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

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.