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WILLIAMSOXS TREATMENT

thtu ccTTtspondmę to B. At the beginning of th* paper be speaks of the rectangles as " representW powers, but later makes the incorrect statement that power = F x S," where F = shearing stress and S the ratę of shear.¹ The areas on the figurę do not really have the dimensions of a power, unless V itself is plotted against stress; but this i_n. accuracy of dimensions hardly affects the argument. On the part of the curve below the linear portion, say at a point G, a linę Gg is drawn parallel to /B. Drawing in the perpendiculars Ge and gM there are the rectangles GMgć representing the power used to overcome viscous resistance, and XMgO to overcome plastic resistance. If a senes of points G are taken, the M-series of points resulting from this construction will give another dotted curve, OMQ. This curve shows the variation of the shearing stress reąuired to overcome plastic resistance with the change in the ratę of shear. The total power is therefore divisible into two parts (areas on the curve), or FS i F_XS + FzS,

where

p-D    pi?    /y

Fi 1 g (plastic); F₂= — (viscous); andS

¹ Notę that WilHamson uses different symbols from those used in the earlier part of this book.

ąV ,    |    . Length*    i

2— has the dimensions of =-—^——r-. = =■.—.

»R*    Time X Length* Time

PR

— has the dimensions of Mass x Length^-1 x Time^-*.

Hence

4^v y PR i

^x Tl ^{= Mass} I    x Time^-*

Whereas

JL materiał* studiod by Williamson it is For _iU und I so that

ud that the curve OMQ is a sunple hyperbola,

/ s

s + S

where s is a measure of the curvaturc of OMQ ; and f s F_x> for an infinite ratę of shear; and F₂ — rj_mS, where « is apparent viscosity at an infinite ratę of shear. Hence

s -f- S

F- /^S

This is a generał eąuation for the case of shear between parallel planes, moving with a uniform velocity difference. The form can be modified to suit the particular type of viscometer used. Using this eąuation, Williamson performs tests on pigments in linseed oil and dynamite nitrocellulose, and finds good agreement with the eąuation.

Van Nieuwenburg shows the Williamson eąuation to be applicable to a large group of materials.

The eąuation reduces to the Newton (Poiseuille) eąuation when / = O, and to the Bingham, when s = O. The treatment is best used when there is considerable curvature in the lower part of the curve, but when the upper region is sufficiently linear to get an accurate asymptote.

When paint is stirred, its viscosity falls and then rises again on standing. This property is called false body, and is similar to thixotropy, except that the latter term should perhaps strictly apply only to cases of truły reversible gel-sol transformations, in which the “ set ” form should have infinite viscosity and finite modulus, and the sol form should have finite (though not necessarily constant) yiscosity, and no appreciable yield-value. Such