Hierarchy of the models of classical mechanics of inhomogeneous fluids

Abstract

The methods of perturbation theory and integral representations are used to analyze the general properties of a system of equations of the mechanics of inhomogeneous fluids including the equations of momentum, mass, and temperature transfer. We also consider various submodels of this system, including the reduced systems in which some kinetic coefficients are equal to zero and degenerate systems in which the variations of density or some other variables are neglected. We analyze both regularly perturbed and singularly perturbed solutions of the system. In the case of reduction or degeneration of solutions, the order of the system decreases. In this case, regularly perturbed solutions are preserved (with certain modifications) but the number of singularly perturbed components participating in the formation of the boundary layers on contact surfaces and their analogs in the bulk of the fluid, i.e., the elongated high-gradient interlayers, decreases. The interaction between all components of the currents is nonlinear, despite the fact that their characteristic scales are different.Together with the experimental and numerical methods, the analytic methods remain one of the basic tools in the investigation of the nature of currents in fluids. In the course of their development, the information variables capable of the reliable characterization of the physical properties of the media and the parameters of currents were selected and the fundamental equations aimed at the description of the mechanics and thermodynamics of fluids were deduced [1, 2]. However, the analysis of the behavior of the entire system and the properties of separate equations, as well as the construction of partial solutions encounter serious difficulties due to the presence of multiscale processes and the nonlinearity of equations and the corresponding boundary and initial conditions. Numerous important results in the theory of slow (as compared with the sound velocity) currents in low-viscous weakly stratified fluids were obtained by the methods of perturbation theory [1,2].Parallel with the fundamental equations, the researchers extensively use constitutive models (various versions of turbulence theory in the hydroaerodynamics of the environment [3] and the theories of boundary layer in the engineering hydromechanics [4]) whose symmetry differs from the symmetry of the fundamental equations [5]. The fact that the constitutive models are not closed stimulated the development of more detailed investigations of the fundamental system of equations and its subsystems. In [6], the analysis of the mechanisms of adaptation of physical fields to rapidly varying external conditions is performed under the assumption of existence of stationary dynamic states of inhomogeneous rotating fluids including the state of rest. The transient wave processes are analyzed in the linear approximation, and the effect of dissipative factors (viscosity, thermal diffusivity, and diffusion) is neglected [6].

Stratified Flow Fine Structure Around a Sloping Plate

“…In the present paper, computations of multiscale flows around a uniformly moving sloping plate are analyzed in a wide range of Reynolds numbers using numerical solution of the fundamental system of fluid mechanics equations, which allows studying flows of both continuously stratified and homogeneous viscous incompressible fluids in a single formulation. These studies complement the previous works performed in the same mathematical formulation for stratified flows around a motionless [4][5][6] and a uniformly moving horizontal strip in the linear [7] and complete nonlinear formulations [8] taking into account the solvability conditions of the system [9,10].…”

“…Generation of new elements with their own kinematics and spatiotemporal scales is due in the dynamic description to a high order and non-linearity of the fundamental system of equations [10]. Complete solutions of the system even in the linear approximation contain several functions [7] which in the non-linear models correspond to flow components interacting with each other and generating new types of perturbations [3,8].…”

“…In the case of potentially homogeneous fluid density variations are so small that cannot be registered by existing technical instruments but the original mathematical formulation with the equations set (1) is retained. In the case of actually homogeneous medium, the fundamental system of equations is degenerated on the singular components [10]. Physically valid initial and boundary conditions are no-slip and no-flux on the surface of the obstacle for velocity components and total salinity respectively, and vanishing of all perturbations at infinity, ( )…”

“…Numerical solution of the fundamental system of equations with satisfaction of the requirements for resolution of all the intrinsic scales without additional hypotheses and relations is the physically reasonable method of analysis which is invariant by values of Reynolds number and consistent with experimental and analytical studies. Classification of stable flow structural components was performed through analysis of physical properties and own scales of the solutions of the linearized system [10]. Numerical solution of the system (1) with the boundary conditions (2) was constructed using new solver stratifiedFoam developed by the author within the frame of the open source computational package OpenFOAM based on the finite volume method [11].…”

Differential Fluid Mechanics—Harmonization of Analytical, Numerical and Laboratory Models of Flows

“…(17). In limiting case of homogeneous fluid both singular solutions become identical, so the problem becomes degenerated and ill-posed [22].…”