str (101)

GE X ERA.

FLOW

J ^E<?cr.4_rj0L

iq>6. tiret ąuoted the eąuation i_n | _{

paper, the calculation having been Dryden ; but the authorehip of the enJSS J generally ascribed to Farrow, Lowe aJJ&⁰*¹ is

equation m ąuestion is

(w ~r 3)^

8®

I

PRy |

LJ

%

where n is the Ostwald exponent, and p rep_re mobility, but, having dimensions which depe^^¹ is best written with a | bar.” .    ¹¹¹‘, I

This eąuation, of course, reduces to the p₀i_se eąuation when n — i. It is useful in practi because, when, for different capillaries, log| is plotted against log(PR/2L), the slope gives n, and I is given by

I =: log^-1 [log(« + 3) ~ Wt]

where i is the intercept of the log-log curve on the stTess axis. In generał, regarding the capillary tubę as a special case, we can wnte

IB log^-1 (log | — I log S),

where i is the ratę of shear, and S the shearing stress.

The dimensional difficulty can be overcome, though admittedly only by a mathematical trick, by taking fjl — — log 1 as the criterion. This is in eftect the log of the viscosity, when the stress is exceedingly smali (i dyne/cm.²), and is, of course, a pure number. The objection to this method is that, in practice, the most useful viscosity is not that found at a very Iow stress, but that which obtains at the kind of stress at włńch the materiał will be used. This objection can be met by ąuoting a true viscosity at some arbitrarily chosen stress. Since viscosities of true fluids all have to be ąuoted at an

GENERAL FLOW EQUAT10N 79

rbitrary temperaturę there is good precedent for

^The^p^ysical significance of this relationship betweenvelocity gradient and shearing stress is not _eaSy to appreciate. Some idea of the processes involved will emerge when the work of Eisenschitz is considered in a later chapter.

The Poiseuille, the Buckingham-Reiner, and the Farrow eąuations are clearly special cases of the generał eąuation relating ratę of shear to stress in a capillary, when the functional relationship between velocity gradient and stress is kept generał. This generał eąuation appears in a footnote in a paper by Rabinowitsch; but it was Mooney who first dis-cussed its importance.¹ Schofield points out that this generał eąuation is no morę than an algebraic simplification of the Schofield-Scott Blair eąuation, which has already been ąuoted (p. 38).

V 2 r^w rw

⁼ w4o^SI/^(S^'^dS>

where W is the stress at the wali, and S the stress at any point (T). This may be written

differentiating

mammm

Schofield gives very interesting grapłucal treat-ments to demonstrate the conditions imder which a

1

The eqnation has been used by others as weli, e.g., Porter and Rao.