ABSTRACT
Thus, given the initial density p(x, t = 0) = po(x) one has to find the density of particles p(x, t) at any moment of time for an arbitrary value of x (the velocity of motion u(t) is given). Using the law of conservation of mass, by counting the balance of substance in a small element of the pipe from x up to x + dx during dt (Fig. 21). From the left, the mass entering the elementary volume is
where Su(t) dt is the volume of matter introduced during time interval dt. At the same time from the right cross-section the outflown mass equals
i.e. the resulting change in mass is
By virtue of the small size of dt, the velocity u(t) is considered as constant. Quantities p(x, t -I-£ dt) and p(x + dx , t + £ dt) are the time averages of the density at cross-sections x and x + dx.
