Basics of Fluid Mechanics, 2014a

13.9. COUNTER–CURRENT FLOW 559
The liquid film thickness is unknown and can be expressed as a function of the
above boundary conditions. Thus, the liquid flow rate is a function of the boundary
conditions. On the liquid side, the gravitational force has to be balanced by the shear
forces as
dτ xy
dx = ρ L g (13.55)
The integration of equation (13.55) results in
τ xy = ρ L gx+ C 1 (13.56)
The integration constant, C 1 , can be found from the boundary condition where τ xy (x =
h) =τ i . Hence,
τ i = ρ L gh+ C 1 (13.57)
The integration constant is then C i = τ i − ρ L gh which leads to
τ xy = ρ L g (x − h)+τ i (13.58)
Substituting the newtonian fluid relationship into equation (13.58) to obtained
or in a simplified form as
μ L
dU y
dx = ρ L g (x − h)+τ i (13.59)
dU y
dx = ρ L g (x − h)
+ τ i
(13.60)
μ L μ L
Equation (13.60) can be integrate to yield
U y = ρ ( )
L g x
2
μ L 2 − hx + τ i x
+ C 2 (13.61)
μ L
The liquid velocity at the wall, [U(x =0)=0], is zero and the integration coefficient
can be found to be
The liquid velocity profile is then
C 2 =0 (13.62)
U y = ρ ( )
L g x
2
μ L 2 − hx
The velocity at the liquid–gas interface is
+ τ i x
μ L
(13.63)
U y (x = h) = τ i h
μ L
− ρ L gh 2
2 μ L
(13.64)