Fractal dimensions in fluid dynamics and their effects on the Rayleigh problem, the Burger's Vortex and the Kelvin–Helmholtz instability

Abstract

In this article, we construct fluid equations in fractal dimensions based on the concept of the product-like fractal measure introduced by Li and Ostoja-Starzewski in their formulation of anisotropic media. Three main problems were discussed and analyzed: the Rayleigh problem, the steady Burger's vortex and the Kelvin–Helmholtz instability. This study confirms the importance of fractal dimensions in fluid mechanics where several mechanisms were revealed. In particular, long-tails took place in Rayleigh problem analogous to those arising in several superdiffusion processes. Besides, the analysis of the steady Burger's vortex has proved that the flux rate of dissipation of energy per unit length of vortex depends on the viscosity of the fluid and it is finite for insignificant viscous effects, a scenario which is detected in small-scale turbulent fluid flows. Moreover, it was proved that both the Rayleigh–Taylor and the Kelvin–Helmholtz instabilities are affected by the fractal dimensions and that, for a particular value of the characteristic length as a function of the wavelength, the Kelvin–Helmholtz instability may be suppressed, a particular scenario which is observed in compressible fluid flows and other physical engineering processes.