224

FLUIDITY AND PLASTICITY

where v is the velocity parallel to the axis at the radius r. The
speed of the material in the variable region is obtained by
integrating Eq. (80) from r = R to r = r or

= JU

41

'P(
4l(

-IB

-4

(81)

The speed of the solid plug is obtained by making r = r0 in
Eq. (81), and is after simplifying

7/2 \

(82)

-r

2-nrvdr

The volume of flow per unit of time is V/t and
V
't

or using Eqs. (81) and (82)
V

7 =

But from Eqs. (78) and (82)

(83)

and from Eq. (81)
2r f rardr = 2^ f ^
°

- r3) - /(fir

- r2)

and introducing the value of r0 from Eq. (78), we have

- -

T ~2P P2 3 P3 ' o

Introducing these values of the separate terms of Eq. (83) and
simplifying, Eq. (83) becomes

V _, (WP R3f,2Pf\
J -**(-% --$-+&»)

or

J_
3P3\R

-t 81 L 3 \Rf
and now introducing the value of p given by Eq. (77),

Y. — ^^ / p __ ^ _±. P 4 \
._ . ._ ^ _ . p ± _^.

(86)