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Abstract
We have presented here a clearly formulated algorithm or semi-analytical solving procedure for obtaining or tracing approximate hydrodynamical fields of flows (and thus, videlicet, their trajectories) for ideal incompressible fluids governed by external large-scale coherent structures of spiral-type, which can be recognized as special invariant at symmetry reduction. Examples of such structures are widely presented in nature in “wind-water-coastline” interactions during a long-time period. Our suggested mathematical approach has obvious practical meaning as tracing process of formation of the paths or trajectories for material flows of fallout descending near ocean coastlines which are forming its geometry or bottom surface of the ocean. In our presentation, we explore (as first approximation) the case of non-stationary flows of Euler equations for incompressible fluids, which should conserve the Bernoulli-function as being invariant for the aforementioned system. The current research assumes approximated solution (with numerical findings), which stems from presenting the Euler equations in a special form with a partial type of approximated components of vortex field in a fluid. Conditions and restrictions for the existence of the 2D and 3D non-stationary solutions of the aforementioned type have been formulated as well.
Highlights
Faculty of Mathematics, Physics and Information Technologies, Odessa I
We will consider here and below, the asymptotics of non-stationary three-dimensional solutions of non-viscous rotational flow, as below: 1 →2 ( u ) + p + φ = const q γ 2 ( x, y) − σ 2 (t), U (z, t) = σ (t cos( A(z, t) t), V (z, t) = σ (t) sin( A(z, t) t), W ( x, y, t) =
The last but not least, we should especially note that we have presented here the appropriate solving procedure for modelling the semi-analytical solutions for ideal fluid flows driven by external large-scale coherent structures of but in cases for which we do not choose to consider any of
Summary
The second of the equations of system (3) (with the additional demand (4)) was proved to have the analytical way for presenting its non-stationary solution [7,8,9] as follows:. − 1 , where ξ = ξ (x, y, z) is some arbitrary function, given by the initial conditions; the realvalued coefficients a(x, y, z, t), b(x, y, z, t) are solutions of the mutual system of two Riccati ordinary differential equations with respect to time t [7,8,9] We should especially note that the system of Equation (9) is the system of two non-linear ordinary differential equations of the 1st order, which could be solved by numerical methods only
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