On the stability of the Ekman boundary layer

We present a result on well-posedness and stability of the Ekman boundary layer problem in the space FM(ℝ², L²(ℝ₊)³), i.e., in the space of L²(ℝ₊)³-valued Fourier transformed finite Radon measures. In particular we obtain stability in the angle velocity of rotation, which is important in the analysis of fast oscillating singular limits. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic–parabolic systems

Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic–parabolic systems including the Navier–Stokes equations of compressible gas, and magnetohydrodynamics with inflow or outflow boundary conditions, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index γ?1. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive (“shock-type”) layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime.     

An analysis of laminar boundary layers on liquid curtains

A simplified analysis of the laminar boundary layer along an isothermal liquid curtain falling under gravity is presented. The analysis uses a von Kármán-Pohlhausen integral method and includes the effects of gravity, pressure differences, surface tension and nozzle exit geometry on the convergence length of liquid curtains which have applications as chemical reactors and as protection systems in laser fusion reactors. It is shown that the effects of the surrounding gases on the curtain shape and convergence length are small, and that good approximations to the liquid curtain shape can be obtained by using inviscid flow analyses.     

On boundary layers

The concept of boundary layers, introduced by A. Volberg in [7], is generalized from subsets of the unit disk to subsets of general non-tangentially accessible (NTA) domains. Capacitary conditions of Wiener type series of both necessary and sufficient type for boundary layers are presented and the connection between boundary layers and minimally thin sets is studied.     

Nonlinear stability of boundary layers for the Boltzmann equation with cutoff soft potentials

The study on the boundary layer is important in both mathematics and physics. This paper considers the nonlinear stability of boundary layer solutions for the Boltzmann equation with cutoff soft potentials when the Mach number of the far field is less than −1. Unlike the collision frequency is strictly positive in the hard potential or hard sphere model, the collision frequency has no positive lower bound for the cutoff soft potentials, so the decay in time cannot be expected. Instead, the present paper proves that the solution will always be in a small region around the boundary layer by noticing the decay property of collision operator in velocity.     

Ekman layers of rotating fluids: The case of general initial data

In this paper we describe the weak solutions of the Navier‐Stokes equations with a large Coriolis term as the Rossby and the Ekman numbers go to zero in a special domain with various boundary conditions. This work extends the results (in the case of well‐prepared initial data) in [19, 30]. © 2000 John Wiley & Sons, Inc.     

Asymptotic analysis of oscillating parametric integrals and ordinary boundary value problems at resonance

This paper deals with boundary value problems whose nonlinear part involves periodic functions and such that the linear part has a one-dimensional solution space. We shall study the existence and multiplicity of solutions using various methods of Nonlinear Analysis such as the Lyapunov-Schmidt reduction and methods of critical point theory. The proofs are based on some general results on the oscillation and asymptotic behavior of certain parametric integrals.     

Three-dimensional periodic boundary layers

The method of successive approximations used by Eichelbrenner and A?kovíc¹ for the study of unsteady three-dimensional boundary layer flow has been extended to analyse the periodic boundary layers in three dimensions. The analysis, which is valid for oscillations of small amplitude, shows some special features such as “steady streaming” flow in the first-order cross-flow similar to the one that has been predicted and observed by Schlichting for two-dimensional periodic boundary layers.     

On magneto-aerodynamic boundary layers

Zusammenfassung Der Einfluss eines senkrecht zur Schicht stehenden magnetischen Feldes auf die Geschwindigkeits- und Temperaturverteilung in einer Grenzschicht wird für einen Sonderfall der ?nderung der elektrischen Leitf?higkeit von Luft untersucht. Die Wirkung des magnetischen Feldes auf den Reibungswiderstand und den W?rmeübergang wird bei relativ zur Platte und zur Flüssigkeit in grosser Entfernung von der Platte festgehaltenem Magnetfeld ermittelt.      

Existence and stability of solutions with boundary layers in multidimensional singularly perturbed reaction-diffusion-advection problems

This paper deals with the boundary-value problem for a nonlinear elliptic equation containing a small parameter multiplying the derivatives and degenerating into a finite equation as the small parameter tends to zero. The existence theorem for the solution with a boundary layer and its Lyapunov stability are proved.     

Linear prediction for generalized oscillating stochastic processes

Translated from Matematicheskie Modeli i Optimizatsiya Vychislitel'nykh Algoritmov, pp. 123–130, 1994.     

Ekman layers of rotating fluids,the case of well prepared initial data

In this paper we study the convergence of weak solutions of the Navier Stokes equations with a large Coriolis term as the Rossby and Ekman numbers go to zero, and in particular the so called Ekman boundary layers, and justify some classical expansions in geophysical fluid dynamics (see [14], chapter 4).     

The direction of the secondary wave motion in the Ekman boundary layer

The secondary wave motion in the free Ekman layer, on which a constant shear stress acts, is examined by an analytic method. The results show, that the waves move away from the center of rotation.     

Spectral boundary homogenization in domains with oscillating boundaries

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with a rapidly oscillating boundary. We consider both cases where the eigenvalues of the limit problem are simple and multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.     

Boundary homogenization in domains with randomly oscillating boundary

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and establish error estimates.