Page 71 - Bird R.B. Transport phenomena

56 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow
(iv) The force exerted by the fluid on the solid surfaces is obtained by summing the
forces acting on the inner and outer cylinders, as follows:
R =

(2.4-18)
The reader should explain the choice of signs in front of the shear stresses above and also
give an interpretation of the final result.
The equations derived above are valid only for laminar flow. The laminar-turbulent
transition occurs in the neighborhood of Re = 2000, with the Reynolds number defined
as Re = 2R(1 - K)(V Z) P//X.

§2.5 FLOW OF TWO ADJACENT IMMISCIBLE FLUIDS 1
Thus far we have considered flow situations with solid-fluid and liquid-gas boundaries.
We now give one example of a flow problem with a liquid-liquid interface (see Fig. 2.5-1).
Two immiscible, incompressible liquids are flowing in the z direction in a horizontal
thin slit of length L and width W under the influence of a horizontal pressure gradient
(Po ~ Pi)/L. The fluid flow rates are adjusted so that the slit is half filled with fluid I (the
more dense phase) and half filled with fluid II (the less dense phase). The fluids are flow-
ing sufficiently slowly that no instabilities occur—that is, that the interface remains ex-
actly planar. It is desired to find the momentum-flux and velocity distributions.
A differential momentum balance leads to the following differential equation for the
momentum flux:
_ po -
dr xz p L
(2.5-1)
dx L
This equation is obtained for both phase I and phase II. Integration of Eq. 2.5-1 for the
two regions gives

(2.5-2)

(2.5-3)

Velocity = (Pn-Pi)b [M
distribution -
Less dense, t
less viscous I
fluid b Plane of zero shear stress
Interface -

i / More dense, x _ 1
more viscous b " 2
fluid

v Shear stress
or momentum-
2L
flux distribution
Fig. 2.5-1 Flow of two immiscible fluids between a pair of horizontal plates under
the influence of a pressure gradient.

1
The adjacent flow of gases and liquids in conduits has been reviewed by A. E. Dukler and M.
Wicks, III, in Chapter 8 of Modern Chemical Engineering, Vol. 1, "Physical Operations/' A. Acrivos (ed.),
Reinhold, New York (1963).