Propagation of disturbances in three-dimensional supersonic boundary layers

The propagation of disturbances in three-dimensional boundary layers under the conditions of a global and a local strong inviscid-viscous interaction is analyzed. A system of subcharacteristics is found based on the condition for the pressure-related subcharacteristic, and an algebraic relation that gives the propagation velocity of disturbances is obtained. The velocity of propagation of disturbances is calculated for two- and three-dimensional flows. The studied problem is of great importance for accurately formulating problems for three-dimensional unsteady boundary-layer equations and for constructing adequate computational models. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 116–127, May–June, 1999.     

A low-dimensional model of separation bubbles

In this work, motivated by the problem of model-based predictive control of separated flows, we identify the key variables and the requirements on a model-based observer and construct a prototype low-dimensional model to be embedded in control applications.Namely, using a phenomenological physics-based approach and dynamical systems and singularity theories, we uncover the low-dimensional nature of the complex dynamics of actuated separated flows and capture the crucial bifurcation and hysteresis inherent in separation phenomena. This new look at the problem naturally leads to several important implications, such as, firstly, uncovering the physical mechanisms for hysteresis, secondly, predicting a finite amplitude instability of the bubble, and, thirdly, to new issues to be studied theoretically and tested experimentally.     

Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems

This paper develops a rigorous notion of dissipation-induced instability in infinite dimensions as an extension of the classical concept implicitly introduced by Thomson and Tait for finite degree of freedom mechanical systems over a century ago. Here we restrict ourselves to a particular form of infinite-dimensional systems—partial differential equations—whose inherent function-analytic differences from finite-dimensional systems make uncovering this notion more intricate. In building the concept of dissipation-induced instability in infinite dimensions we found Arnold’s and Yudovich’s nonlinear stability methods, for conservative and dissipative systems respectively, along with some new existence theory for solutions, to be the essential foundation. However, when proving the results for classical solutions, as motivated by their direct physical significance, we had to overcome a number of fundamental difficulties associated with existing stability analysis methods, which has led to new techniques. In particular, in this work we establish the connection of existence and general stability theories in strong and weak topologies and provide new insights into the physics and geometry of the dissipation-induced instability phenomena in infinite-dimensional systems. As a paradigm and the first infinite-dimensional example to be rigorously analyzed, we use a two-layer quasi-geostrophic beta-plane model, which describes the fundamental baroclinic instability in atmospheric and ocean dynamics; early formal linear approximate studies suggested that this system can be destabilized after the introduction of dissipation.     

Stability on Time-Dependent Domains

We explore the key differences in the stability picture between extended systems on time-fixed and time-dependent spatial domains. As a paradigm, we take the complex Swift–Hohenberg equation, which is the simplest nonlinear model with a finite critical wavenumber, and use it to study dynamic pattern formation and evolution on time-dependent spatial domains in translationally invariant systems, i.e., when dilution effects are absent. In particular, we discuss the effects of a time-dependent domain on the stability of spatially homogeneous and spatially periodic base states, and explore its effects on the Eckhaus instability of periodic states. New equations describing the nonlinear evolution of the pattern wavenumber on time-dependent domains are derived, and the results compared with those on fixed domains. Pattern coarsening on time-dependent domains is contrasted with that on fixed domains with the help of the Cahn–Hilliard equation extended here to time-dependent domains. Parallel results for the evolution of the Benjamin–Feir instability on time-dependent domains are also given.     

On the Moving Contact Line Singularity

Doklady Physics - Given that contact line between liquid and solid phases can move regardless how negligibly small are the surface roughness, Navier slip, liquid volatility, impurities, deviations...     

Problems on Time-Varying Domains: Formulation,Dynamics, and Challenges

Acta Applicandae Mathematicae - The purpose of this article is to introduce the reader to phenomena on time-varying spatial domains and to highlight the differences from their counterpart on...