On the Semi-Analytical Solutions in Hydrodynamics of Ideal Fluid Flows Governed by Large-Scale Coherent Structures of Spiral-Type

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.

Note on semi‐analytical nonstationary solution for the rivulet flows of non‐Newtonian fluids

“…Also, some remarkable articles should be cited, which concern the problem under consideration. [15][16][17][18][19][20][21][22][23][24] In Ershkov et al, 22 the inverse approximation of creeping flows considered herein was used, i.e., disregarding the terms, associated with viscous forces, with respect to the convective terms in momentum equations of fluid flows (the same approximation was used by one of us in recent researches 23,24 ).…”

Flow of a Viscous Incompressible Fluid from a Moving Point Source

“…The obtained solutions are governed by hydrodynamical fields of flows which can be recognized as special invariants at symmetry reduction. The final state of such maximal symmetry reduction can be presented as Hopf bifurcation of zeroth order which can be considered to be a prototype for the study of a dynamical chaos for the trajectories of non-stationary solutions [14][15][16][17].…”