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We present a model for the relative velocity of inertial particles in turbulent flows. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the generalized acceleration and generalized…
We study a class of processes that are akin to the Wright-Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the…
The theory describing the evolution of inhomogeneous vortex tangle at zero temperature is developed on the bases of kinetics of merging and splitting vortex loops. Vortex loops composing the vortex tangle can move as a whole with some drift…
We present an extended version of an invited talk given on the International Conference "Turbulent Mixing and Beyond". The dynamical and statistical description of stably stratified turbulent boundary layers with the important example of…
This paper is devoted to a statistical analysis of the fluctuations of velocity and acceleration produced by a random distribution of point vortices in two-dimensional turbulence. We show that the velocity probability density function…
The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly…
The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different however. The diffusion coefficient can be made responsible for the size of jumps or…
A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is also…
In this paper, we aim to study the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we prove the weak convergence of slow process $X^\varepsilon$ in $C([0,T];\mathbb{R}^n)$ towards the…
The advection-diffusion equation can be approximated by a one-dimensional diffusion equation in Lagrangian coordinates along the directions of compression of fluid elements (the stable manifold). This result holds in any number of…
Time fractional advection-dispersion equations arise as generalizations of classical integer order advection-dispersion equations and are increasingly used to model fluid flow problems through porous media. In this paper we develop an…
A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time…
We study the long-time asymptotic behavior of the position distribution of a run-and-tumble particle (RTP) in two dimensions and show that the distribution at a time $t$ can be expressed as a perturbative series in $(\gamma t)^{-1}$, where…
We report new measurements of single particle dispersion in turbulent two-dimensional (2D) flows. Laboratory experiments in electromagnetically driven and Faraday wave driven turbulence reveal a transition from weakly dispersing…
Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. But a well-known defect of diffusive approximation whereby some of the particles propagate unrealistically fast has directed…
We consider the random evolution described by the motion of a particle moving on a circle alternating the angular velocities $ \pm c $ and changing rotation at Poisson random times, resulting in a telegraph process over the circle. We study…
In turbulent flows the $n$'th order structure functions $S_n(R)$ scale like $R^{\zeta_n}$ when $R$ is in the "inertial range". Extended Self-Similarity refers to the substantial increase in the range of power law behaviour of $S_n(R)$ when…
This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion on the Euclidean space, which is deeply related with a family of fractional Gagliardo-Nirenberg-Sobolev inequalities. Generically,…
The problem of the emergence of wave dispersion due to the heterogeneity of a transmission line (TL) is considered. An exactly solvable model helps to better understand the physical process of a signal passing through a non-uniform section…
The high order fluid model developed in the preceding paper is employed here to study the propagation of negative planar streamer fronts in pure nitrogen. The model consists of the balance equations for electron density, average electron…