The second equation of continuity

Ya. G. Kluyshin,

St. Petersburg Academy of Civil Aviaion, faculty of applied mathematics
5-3-241, Budapeshtskaya str., 192242, St. Petersburg, Russia
Telephone: 7-812-174-88-48, e-mail: Klyushin@shaping.org

Abstract. An equation generalizing the classical continuity equation for the case of accelerated movement is proposed.

Let v be liquid flow velocity and r be its density. The velocity of the liquid leaking through a surface s

, (1)

where vn is v projection on external normal u , to s. On the other hand the velocity of liquid changing in the volume with surface s consists of

.

Here and below the lower index t means private derivative with respect to t. One gets from here with the help of Gauss theorem:

, (2)

for any volume u , or

, (3)

which is the classical continuity equation of the flow that is accelerated. Then the second total derivative with respect to t in (1) will be also non zero. One has:

(4)

On the other hand acceleration, with which density r changes in volume u , is

. (5)

i.e

. (6)

for any u , i.e.

(7)

of the flow is steady, i.e r tt = 0, vt = 0, one can easily verify that (7) comes to (3). One the whole these both equations should be valid simultaneously and (3) can be used to simplify (7). One gets

(8)

(3) and (8) must be valid simultaneous for accelerated processes. (8) becomes identify for nonaccelerated processes. Both (3) and (8) are kinematics facts and are independent with respect to any assumptions except assumption that there is no sources of liquid inside the volume under consideration. Just analogous correlation could be received for higher rank derivatives if necessary.