Page 164 - Bird R.B. Transport phenomena

148 Chapter 4 Velocity Distributions with More Than One Independent Variable

Show that the problem can now be restated as follows:
-OR + + Ml-f]
(bob) ^ = -6 R Ц-5Г at t = 0, B = 0 (4C2-11)
—~f=
R
dr z \ d x
л « ,21 I at т = 0 7 ф = 0
v
(fluid) -^ = M— - atx = 0, ф = A(dв /dt) (4С.2-12)
R
dx [at x = 1, ф = A(dO /dt)
aR
From these two equations we want to get 9 and ф as functions of x and r, with M and A as
R
parameters.
(e) Obtain the "sinusoidal steady-state" solution by taking the input function 6 (the dis-
aR
placement of the cup) to be of the form
0a (r) = 0° Me>™\ ( C is real) (4С.2-13)
R
a
in which a> = a)/o) = co\/T/k is a dimensionless frequency. Then postulate that the bob and
0
fluid motions will also be sinusoidal, but with different amplitudes and phases:
17
6 (T) = Ш{в° е ~ } (в° is omplex) (4C.2-14)
c
от
к
к
R
ф(х, г) = д\{ф°(хУ" } (ф°{х) is complex) (4C.2-15)
т
Verify that the amplitude ratio is given by \0° \ /0° , where |- • -| indicates the absolute magni-
aR
R
tude of a complex quantity. Further show that the phase angle a is given by tan a =
3(0°<(/№(0°<), where Ш and 3 stand for the real and imaginary parts, respectively.
(f) Substitute the postulated solutions of (e) into the equations in (d) to obtain equations for
the complex amplitudes B° and ф°.
R
(g) Solve the equation for ф°(х) and verify that
c
dф o А(ш) 3/2 (e° R o s h V i w / M - 6° aR (4C.2-16)
~dx~ VM
(h) Next, solve the 6° equation to obtain
R
OR АМШ
. . (4C.2-17)
VaR 2
(1 - п ) " У ' " + АМш coshVi п/М
/ш/М
from which the amplitude ratio \0° \ /0° and phase shift a can be found.
R
R
(i) For high-viscosity fluids, we can seek a power series by expanding the hyperbolic func-
tions in Eq. 4C.2-17 to get a power series in 1 /M. Show that this leads to
From this, find the amplitude ratio and the phase angle.
2
2
(j) Plot \6° \ /6° versus п for fi/p = 10 cm /s, L = 25 cm, R = 5.5 cm, I = 2500 gm/cm , к = 4 X
aR
R
6
10 dyn cm. Where is the maximum in the curve?
4C.3 Darcy's equation for flow through porous media. For the flow of a fluid through a porous
medium, the equations of continuity and motion may be replaced by
dp
smoothed continuity equation e — = -(V • pv ) (4C.3-1)
0
Darcy's equation 5 v = ~ (Vp - pg) (4C.3-2)
0
Henry Philibert Gaspard Darcy (1803-1858) studied in Paris and became famous for designing
5
the municipal water-supply system in Dijon, the city of his birth. H. Darcy, Les Fontaines Publiques de la
Ville de Dijon, Victor Dalmont, Paris (1856). For further discussions of "Darcy's law/' see J. Happel and
H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nihjoff, Dordrecht (1983); and H. Brenner
and D. A. Edwards, Macrotransport Processes, Butterworth-Heinemann, Boston (1993).