A closure model for turbulent flows is developed based on a dynamical system theory. An appropriately discretized formulation of the governing equations considered this process. The key ingredient an approximation system’s attractor, where all trajectories in phase space are confined. This approximate inertial manifold approach provides path to track lower-dimensional subspace. Unlike conventio...

Phase transitions can be modeled by the motion of an interface between two locally stable phases. A modiied Kuramoto-Sivashinsky equation, h t + r 2 h + r 4 h = (1 ?)jrhj 2 (r 2 h) 2 + (h xx h yy ? h 2 xy); describes near planar interfaces which are marginally long-wave unstable. We study the question of nite-time singularity formation in this equation in one and two space dimensions on a perio...

Phase transitions can be modeled by the motion of an interface between two locally stable phases. A modified Kuramoto-Sivashinsky equation, h, + 72h + Wh = (1 a)lVhl ~ ± X(V~h): + ~A(hx~hyy h~y) , describes near planar interfaces which are marginally long-wave unstable. We study the question of finite-time singularity formation in this equation in one and two space dimensions on a periodic doma...

The (Michelson) Sivashinsky equation of premixed flames is studied in a rectangular domain in two dimensions. A huge number of two-dimensional (2D) stationary solutions are trivially obtained by the addition of two 1D solutions. With Neumann boundary conditions, it is shown numerically that adding two stable 1D solutions leads to a 2D stable solution. This type of solution is shown to play an i...

Abstract Using approximation techniques, long-wave length framework and boundary-layer, the effects of electrostatic force induced shear stress on flow behavior down an inclined solid substrate are investigated. In general case, considered model accounts in presence inertia regime streamwise viscous diffusion with influence normal electric field imposed stress. Galerkin weighted residual, two c...

We study the dynamics of a ferrofluid thin film confined in Hele-Shaw cell, and subjected to tilted nonuniform magnetic field. It is shown that interface between an inviscid outer fluid (air) supports traveling waves, governed by novel modified Kuramoto--Sivashinsky-type equation derived under long-wave approximation. The balance energy production dissipation this equations allows for existence...

We first show asymptotic L bounds for a class of equations, which includes the Burger-Sivashinsly model for odd solutions with periodic boundary conditions. We consider the conditional stability of stationary solutions of Kuramoto-Sivashinsky equation in the periodic setting. We establish rigorously the general structure of the spectrum of the linearized operator. In addition, we show condition...

We revisit the Near Equidiffusional Flames (NEF) model introduced by Matkowsky and Sivashinsky in 1979 and consider a simplified, quasisteady version of it. This simplification allows, near the planar front, an explicit derivation of the front equation. The latter is a pseudodifferential fully nonlinear parabolic equation of the fourth-order. First, we study the (orbital) stability of the null ...