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₃o BUCKINGHAM-REINER TREATMENT

Eąuation (2), p. 8), and Pr/2L will first exceed S„ where r is at its maximum (i.e., at the wali) ; pand as the stress rises, the region in which thiss. has taken place, and in which there is flow, will iiacrease in volume. At pressures above 2LS_Q/R, How will occur in a sheath near the wali, the region near the centre, in which Pr/2L is still less than hioving

~ ------ -T"

r|

fi

Fig. 6.

forward as a solid ping The extrusion figurę is shown in Fig. 6.

These conditions were taken into account by Buckingham and Reiner, and by means of a suitably modified Poiseuille integration, they derived^E the equation :—

ttR⁴

8LV

1

$

where p is the pressure corresponding to S₀, so that S„ = ^R/2L.

This integration will be followed, in a^modified form, later in this chapter.

When P is fairly large compared with p, and the last term can therefore be neglected, Equation (2) may be written in this form :—

¹ The fuli integration is given in Bingham's book.

The curve for such an eąuation would be of the generał form of that shown for the Bingham system in Fig. 4, but p the extrapolated intercept, would

be equal to %l, a being the true intercept. Part of

the curvature, which i? found with so many materials at Iow pressures, is thus explained.

Reiner did not immediately carry his work on these lines any further, but Buckingham in doing so found that at very Iow stresses, when the whole materiał is at a stress below S₀, it sometimes slides through the tubę as a solid pług. If this pług is regarded as being lubricated by a thin, fluid envelope of mean thickness e, the velocity of this pług would be given byli—

v_p = <f>. S . e, where 4 is the fluidity of the water envelope,

or

hence

Vp = tf>. PRe/2L

V = 4. PR³t7£/2L.....(4)

Green showed that this " pług ” type of flow really exists; but it was Scott Blair and Crowther who first tested the eąuation on actual materials. For clay and soil pastes, it was found that it was necessary to introduce only one further complication in order to explain the data over a fairly wide rangę. The water envelope had hitherto been assumed to be a true fluid; but actually the water is only partly free, as it is partly bound to the glass, and to the clay particles. It does not, therefore, take the form of a simple envelope, and is not truły fluid. If a yield-value for this envelope is introduced, the whole eąuation may be written