Fully nonlinear interfacial waves in a bounded two-fluid system

We study the nonlinear flow which results when two immiscible inviscid incompressible fluids of different densities and separated by an interface which is free to move and which supports surface tension, are caused to flow in a straight infinite channel. Gravity is taken into consideration and the velocities of each phase can be different, thus giving rise to the Kelvin-Helmholtz instability. Our objective is to study the competing effects of the Kelvin-Helmholtz instability coupled with a stably or unstably stratified fluid system (Rayleigh-Taylor instability) when surface tension is present to regularize the dynamics. Our approach involves the derivation of two-and three-dimensional model evolution equations using long-wave asymptotics and the ensuing analysis and computation of these models. In addition, we derive the appropriate Birkhoff-Rott integro-differential equation for two-phase inviscid flows in channels of arbitrary aspect ratios. A long wave asymptotic analysis is undertaken to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across the interface. Linear stability analysis reveals that capillary forces stabilize short-wave disturbances in a dispersive manner and we study their effect on the fully nonlinear dynamics described by our models. In the case of two-dimensional interfacial deflections, traveling waves of permanent form are constructed and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves are expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9 where 2h is the channel thickness. In the absence of gravity, solitary waves are not possible but periodic ones are. Numerically constructed traveling and solitary waves are given for representative physical parameters. The initial value problem for the partial differential equations is also addressed numerically in periodic domains, and the regularizing effect of surface tension is investigated. In particular, when surface tension is absent it is shown that the system of governing evolution equations terminates in a singularity after a finite time. This is achieved by studying a 2 x 2 system of nonlinear conservation laws in the complex plane and by numerical solution of the evolution equations. The analysis shows that a sinusoidal perturbation of the flat interface and a cosine perturbation to the unit velocity jump across the interface, develop a singularity at time tc = ln 1/ε+0 (ln(ln 1/ε)) where ε is the initial amplitude of the disturbances. This result is asymptotic for small ε and is derived by studying the asymptotic form of the flow characteristics in the complex plane. We also derive the analogous three-dimensional evolution equations by assuming that the wavelengths in the principal horizontal directions are large compared to the channel thickness. Surface tension is again incorporated to regularize short-wave Kelvin-Helmholtz instabilities and the equations are solved numerically subject to periodic boundary conditions. Evidence of singularity formation is found. In particular, we observe that singularities occur at isolated points starting from general initial conditions. This finding is consistent with numerical studies of unbounded three-dimensional vortex sheets (see Introduction for a discussion and references). In the final part of this work we consider the vortex-sheet formulation of the exact nonlinear two-dimensional flow of a vortex sheet which is bounded in a channel. We derive a Birkhoff-Rott type integro-differential evolution equation for the velocity of the interface in terms of the vorticity as well as the evolution equation for the unnormalized vortex sheet strength. For the case of a spatially periodic vortex sheet, this Birkhoff-Rott type equation is written in terms of Jacobi\u27s functions. The equation is shown to recover the limits of unbounded and non-periodic flows which are known in the literature

On Artifacts in Limited Data Spherical Radon Transform: Curved Observation Surface

In this article, we consider the limited data problem for spherical mean transform. We characterize the generation and strength of the artifacts in a reconstruction formula. In contrast to the third's author work [Ngu15b], the observation surface considered in this article is not flat. Our results are comparable to those obtained in [Ngu15b] for flat observation surface. For the two dimensional problem, we show that the artifacts are $k$ orders smoother than the original singularities, where $k$ is vanishing order of the smoothing function. Moreover, if the original singularity is conormal, then the artifacts are $k+\frac{1}{2}$ order smoother than the original singularity. We provide some numerical examples and discuss how the smoothing effects the artifacts visually. For three dimensional case, although the result is similar to that [Ngu15b], the proof is significantly different. We introduce a new idea of lifting the space

