Page 56 - Bird R.B. Transport phenomena

§2.1 Shell Momentum Balances and Boundary Conditions 41

\ Fluid containing Fig. 2.0-1 (a) Laminar flow, in which fluid
tiny particles layers move smoothly over one another in
the direction of flow, and (b) turbulent
flow, in which the flow pattern is complex
Direction and time-dependent, with considerable
(a)
of flow motion perpendicular to the principal flow
direction.

Direction
(b)
of flow

This chapter is concerned only with laminar flow. "Laminar flow" is the orderly flow
that is observed, for example, in tube flow at velocities sufficiently low that tiny particles
injected into the tube move along in a thin line. This is in sharp contrast with the wildly
chaotic "turbulent flow" at sufficiently high velocities that the particles are flung apart
and dispersed throughout the entire cross section of the tube. Turbulent flow is the sub-
ject of Chapter 5. The sketches in Fig. 2.0-1 illustrate the difference between the two flow
regimes.

§2.1 SHELL MOMENTUM BALANCES AND BOUNDARY
CONDITIONS

The problems discussed in §2.2 through §2.5 are approached by setting up momentum
balances over a thin "shell" of the fluid. For steady flow, the momentum balance is

rate of rate of rate of [rate of
momentum in | momentum out momentum in | I momentum out Г force of gravity 1 _ n
by convective I by convective by molecular I by molecular [acting on system] (2.1-1)
transport transport transport [transport
This is a restricted statement of the law of conservation of momentum. In this chapter we
apply this statement only to one component of the momentum—namely, the component
in the direction of flow. To write the momentum balance we need the expressions for the
convective momentum fluxes given in Table 1.7-1 and the molecular momentum fluxes
given in Table 1.2-1; keep in mind that the molecular momentum flux includes both the
pressure and the viscous contributions.
In this chapter the momentum balance is applied only to systems in which there is
just one velocity component, which depends on only one spatial variable; in addition,
the flow must be rectilinear. In the next chapter the momentum balance concept is ex-
tended to unsteady-state systems with curvilinear motion and more than one velocity
component.
The procedure in this chapter for setting up and solving viscous flow problems is as
follows:
• Identify the nonvanishing velocity component and the spatial variable on which it
depends.
• Write a momentum balance of the form of Eq. 2.1-1 over a thin shell perpendicular
to the relevant spatial variable.
• Let the thickness of the shell approach zero and make use of the definition of the first
derivative to obtain the corresponding differential equation for the momentum flux.