Exact solutions to the Navier–Stokes equations describing stratified fluid flows

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Abstract

The paper considers the necessity of constructing exact solutions to the equations of dynamics of a viscous fluid stratified in terms of several physical characteristics, with density and viscosity taken as an example. The application of the families of exact solutions constructed for stratified fluids to modeling various technological processes dealing with moving viscous fluid media is discussed. Based on Lin’s exact solutions, linear in some coordinates, a class of exact solutions to the Navier–Stokes equations is constructed for viscous multilayer media in a mass force field. The class is then extended to the case of the arbitrary relation of kinetic force fields to all three Cartesian coordinates and time. The issues of overdetermination and solvability of the reduced (based on the families under study) Navier–Stokes equation system supplemented by the incompressibility equation are discussed. The case of isobaric shearing flow outside the mass force field is considered in detail as an illustration. Three approaches to obtaining consistency conditions for the overdetermined reduced system of motion equations are discussed, and their interrelation is shown.

About the authors

Natalya V. Burmasheva

Institute of Engineering Science, Urals Branch, Russian Academy of Sciences; Ural Federal University named after the First President of Russia B. N. Yeltsin

Email: nat_burm@mail.ru
ORCID iD: 0000-0003-4711-1894
SPIN-code: 7927-5530
Scopus Author ID: 57193346922
ResearcherId: E-3908-2016
http://www.mathnet.ru/person52636

Cand. Tech. Sci.; Senior Researcher; Sect. of Nonlinear Vortex Hydrodynamics1; Associate Professor; Dept. of Theoretical Mechanics2

34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation; 19, Mira st., Ekaterinburg, 620002, Russian Federation

Evgeniy Yu. Prosviryakov

Institute of Engineering Science, Urals Branch, Russian Academy of Sciences; Ural Federal University named after the First President of Russia B. N. Yeltsin

Author for correspondence.
Email: evgen_pros@mail.ru
ORCID iD: 0000-0002-2349-7801
SPIN-code: 3880-5690
Scopus Author ID: 57189461740
ResearcherId: E-6254-2016
http://www.mathnet.ru/person41426

Dr. Phys. & Math. Sci.; Head of Sector; Sect. of Nonlinear Vortex Hydrodynamics1; Professor; Dept. of Theoretical Mechanics2

34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation; 19, Mira st., Ekaterinburg, 620002, Russian Federation

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