On exact solutions of the nonlinear heat equation

A method for construction of exact solutions to the nonlinear heat equation ut = (F (u)ux)x + G (u)ux + H (u), which is based on the ansatz p(x) = ω₁(t) φ(u) + ω₂(t), is proposed. The function p(x) is a solution of the equation (p′)² = Ap² + B, and the functions ω₁(t), ω₂(t) and ϕ(u) can be found from the condition that this ansatz reduces the nonlinear heat equation to a system of two ordinary differential equations with unknown functions ω₁(t) and ω₂(t).Запропоновано метод побудови точних розв’язків нелінійного рівняння теплопровідності ut = (F(u)ux)x + + G(u)ux + H(u), який ґрунтується на використанні підстановки p(x) = ω₁(t) φ(u) + ω₂(t), де функція p(x) є розв’язком рівняння (p′)² = Ap² + B, а функції ω₁(t), ω₂(t) та ϕ(u) знаходяться з умови, що дана підстановка редукує рівняння до системи двох звичайних диференціальних рівнянь з невідомими функціями ω₁(t) та ω₂(t).Предложен метод построения точных решений нелинейного уравнения теплопроводности ut = (F(u)ux)x + + G(u)ux + H(u), основанный на использовании подстановки p(x) = ω₁(t) φ(u) + ω₂(t), где функция p(x) является решением уравнения (p′)² = Ap² + B, а функции ω₁(t), ω₂(t) и ϕ(u) находятся из условия, что данная подстановка редуцирует уравнение к системе двух обыкновенных дифференциальных уравнений с неизвестными функциями ω₁(t) и ω₂(t)

Бюджетна безпека в Україні в контексті соціальної стабільності (Security budget in ukraine in the context of social stability)

Стаття присвячена питанням бюджетної безпеки, як фактору соціального розвитку та головній умові на шляху досягнення соціальної стабільності. Представлено міжнародні рейтинги, які свідчать про кризові тренди в трансформаційному русі країни. Показано, що для бюджетної безпеки необхідно скоротити в країні бідність шляхом зміни податкової системи, пенсійної реформи, переходу на страхові принципи роботи медичної галузі, на- дання більшої самостійності органам місцевого самоврядування у формуванні місцевих бюджетів та проведенні соціальної політики та ін. (The article is devoted to the security budget, as a factor of social development and the main condition to achieving social stability. Author presents the international rankings, showing the crisis trends in the transformation of the country moving. It is shown that for the security budget is necessary to reduce poverty in the country by changing the tax system, the pension reform, and transition to insurance principles of the medical sphere, providing greater autonomy to local authorities in the shaping of local budgets and social policy and some others.

Medical rehabilitation of blood flow disorders in patients with one-sided pathological kidney lever

The release of vasoactive substances into the bloodstream causes a number of vascular reactions, alternating vasoconstriction and vasodilation disrupt the course of adequate adaptive responses to the restoration of blood circulation in the kidneys [3, 5]. The additional impact of surgery also affects the adequate restoration of total renal function [3]. There are two ways to positively affect the state of blood circulation: improving the rheological properties of blood and preventing or reducing vascular spasm of the renal parenchyma, which should be effectively performed during the perioperative period and in the long term after surgery [6]. The aim of the study. To analyze and clinically evaluate the method of perioperative correction of renal blood flow in patients with unilateral kidney damage. Material and methods of research. The clinical study was performed in 58 patients aged 18 to 65 years with unilateral kidney damage who received surgical treatment according to the protocols of medical care for a specific pathology, as well as additional measures of perioperative improvement of blood flow in the parenchyma of both kidneys

Nomadic Transformation of Mountain-Meadow Brown Soils (Dystric Cambisols) of the Svydovets Array of the Ukrainian Carpathians

In the highlands of the Ukrainian Carpathians, brown soil formation process is supplemented by turf, which significantly affects the soil properties and composition formation – this way unique shallow, rubble, highly acidic, base unsaturated, mountain-meadow brown soils with high Corg content (Dystric Cambisols) are formed. Irrational and haphazard farming in the valleys leads to the soil degradation and formation of specific anthropogenically altered mountain-meadow brown soil. To study the features of mountain-meadow brown soil formation and to analyze changes in properties, as a result of economic activity, we conducted detailed soil-geographic research within the valleys of the Svydovets array. Research has established that anthropogenically altered mountain-meadow brown soils are characterized by lower Corg rates, higher acidic reaction of the soil solution, high hydrolytic acidity and the dominance of exchange calcium in the absorbing soil complex, different qualitative humus composition – humate-fulvate. As a result of anthropogenic soil transformation, composition indicators underwent distinct changes – soil of the genetic horizons with a decrease in the total porosity; structure of the humus-accumulative horizon of virgin soil has undergone transformation and is characterized as prismoidal

Exact Travelling Wave Solutions of Some Nonlinear Nonlocal Evolutionary Equations

Direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some spatially nonlocal hydrodynamic-type model. Special attention is paid to the construction of the kink-like and soliton-like solutions.Comment: 13 pages, LaTeX2