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Chapter 4
Velocity Distributions with More
Than One Independent Variable
§4.1 Time-dependent flow of Newtonian fluids
§4.2° Solving flow problems using a stream function
§4.3° Flow of inviscid fluids by use of the velocity potential
§4.4° Flow near solid surfaces by boundary-layer theory
In Chapter 2 we saw that viscous flow problems with straight streamlines can be solved
by shell momentum balances. In Chapter 3 we introduced the equations of continuity
and motion, which provide a better way to set up problems. The method was illustrated
in §3.6, but there we restricted ourselves to flow problems in which only ordinary differ-
ential equations had to be solved.
In this chapter we discuss several classes of problems that involve the solutions of
partial differential equations: unsteady-state flow (§4.1), viscous flow in more than one
direction (§4.2), the flow of inviscid fluids (§4.3), and viscous flow in boundary layers
(§4.4). Since all these topics are treated extensively in fluid dynamics treatises, we pro-
vide here only an introduction to them and illustrate some widely used methods for
problem solving.
In addition to the analytical methods given in this chapter, there is also a rapidly ex-
1
panding literature on numerical methods. The field of computational fluid dynamics is
already playing an important role in the field of transport phenomena. The numerical
and analytical methods play roles complementary to one another, with the numerical
methods being indispensable for complicated practical problems.
§4.1 TIME-DEPENDENT FLOW OF NEWTONIAN FLUIDS
In §3.6 only steady-state problems were solved. However, in many situations the veloc-
ity depends on both position and time, and the flow is described by partial differential
equations. In this section we illustrate three techniques that are much used in fluid
dynamics, heat conduction, and diffusion (as well as in many other branches of physics
and engineering). In each of these techniques the problem of solving a partial differ-
ential equation is converted into a problem of solving one or more ordinary differential
equations.
1
R. W. Johnson (ed.), The Handbook of Fluid Dynamics, CRC Press, Boca Raton, Fla. (1998);
C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press (1997).
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