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Transformation of continuity equation in nonlinear models

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Transformation of continuity equation in nonlinear models of potential flows of continua

S. B. Karavashkin

Special Laboratory for Fundamental Elaboration SELF

e-mail: selftrans@yandex.ru , selflab@mail.ru

We will prove the theorem of divergence of vector for deformed continua

Keywords: mathematical physics, wave physics, vector algebra, mechanics of deformed continua, divergence of vector

Classification by MCS 2000: 76A02, 76A10, 76D33

Classification by PASC 2001: 47.10.+g, 47.15.-x, 47.20.Ky, 47.35.+i

In [1], where we developed the basic equations of EM field theory, we also studied the behaviour of flux of vector in some fixed continuum of field. With it for the region of dynamic processes in the flow, we have improved the known constructions based on conventional statement of divergence problem. In [1] we applied this result immediately to the conditions and postulates of field theory; in this connection in the course of consideration we made some simplifications as to linearity of field parameters. But the problem stated completely and solved with account of the nonlinearity of continuum parameters has a wider application and allows yielding considerable changes in the known regularities.

  Consider some potential flow of deformable continuum produced by the source alternating in power, and let the specific flux of vector of this potential flow varies as

To reduce the problem to the mass conservation law, let us assume

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where Q = dm/dt  is the source productivity and T is the period of full cycle of source productivity variation. In other words, if averaging over the period, the source does not change the mass of material continuum.

Furthermore, it follows from the statement of problem that, as the flow is potential, the difference in momentary density of flow along the equipotential lines is absent, i.e.

where 

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Before we start to construct the solution, we would like to mention the following. Despite many limitations in the statement of problem, this model can be applied to different and often-used variants of schemes containing continuous systems of stationary point spherical either cylindrical sources of variable power in the infinite material space with distributed sinks at the infinity.

Fig. 1. General shape of the power tube of time-variable flow

To solve the problem, consider some connective source-free volume of continuum limited from the sides by the surface coinciding with the force lines of current and from the ends - by the equiphase surfaces (see Fig. 1). In distinct from [1], we may not express directly the length of force lines of the selected volume with respect to the time shift of vector flux (1) at the boundaries of selected volume, because of dependence (). None the less, because of flow potentiality we may state that on any potential surface within the selected volume