Page 72 - Bird R.B. Transport phenomena

§2.5 Flow of Two Adjacent Immiscible Fluids 57

We may immediately make use of one of the boundary conditions—namely, that the
momentum flux r xz is continuous through the fluid-fluid interface:

B.C. 1: atx = 0, 4 = 4 (2.5-4)
This tells us that C\ = C"; hence we drop the superscript and call both integration con-
stants Q.
When Newton's law of viscosity is substituted into Eqs. 2.5-2 and 2.5-3, we get
(2.5-5)

p L C, (2.5-6)
dx \ L
These two equations can be integrated to give
, (Р0-Р1У
(2.5-7)

C l r 1 (2.5-8)
о Иг II X
2/LL L /л
l
The three integration constants can be determined from the following no-slip boundary
conditions:
B.C. 2: at x = 0, v\ = v\ l (2.5-9)
B.C. 3: at x = -b, v\ = 0 (2.5-10)
1
B.C. 4: atx=+b, Ы = 0 (2.5-11)
When these three boundary conditions are applied, we get three simultaneous equations
for the integration constants:
from B.C. 2: C\ = C\ l (2.5-12)

from B.C. 3: 0 = - (2.5-13)
2/x'L
from B.C. 4: 0 = - (2.5-14)
M 11
From these three equations we get
(Po ~ Pdb /V - д 11
C, = - (2.5-15)
2L \p
2
(Po-Pi)b ( 2fi ]
C\= + = cv (2.5-16)
The resulting momentum-flux and velocity profiles are

(2.5-17)

(2.5-18)

_ ft, - ^)b 2 Г/ у \ / ' - (2.5-19)
M
11
V'L L V + M / V +