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Transformation of continuity equation in nonlinear models

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Taking in (9) the limit  t 0 , _kl 0 , we yield

Substituting (10) into (7), yield 

and, as v = const along the equipotential lines,

Returning to (5), substitute to it the yielded value of right-hand integral (11). Noting that the second integral in (11) in constriction of the selected volume has an order of smallness higher that that of the differential over V, yield

or

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The differential equation (12) differs from the known equation that describes the flux of vector conservation in the selected volume by the summand

which takes into account the time delay of propagation of flux of continuum in change of the source productivity.

If now we substitute into (12) the value (, t) of (1), we will yield

Proceeding from the condition of potential flow, we can state that on the surface S₂ of the selected flow

Hence, (13) will take the following appearance:

Since after the statement of problem v depends on the time mediately the continuum density at the studied point of volume,

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and (15) is transformed to the appearance  

The expression (17) differs from the known continuity equation by the term 

which in case of small excitations (v = const) vanishes, returning (17) to the conventional equation. In the domain of nonlinear regularity we must not neglect the term (18).

Thus, we have proven that in transition to the domain of nonlinear characteristics of processes in continuum, the continuity equation does not remain its appearance, as it was thought before, but transforms with the type of nonlinearity of the studied process. And the most general form of continuity equation will differ also from the yielded expression, as this last accounts only the nonlinearity of parameter v with linearity of the other parameters and potentiality of the continuum flow.

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Reference:

  1. Karavashkin, S.B. On longitudinal electromagnetic waves. Chapter 1. Lifting the bans. SELF Transactions, 1 (1994), 15- 48.