In this work, the exact analytic solutions for some unsteady oscillating flows of an incompressible Burgers' fluid in a duct of rectangular cross-section are considered. Here, two types of flows, namely, (1) flow due to the oscillating pressure gradient, and (2) flow due to the oscillation of duct parallel to its length, were considered. The exact analytical expressions for the velocity field a...

It has been known for a while that a nonlinear equation driven by singular noise must be interpreted in the re-normalized, or Wick, form. For the stochastic Burgers equation, Wick nonlinearity forces the solution to be a generalized process no matter how regular the random perturbation is, whence the curse. On the other hand, certain multiplicative random perturbations of the deterministic Burg...

We link the appearance of universal kernels in random matrix ensembles to the phenomenon of shock formation in some fluid dynamical equations. Such equations are derived from Dyson's random walks after a proper rescaling of the time. In the case of the gaussian unitary ensemble, on which we focus in this paper, we show that the characteristics polynomials and their inverse evolve according to a...

The presentation will briefly review a Lagrangian model for deforming droplets in turbulence, in which the cumulative deformation of fluid elements at small scales is crucial. In particular, we show how knowledge about the statistical distribution of finite-time Lyapunov exponents (FTLEs) can be used to find a critical Capillary number above which droplets will break up. Probability distributio...

An overview of methods of pseudo-analysis in applications on important classes of nonlinear partial differential equations, occurring in different fields, is given. Hamilton-Jacobi equations, specially important in the control theory, are for important models usually with non-linear Hamiltonian H which is also not smooth, e.g., the absolute value, min or max operations, where it can not apply t...

The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503–3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singula...

Wick calculus in Gaussian analysis is investigated. It is shown that this calculus can be developed in a space of regular generalized functions. The results are applied to the discussion of solutions for Wick type stochastic (partial) diierential equations. In particular, the viscous Burgers equation with a stochastic source is studied. Its solution is shown to be a regular generalized process ...

The Burgers’ equation is an important and basic nonlinear partial differential equation in fluid dynamics, and has been used as a model equation in other fields, such as modeling of shock waves, gas dynamics, turbulence, and large bubble structures consisting of clusters of galaxies in space. Many researchers have proposed various numerical methods for solving the Burgers’ equation, such as the...