Scientific Papers - Vi

506 ON THE PRESSURE DEVELOPED DURING THE
The following values of P/Q are calculated from (8):
[423
z -P/Q 1 !
T^ 6-9147 1 arbitrary
TffiT 4-6517 ! 2 ' 0-6931
1\> 2-5584 4 0-4621
4 1-8484 10 ! 0-2558
| 1-3863 100 0-0465
1 arbitrary 1000 ' • 0-0069
Reverting to the case where the. pressure inside the cavity is zero, or at any rate constant, we may proceed to calculate the pressure at any internal point. The general equation of pressure is
1 dp __ Du _ du du o dr Dt dt dr'
.(9)
u being a function of r and t, reckoned positive in the direction of increasing r. As in (!),« = UR2/r2, and
du 1 d
Also
and by (4)
so that
dt
dt
dU dt
' dt
l?l p R*'
P
P
dt
Thus, suitably determining the constant of integration, we get
At the first moment after release, when E= R0, we have
(11)
When r = R, that is on the boundary, p = 0, whatever R may be, in accordance with assumptions already made.
Initially the maximum p is at infinity, but as the contraction proceeds, this ceases to be true. If we introduce z to represent RQS/R3, (10) may be written
J^ _ 1 _ _ /2 _. 4,\ _ /,, __ -i \ /ION
P 3r^ ' 3ra^ '' .....................^ '
7 / 7-1
and
_ A2 *)to a permanent gas obedient to Boyle's law. Then, if the initial pressure be Q, the work of compression is 4>7rQRo8 log (R0/R), which is to be subtracted from (3). Hence