Page 57 - Bird R.B. Transport phenomena

42 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow
• Integrate this equation to get the momentum-flux distribution.
• Insert Newton's law of viscosity and obtain a differential equation for the velocity.
• Integrate this equation to get the velocity distribution.
• Use the velocity distribution to get other quantities, such as the maximum veloc-
ity, average velocity, or force on solid surfaces.
In the integrations mentioned above, several constants of integration appear, and these
are evaluated by using "boundary conditions"—that is, statements about the velocity or
stress at the boundaries of the system. The most commonly used boundary conditions
are as follows:
a. At solid-fluid interfaces the fluid velocity equals the velocity with which the solid
surface is moving; this statement is applied to both the tangential and the normal
component of the velocity vector. The equality of the tangential components is
referred to as the "no-slip condition."
b. At a liquid-liquid interfacial plane of constant x, the tangential velocity compo-
nents v and v are continuous through the interface (the "no-slip condition") as
y z
are also the molecular stress-tensor components p + т , т , and r .
хх
xz
ху
c. At a liquid-gas interfacial plane of constant x, the stress-tensor components r xy
and T are taken to be zero, provided that the gas-side velocity gradient is not too
XZ
large. This is reasonable, since the viscosities of gases are much less than those of
liquids.
In all of these boundary conditions it is presumed that there is no material passing
through the interface; that is, there is no adsorption, absorption, dissolution, evapora-
tion, melting, or chemical reaction at the surface between the two phases. Boundary con-
ditions incorporating such phenomena appear in Problems 3C.5 and 11C.6, and §18.1.
In this section we have presented some guidelines for solving simple viscous flow
problems. For some problems slight variations on these guidelines may prove to be
appropriate.

§2.2 FLOW OF A FALLING FILM
The first example we discuss is that of the flow of a liquid down an inclined flat plate of
length L and width W, as shown in Fig. 2.2-1. Such films have been studied in connection
with wetted-wall towers, evaporation and gas-absorption experiments, and applications
of coatings. We consider the viscosity and density of the fluid to be constant.
A complete description of the liquid flow is difficult because of the disturbances at
the edges of the system (z = 0, z = L, у = 0, у = W). An adequate description can often be

Entrance disturbance -

Liquid film
- Liquid in
Reservoir
Exit disturbance
Fig. 2.2-1 Schematic
diagram of the falling
Direction of film experiment, show-
gravity
ing end effects.