors:

s(w, w, t) = Vg(u, w, 0, t) = (20.4-6)