The rapid expansion of numbers of fluid flows irreducible mathematical models has necessitated supplementing the principles Aristotelian-Occam-Leibniz logic with the requirement for definability of the subject, causality and completeness. All fluids are heterogeneous, characterized by non-uniform distributions of substances, pressure, and temperature, stratified in acting fields, and non-stationary. A fluid medium is discrete and characterized by geometric, hydrodynamic, and physical parameters. Due to the atomic and molecular properties inside and on boundaries of liquids and gases, various aggregates of physical and chemical nature of different sizes from nano- to micron scales are continuously formed and decay eliminating their boundaries. Conversion processes accompany this structural reorganization include the capture of thermal and mechanical energy during cell formation and the release of potential surface energy during cell destruction. Continuously rebuilding flow structures outlined by sharp boundaries are visualized in scales extending from light years in space to fractions of a millimeter in the laboratory.

Fluid flow is defined as the transfer of continuous momentum, energy, and matter, causing changes in the medium parameters. The parametrically and scale-invariant system of equations for the transport of density, momentum, total energy, and matter is selected to describe the dynamics and structure of liquid or gas flows. This system is closed by equations of state for the Gibbs potential and density and supplemented by initial and boundary conditions. The analysis of motion flows in a stratified medium, which is carried out taking into account the compatibility condition, involves calculating fluid state in the absence of external forcing. In a stratified medium, thin flow, known as « diffusion induced flows on topography, » form in a gravitational field near inclined impermeable boundaries of fixed and moving bodies.

Infinitesimal periodic flows are calculated using unified perturbation theory, immersing the problem in the algebra of complex numbers. When transitioning to complex numbers, frequency remains real and positive definite, and the wave number supposes to be complex. Analysis revealed that regular solutions of the dispersion relations describe known waves – gravity waves at interfaces and internal waves, capillary, inertial, acoustic, and hybrid ones. Families of singular solutions characterize sets of ligaments, which correspond to high-gradient interfaces and fibers in the flow patterns. These flow components decay at different rates with increasing distance from the source under the influence of dissipative factors.

The general properties of the solutions illustrate comparisons with schlieren patterns of diffusion induced flows, as well as wakes, vortices and internal waves generated by oscillating or moving bodies (plate, cylinder, sphere) and convective flows in a laboratory tank, as well as with observation of individual phenomena in natural conditions – in the ocean and atmosphere.

Yuli Chashechkin (IPMech RAS, Moscow)

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