Page 113 - Bird R.B. Transport phenomena

98 Chapter 3 The Equations of Change for Isothermal Systems

In these dimensionless equations, the four scale factors / v , p, and /x appear in one dimen-
0/
0
sionless group. The reciprocal of this group is named after a famous fluid dynamicist 3
Ц
Re = | = Reynolds number (3.7-10)

The magnitude of this dimensionless group gives an indication of the relative impor-
tance of inertial and viscous forces in the fluid system.
From the two forms of the equation of motion given in Eq. 3.7-9, we can gain some
perspective on the special forms of the Navier-Stokes equation given in §3.5. Equation
3.7-9a gives the Euler equation of Eq. 3.5-9 when Re —> °° and Eq. 3.7-9b gives the creep-
ing flow equation of Eq. 3.5-8 when Re < < 1. The regions of applicability of these and
other asymptotic forms of the equation of motion are considered further in §§4.3 and 4.4.
Additional dimensionless groups may arise in the initial and boundary conditions;
two that appear in problems with fluid-fluid interfaces are

Fr = к-н = Froude number (3.7-11) 4

We = = Weber number (3.7-12) 5

The first of these contains the gravitational acceleration g, and the second contains the in-
terfacial tension cr, which may enter into the boundary conditions, as described in Prob-
lem 3C.5. Still other groups may appear, such as ratios of lengths in the flow system (for
example, the ratio of tube diameter to the diameter of the hole in an orifice meter).

EXAMPLE 3.7-1 The flow of an incompressible Newtonian fluid past a circular cylinder is to be studied exper-
imentally. We want to know how the flow patterns and pressure distribution depend on the
Transverse Flow cylinder diameter, length, the approach velocity, and the fluid density and viscosity. Show
around a Circular how to organize the work so that the number of experiments needed will be minimized.
Cylinder 6
SOLUTION
For the analysis we consider an idealized flow system: a cylinder of diameter D and length L,
submerged in an otherwise unbounded fluid of constant density and viscosity. Initially the
fluid and the cylinder are both at rest. At time t = 0, the cylinder is abruptly made to move
with velocity v x in the negative x direction. The subsequent fluid motion is analyzed by using
coordinates fixed in the cylinder axis as shown in Fig. 3.7-1.
The differential equations describing the flow are the equation of continuity (Eq. 3.7-1)
and the equation of motion (Eq. 3.7-2). The initial condition for t = 0 is:
I.C. ifx 2 + y > ^ D 2 or if \z\ >{L, (3.7-13)
2
The boundary conditions for t > 0 and all z are:
B.C. 1 as x 2 + y 2 + z 2 v—> о, (3.7-14)
B.C. 2 if x 2 + y 2 < ^C v = 0 (3.7-15)
B.C. 3 as x —> - oo at у = О, (3.7-16)

3
See fn. 1 in §2.2.
4 William Froude (1810-1879) (rhymes with "food") studied at Oxford and worked as a civil
engineer concerned with railways and steamships. The Froude number is sometimes defined as the
square root of the group given in Eq. 3.7-11.
5 Moritz Weber (1871-1951) (pronounced "Vayber") was a professor of naval architecture in Berlin;
another dimensionless group involving the surface tension in the capillan/ number, defined as Ca = \^ь^/а\
6 This example is adapted from R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures
on Physics, Vol. II, Addison-Wesley, Reading, Mass. (1964), §41-4.