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38    SCHOFIELD‘S TREATMENT

The " o-phenomenon, ’ ’ as it has been called, is mathematically similar to the pług flow already discussed ; but it is of quite a different order, the amount of flow involved being about a hundred tira es as great in the fornier as in the latter.

The Schofield treatment is as follows :—

" If condition (x) of the above be granted, so that the particles are not accelerated, the stress W at the j wali of a tubę of łength L and radius R, to which a j pressure difference P is applied, is PR/2L ; while I the stress S at a point T within the tubę and distant I r from the axis is Pr/zL. Conseąuently r can be expressed in terms of S thus :—

r = R/W.S .    .    .    . . ,₍₆₎ j

If v be the velocity at T, we may write in accotdance I with condition (3)

dv/dr = — /(S).

" Substituting the vałue of r given by Eąuation I (6), and integrating,

j/=R/W.j"7(S)<*S . . . j

if, in accordance with (2), v — o when S = W. The I flow dV, between r and (r + dr) — 2mdr . v. Sub- I stituting for r and v from Eąuations (6) and (7) and I integrating,

From this it is elear that, for any given materiał, I V/trR⁸ should depend on W only if the three eon- I ditions are fulfilled. By making specific assumptions I about the form of /(S), V/ttR³ can be evaluated. I ¹

SCHOFIELD’S TREATMENT

39

Thus, using MaxwelTs assumption that

1§mM.

where fi (the fluidity) is the reciprocal of the riscosity,¹ Eąuation (8) reduces to PoiseuiIle’s equa-tion in the form

V

ttR³

"In the same way, the expression based on the Bingham assumption can be deduced. Here it is supposed that the materiał does not flow unless a stress exceeding a critical value, S₀, be applied to it, and that, at stresses higher than S₀, the velocity gradient eąuals fi(S — S₀). Again fi is a. constant having the dimensions of a reciprocal viscosity, and is usually called the mobility. When such a materiał is forced through a tubę, a central cylinder of radius RS₀/W, within which the stress is less than S₀, moves as a solid pług, and only the materiał outside this cylinder flows. When W is less than S₀, no flow occurs, and V = o. In substituting in Eąuation (8) to obtain the value of V when W exceeds S₀, it must be remembered that, sińce /(S) is discontinuous, heing zero from o to S„, and /x(S — S₀) from S₀ to W, the integrations must be carried out in two stages. This has the effect of splitting V into two terms, thus

i^M^^s-^s;,dS-^dS+

n fW fW

The first is the contribution of the pług, the second

¹ Fluidity is here regarded as a “ special case ” of mobility, namely, the case where Poiseuille’s law is obeyed. The numbers of the eąuations have been chauged to suit the preseut text.

1

Thia ffnnAra l flfiiiation will be considered morę fully later.