Page 55 - Bird R.B. Transport phenomena

Chapter 2

Shell Momentum Balances

and Velocity Distributions

in Laminar Flow

§2.1 Shell momentum balances and boundary conditions
§2.2 Flow of a falling film
§2.3 Flow through a circular tube
§2.4 Flow through an annulus
§2.5 Flow of two adjacent immiscible fluids
§2.6 Creeping flow around a sphere

In this chapter we show how to obtain the velocity profiles for laminar flows of fluids in
simple flow systems. These derivations make use of the definition of viscosity, the ex-
pressions for the molecular and convective momentum fluxes, and the concept of a mo-
mentum balance. Once the velocity profiles have been obtained, we can then get other
quantities such as the maximum velocity, the average velocity, or the shear stress at a
surface. Often it is these latter quantities that are of interest in engineering problems.
In the first section we make a few general remarks about how to set up differential
momentum balances. In the sections that follow we work out in detail several classical
examples of viscous flow patterns. These examples should be thoroughly understood,
since we shall have frequent occasions to refer to them in subsequent chapters. Although
these problems are rather simple and involve idealized systems, they are nonetheless
often used in solving practical problems.
The systems studied in this chapter are so arranged that the reader is gradually in-
troduced to a variety of factors that arise in the solution of viscous flow problems. In §2.2
the falling film problem illustrates the role of gravity forces and the use of Cartesian co-
ordinates; it also shows how to solve the problem when viscosity may be a function of
position. In §2.3 the flow in a circular tube illustrates the role of pressure and gravity
forces and the use of cylindrical coordinates; an approximate extension to compressible
flow is given. In §2.4 the flow in a cylindrical annulus emphasizes the role played by the
boundary conditions. Then in §2.5 the question of boundary conditions is pursued fur-
ther in the discussion of the flow of two adjacent immiscible liquids. Finally, in §2.6 the
flow around a sphere is discussed briefly to illustrate a problem in spherical coordinates
and also to point out how both tangential and normal forces are handled.
The methods and problems in this chapter apply only to steady flow. By "steady" we
mean that the pressure, density, and velocity components at each point in the stream do
not change with time. The general equations for unsteady flow are given in Chapter 3.

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