Page 387 - Bird R.B. Transport phenomena

Problems 369

(b) Choose appropriate values of v , v , and y 0 to convert the equations in (a) into Eqs. 11.4-44
z0
yQ
to 46, and show that the definitions in Eqs. 11.4-41 to 43 follow directly.
(c) Why is the choice of variables developed in (b) preferable to that obtained by setting the
three dimensionless groups in Eqs. HB.15-1 and 2 equal to unity?
11C.1. The speed of propagation of sound waves. Sound waves are harmonic compression waves
of very small amplitude traveling through a compressible fluid. The velocity of propagation
of such waves may be estimated by assuming that the momentum flux tensor т and the heat
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flux vector q are zero and that the velocity v of the fluid is small. The neglect of т and q is
equivalent to assuming that the entropy is constant following the motion of a given fluid ele-
ment (see Problem 11 D.I),
(a) Use equilibrium thermodynamics to show that

Л I П I (llC.1-1)
Л
in which у = C /C .
v
p
(b) When sound is being propagated through a fluid, there are slight perturbations in the
pressure, density, and velocity from the rest state: p = p 0 + p', p = p 0 + p', and v = v 0 + v',
the subscript-zero quantities being constants associated with the rest state (with v 0 being
zero), and the primed quantities being very small. Show that when these quantities are substi-
tuted into the equation of continuity and the equation of motion (with the т and g terms omit-
ted) and products of the small primed quantities are omitted, we get
dp
Equation of continuity — = —p (V • v) (11C.1-2)
o
Equation of motion p -^ = -Vp (11C.1-3)
0
(c) Next use the result in (a) to rewrite the equation of motion as

in which v\ = у(др/др) .
т
(d) Show how Eqs. И С1-2 and 4 can be combined to give
2
Ц = vlV p (llC.1-5)
dt 1
(e) Show that a solution of Eq. 11 C.I-5 is

p = p 0 1 + A sin ( ^ (z - v t)) (llC.1-6)
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L \ л /J
This solution represents a harmonic wave of wavelength Л and amplitude PQA traveling in the
z direction at a speed v . More general solutions may be constructed by a superposition of
s
waves of different wavelengths and directions.
11C.2. Free convection in a slot. A fluid of constant viscosity, with density given by Eq. 11.3-1, is
confined in a rectangular slot. The slot has vertical walls at x = ±B, у = ± W, and a top and
bottom at z = ±H, with H » W » B. The walls are nonisothermal, with temperature dis-
tribution T = T + Ay, so that the fluid circulates by free convection. The velocity profiles
w
are to be predicted, for steady laminar flow conditions and small deviations from the mean
density p.

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See L. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd edition, Pergamon, Oxford (1987), Chapter
VIII; R. J. Silbey and R. A. Alberty, Physical Chemistry, 3rd edition, Wiley, New York (2001), §17.4.