Basics of Fluid Mechanics, 2014a

562 CHAPTER 13. MULTI–PHASE FLOW
Equation (13.76) can be integrated twice to yield
U G = ΔP
μ G L x2 + C 1 x + C 2 (13.77)
This velocity profile must satisfy zero velocity at the right wall. The velocity at the
interface is the same as the liquid phase velocity or the shear stress are equal. Mathematically
these boundary conditions are
and
U G (x = D) =0 (13.78)
U G (x = h) =U L (x = h) (a) or (13.79)
τ G (x = h) =τ L (x = h)
(b)
Applying B.C. (13.78) into equation (13.77) results in
U G =0= ΔP
μ G L D2 + C 1 D + C 2 (13.80)
↩→ C 2 = − ΔP
μ G L D2 + C 1 D
Which leads to
U G = ΔP
μ G L
(
x 2 − D 2) + C 1 (x − D) (13.81)
At the other boundary condition, equation (13.79)(a), becomes
ρ L gh 2
= ΔP (
h 2 − D 2) + C 1 (h − D) (13.82)
6 μ L μ G L
The last integration constant, C 1 can be evaluated as
C 1 =
ρ L gh 2 ΔP (h + D)
−
6 μ L (h − D) μ G L
(13.83)
With the integration constants evaluated, the gas velocity profile is
U G = ΔP
μ G L
(
x 2 − D 2) + ρ L gh 2 (x − D)
6 μ L (h − D)
−
ΔP (h + D)(x − D)
μ G L
(13.84)
The velocity in Equation (13.84) is equal to the velocity equation (13.64) when (x = h).
However, in that case, it is easy to show that the gas shear stress is not equal to the
liquid shear stress at the interface (when the velocities are assumed to be the equal).
The difference in shear stresses at the interface due to this assumption, of the equal
velocities, cause this assumption to be not physical.