SELF

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S.B. Karavashkin

70

As the volume is limited by equipotential surfaces, we can state that for any force line within the volume

where L means some arbitrary force line within the selected volume.

The phase shift between the flux of vector (, t) that enters and leaves the selected volume we will introduce through the time variable.

We have to account the phase shift in this form because the values of specific flux at the equipotential boundaries of selected volume correspond to the source power at different moments of time. So, if we disregard the phase time shift, the above values, with the time-stationary selected volume, will not reveal the pattern of mass variation in the whole continuum related to the time-varying source power. But the account of shift over the spatial independent variables can characterise the process in the flow only from the time-stationary source, as it determines the force lines density transformation in the selected volume resulting from the interaction of internal and external forces of continuum in scattering the flow in the spatial continuum.

Due to the said, the phase shift introduced through the time variable will not mean the non-simultaneity of flux variation through the surfaces of selected volume. It will account the variation of power of the source of continuum during its passing the instantaneous value of flux of vector along the force lines limited by equipotential surfaces of that continuum, and so in Fig. 1 we introduce t with the minus.

71

Determine the divergence (, t) in constriction of the selected volume into a point, basing, as we did it in [1], on the standard representation

In accordance with the statement of problem, the full surface S consists of three components:

where S₁ and S₂ are the butt surfaces of the selected volume and S_l is the lateral surface of the selected volume. With (6), the condition at which the lateral surface coincided with the force lines will be the following: 

where

72

The first integral of the right-hand sum of (7) does not contain the phase shift t; in absence of sources within the selected volume, it turns to zero, while the second integral is non-zero and can be easily transformed into the integral over the volume. For it, let us divide the selected volume in equipotential surfaces into n volumes (n >> 1) . In each of these little volumes, we can take v_k = const, 1 k n with the accuracy to the second order of smallness. With it we can present (4) with enough accuracy as

where

(Here and further _k , _l   determine the smallness of value, not variation). Noting (8),