Page 100 - Bird R.B. Transport phenomena

§3.5 The Equations of Change in Terms of the Substantial Derivative 85

Fig. 3.5-1. The equation of state for a slightly com-
^ Slightly compressible pressible fluid and an incompressible fluid when T
I fluid with
| p-p o = K(p-p o ) is constant.
where К = constant

^ Incompressible fluid
with p = p 0

Po

potential energy per unit mass. Equation 3.5-6 is a standard starting point for describing
isothermal flows of gases and liquids.
It must be kept in mind that, when constant p is assumed, the equation of state (at
constant T) is a vertical line on a plot of p vs. p (see Fig. 3.5-1). Thus, the absolute pres-
sure is no longer determinable from p and T, although pressure gradients and instanta-
neous differences remain determinate by Eq. 3.5-6 or Eq. 3.5-7. Absolute pressures are
also obtainable if p is known at some point in the system.
(ii) When the acceleration terms in the Navier-Stokes equation are neglected—that is,
when p(Dv/Dt) = 0—we get

2
0 = -Vp + fiV v + pg (3.5-8)
which is called the Stokes flow equation. It is sometimes called the creeping flow equation, be-
cause the term p[v • Vv], which is quadratic in the velocity, can be discarded when the
flow is extremely slow. For some flows, such as the Hagen-Poiseuille tube flow, the term
p[v • Vv] drops out, and a restriction to slow flow is not implied. The Stokes flow equation
is important in lubrication theory, the study of particle motions in suspension, flow
through porous media, and swimming of microbes. There is a vast literature on this
subject. 3
(iii) When viscous forces are neglected—that is, [V • т] = 0—the equation of motion
becomes
P§y=-Vp + pg (3.5-9)

which is known as the Euler equation for "inviscid" fluids. 4 Of course, there are no truly
"inviscid" fluids, but there are many flows in which the viscous forces are relatively
unimportant. Examples are the flow around airplane wings (except near the solid
boundary), flow of rivers around the upstream surfaces of bridge abutments, some prob-
lems in compressible gas dynamics, and flow of ocean currents. 5

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, The Hague
3
(1983); S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth-
Heinemann, Boston (1991).
L. Euler, Mem. Acad. Sci. Berlin, 11, 217-273, 274-315, 316-361 (1755). The Swiss-born
4
mathematician Leonhard Euler (1707-1783) (pronounced "Oiler") taught in St. Petersburg, Basel, and
Berlin and published extensively in many fields of mathematics and physics.
5
See, for example, D. J. Acheson, Elementary Fluid Mechanics, Clarendon Press, Oxford (1990),
Chapters 3-5; and G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press (1967),
Chapter 6.