N. V. Burmasheva, E. Yu. Prosviryakov
EXACT SOLUTIONS OF THE NAVIER–STOKES EQUATIONS FOR DESCRIBING AN ISOBARIC ONEDIRECTIONAL VERTICAL VORTEX FLOW OF A FLUID
DOI: 10.17804/24109908.2021.2.030051 The article proposes a family of exact solutions to the Navier–Stokes equations for describing isobaric inhomogeneous unidirectional fluid motions. Due to the incompressibility equation, the velocity of the inhomogeneous Couette flow depends on two coordinates and time. The expression for the velocity field has a wide functional arbitrariness. This exact solution is obtained by the method of separation of variables, and both algebraic operations (additivity and multiplicativity) are used to substantiate the importance of modifying the classical Couette flow. The article contains extensive bibliographic information that makes it possible to trace a change in the exact Couette solution for various areas of the hydrodynamics of a Newtonian incompressible fluid. The fluid flow is described by a polynomial depending on one variable (horizontal coordinate). The coefficients of the polynomial functionally depend on the second (vertical) coordinate and time; they are determined by a chain of the simplest homogeneous and inhomogeneous partial differential parabolictype equations. The chain of equations is obtained by the method of undetermined coefficients after substituting the exact solution into the Navier–Stokes equation. An algorithm for integrating a system of ordinary differential equations for studying the steady motion of a viscous fluid is presented. In this case, all the functions defining velocity are polynomials. It is shown that the topology of the vorticity vector and shear stresses has a complex structure even without convective mixing (creeping flow).
Keywords: exact solution, Couette flow, Navier–Stokes equation, inhomogeneous unidirectional flow, method of separation of variables, shear stress References:
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Article reference
Burmasheva N. V., Prosviryakov E. Yu. Exact Solutions of the Navier–stokes Equations for Describing An Isobaric OneDirectional Vertical Vortex Flow of a Fluid // Diagnostics, Resource and Mechanics of materials and structures. 
2021.  Iss. 2.  P. 3051.  DOI: 10.17804/24109908.2021.2.030051. 
URL: http://eng.dreamjournal.org/issues/20212/20212_316.html (accessed: 04/14/2024).
