Page 111 - Bird R.B. Transport phenomena

96 Chapter 3 The Equations of Change for Isothermal Systems

This is an "equidimensional equation/' which may be solved by assuming a trial solution
/ = r" (see Eq. C.l-14). Substitution of this trial solution into Eq. 3.6-51 gives ;/ = 1, -2. The so-
lution of Eq. 3.6-51 is then
/=С,г + ^ (3.6-52)

so that
v^r, в) = • — } sin I (3.6-53)

3
Application of the boundary conditions shows that Cj = 0 and C = (1R . Therefore the final
2
expression for the velocity distribution is
v sin ( (3.6-54)
Next we evaluate the torque needed to maintain the rotation of the sphere. This will be the in-
tegral, over the sphere surface, of the tangential force (т \ )Я sin в(1в(1ф exerted on the fluid
2
гф г=к
by a solid surface element, multiplied by the lever arm R sin в for that element:
7\ = \ llT V (r )\ r=R (R sin $)R sin вйвйф
2
r(l
Jo Jo
Г2тт Гтт
2
(3/хП sin 6)(R sin 0)R sin вйвйф
Jo Jo
si
Г sin ed6
3
Jo
(3.6-55)
In going from the first to the second line, we have used Table B.I, and in going from the sec-
ond to the third line we have done the integration over the range of the ф variable. The inte-
gral in the third line is |.
As the angular velocity increases, deviations from the "primary flow" of Eq. 3.6-54 occur.
Because of the centrifugal force effects, the fluid is pulled in toward the poles of the sphere
and shoved outward from the equator as shown in Fig. 3.6-8. To describe this "secondary
flow/' one has to include the [v • Vv] term in the equation of motion. This can be done by the
use of a stream-function method. 6

Fig. 3.6-8. Rough sketch showing the secondary flow
which appears around a rotating sphere as the Reynolds
number is increased.

6 See, for example, the development by O. Hassager in R. B. Bird, R. C. Armstrong, and O. Hassager,
Dynamics of Polymeric Liquids, Vol. 1., Wiley-Interscience, New York, 2nd edition (1987), pp. 31-33. See also
L. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 2nd edition (1987), p. 65; and L. G. Leal,
Laminar Flow and Convective Transport Processes, Butterworth-Heinemann, Boston (1992), pp. 180-181.