Page 660 - Bird R.B. Transport phenomena

640 Chapter 20 Concentration Distributions with More Than One Independent Variable
In these interfacially embedded coordinates, the mass average velocity V relative to sta-
tionary coordinate axes takes the form

V(M, W, y, t) = V(M, W, y,f) + y r(w, w, y, t) (20.4-7)

In this section, v is the mass average fluid velocity relative to an observer at (w, zv, y), and
{d/dt)r{u, zv, y, t) is the velocity of that observer relative to the stationary origin. Taking
the divergence of this equation gives the corollary 2 (see Problem 20D.5)
д In Ve(w, zv, у, t)
(V • V(r, 0) = (V • V(M, W, y, t)) + 6 d t J (20.4-8)

This equation states that the divergence of V differs from that of v by the local rate of ex-
pansion or contraction of the embedded coordinate frame.
The last term in Eq. 20.4-8 arises when interfacial deformation occurs. Its omission in
such problems gives inaccurate predictions, which Higbie 5 and Danckwerts ' 6 7 then ad-
justed by introducing hypothetical surface residence times ' or surface rejuvenation. 7
5 6
Such hypotheses are not needed in the present analysis.
Application of Eq. 20.4-8 at у = 0 and use of the constant-density condition

(V • V) = 0 (20.4-9)
along with the no-slip condition on the tangential part of v, gives the derivative
д In s(u, zv, t)
(20.4-10)
y-0 dt
Ol
Hence, the truncated Taylor expansion

describes the normal component of v in an incompressible fluid near a deforming
interface.
The corresponding expansion for the tangential part of v gives
2
v || = yB || (и, w, t) + O(y ) (20.4-12)
in which В ц (и, w, t) is the interfacial y-derivative of v ц. With these results (neglecting
the O(y ) terms) and approximation (i), we can write Eq. 20.4-1 for ш (и, w, y, t) as
2
А
^ + (yB, • V..) ( ^ ) %
[ ^ 1 ^ ] I r. (20.4-13)

Here (V o • n) is the surface divergence of n at the nearest interfacial point and is the sum
of the principal curvatures of the surface there. The + • • • stands for terms of higher
order, which are here neglected.
To select the dominant terms in Eq. 20.4-13, we introduce a dimensionless coordinate
У = у/к& ) (20.4-14)
АВ

5 R. Higbie, Trans. AlChE, 31, 365-389 (1935).
6 P. V. Danckwerts, Ind. Eng. Chem., 43,1460-1467 (1951).
7
P. V. Danckwerts, AIChE Journal, 1,456-463 (1955).