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10

POISEUILLE-IIAGEN LAW

The " figurę of extrusion " is a paraboloid, and its cquation is easily derived. Consider it as being madę up of a series of very smali strips, like the concentric tubes of a telescope. If the thickness of each tubę is dr, the volume will be, as a close approximation, 2tn (the periphery) x height (which is v) X dr, and the total volume of the figurę will be:—

f^R

V = | 2irt .v .dr,

Jo

where V is the volume extruded per second.

Or, from Equation (3), sińce

v = P(R2 - y²)/4ijL,

we have

V = P7r/2Lł;f (R²y — r^)dr Jo

= P7rR⁴/8L^.......w

This is the Poiseuille-Hagen eąuation. To derive n in absolute units, P is quoted in dynes/cm.², i.e,, pressure in cm. Hg x density of Mercury x 981.

In order to derive this, several assumptions have been madę:—

(1)    That the fluid flows in a telescopic manner and that there are no vortices. Later it will be shown that this is only true for certain conditions, and below certain rates of flow.

(2)    No correction has been madę to allow for what takes place when the fluid is squeezed into the tubę from the Container in which it is stored. There are two corrections involved here. First there is the end-effect correction, which allows for the work done to deform the materiał round the shoulder of the tubę, i.e., that work which would be needed to

extrude materiał through an orifice of radius R, I the tubę were cut off at “a” (Fig. 3). This is generally quite a smali correction, and can usually be neglected if a capillary of reasonable length and radius is used. Any correction necessary is usually

Fig. 3.

madę by adding a hypothetical amount to the value for the length of the tubę, this amount being taken as proportional to the radius of the tubę. This expression was worked out by Couette.

Secondly there is the kinetic energy correction, which is morę important.¹ When the materiał which is moving slowly along the wide tubę suddenly finds itself in the capillary, it has to move much morę quickly, and for this it requires additional kinetic energy. Some of the pressure applied is used up in this, instead of its all being employed in shearing the materiał. Bernouilli calculates that if | is the density and z the vertical co-ordinate of gravity, along a simple streamline,

§|| 1 gz ii = constant ... (5) When the velocity H suddenly increases, the effective pressure falls. Hagenbach (1860) was the first person to. deduce an expression to correct Poiseuille’s law for this effect. His expression is

p = _PV²lj/2 . ir²R^ł,

i See also Dorsey and Stone.