Free convection from a horizontal line source of heat

Summary In this paper a study is made of finite Grashof number effects upon the flow, in an unbounded domain, in the buoyant plume which forms over a horizontal line source of heat. The consequences of introducing a plane boundary into the domain are also assessed for two particular configurations.

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Zusammenfassung In dieser Arbeit wird der Effekt der endlichen Grashof-Zahl auf die Strömung untersucht, in einem unbegrenzten Medium mit natürlicher Konvektion, die durch eine horizontale linienförmige Quelle erzeugt wird. Die Folgen einer ebenen Begrenzungsfläche werden für zwei besondere Konfigurationen untersucht.

     

Asymptotic properties of the spectrum in the problem on waves in a bounded volume on a two-layer fluid

The asymptotic of eigen frequencies and corresponding waves on the free surface and interface of a two-layer ideal heavy fluid is constructed in two cases: the fluid is almost uniform and the upper layer has a low density. The asymptotic formulae are jusitified under the condition that the volume of the fluid is bounded. For the problem of surface waves, travelling in a submerged or surface-piercing infinite cylinder, the sufficient conditions for localized solutions of the limit problems to exist are indicated, and the hypothesis on the inevitable trapping of a wave by the body, which does not intersect both surfaces, is also formulated.     

Recovering the surface temperature history of a two-layer composite body

We consider the problem of recovering the surface temperature history of a two-layer composite body from interior measurements. The problem is ill-posed. We shall use the Fourier transform and the method of truncated integration to regularize the problem. Error estimate is given.     

Modified KdV equation with a source term in a bounded domain

A mixed problem for the modified KdV equation in Q = (0,1) × (0,∞) (1) where a(t)>0, is considered. No restriction on the sign of d(t) is imposed. We prove the existence and uniqueness of strong and weak global solutions as well as the exponential decay of solutions as t→+∞. Copyright © 2006 John Wiley & Sons, Ltd.     

Nonlinear diffusion with a bounded stationary level surface

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C²$C^2$. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C²$C^2$-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole RN${\mathbb{R}}^N$ of some substance whose density is initially a characteristic function of the complement of a domain with bounded C²$C^2$ boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SN${\mathbb{S}}^N$ and the hyperbolic space HN${\mathbb{H}}^N$.     

On global motion of a compressible viscous heat-conducting fluid bounded by a free surface

We consider the motion of a viscous compressible heat-conducting fluid in ³ bounded by a free surface which is under constant exterior pressure. We present the global existence theorems in two cases: when the free surface is under the surface tension and without it.     

Nonlinear interaction of internal and surface gravity waves in a two-layer fluid with free surface

A new nonlinear model of the propagation of wave packets in the system “liquid layer with solid bottom–liquid layer with free surface” is considered. With the use of the method of multiple-scale expansions, the first three linear approximations of the nonlinear problem are obtained. Solutions of problem of the first approximation are constructed and analyzed in detail. It is shown that there exist internal and surface components of the wave field, and their interaction is analyzed.     

Unsteady motion of a finite mass of fluid,bounded by a free surface

It is proved that the problem with a free boundary for the Navier-Stokes equations, describing the motion of a finite mass of viscous, incompressible capillary fluid, has a unique solution for all t > 0 if the domain occupied by the fluid is nearly a ball and the velocity vector field is small at the initial moment.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 137–157, 1986.     

The combined method for calculating the field generated by a point source in a surface waveguide

The two-dimensional problem of propagation of waves generated by a point source in a plane boundary-layer waveguide is investigated. The Dirichlet condition is given on the surface of the waveguide. With the help of cumbersome transformations, the solution is represented as a sum of geometric-optical waves, normal waves, and a remainder. Sufficient conditions on the general amount of detached normal and geometric-optical waves are obtained. The remainder is expressed by a simple formula. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 110–120.     

Mixed convection similarity solutions on a horizontal surface

The mixed convection boundary layer on a horizontal plate is considered for the two separate cases when there is a uniform free stream with the plate held fixed and when there is no outer flow but the plate is moving continuously with a uniform velocity along its length. In both cases we assume that power law temperature distribution on the plate which enables the governing equations to be reduced to similarity form. For the first problem we consider the range of buoyancy parameter for which there are dual solutions, showing how these dual solutions arise from a bifurcation and how the lower branch of solutions terminate as the buoyancy parameter tends to zero. For the second problem we show that there is a unique solution for all positive values of the buoyancy parameter and that for negative values the solution terminates at a singular solution with algebraic decay.

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Zusammenfassung Die Grenzschicht an einer horizontalen Platte mit gemischter Konvektion wird für zwei Fälle untersucht: Für eine ruhende Platte bei gleichförmiger Außenströmung und für eine Platte, die sich kontinuierlich mit konstanter Geschwindigkeit in ihrer eigenen Ebene bewegt, ohne Außenströmung. Für beide Fälle wählen wir ein Potenzgesetz für die Temperaturverteilung entlang der Wand, für welche die Grundgleichungen eine Ähnlichkeitslösung zulassen. In der ersten Problemstellung betrachten wir einen Bereich des Auftriebsparameters, für den zweifache Lösungen existieren. Wir zeigen, wie diese doppelten Lösungen aus einer Bifurkation entstehen und wie der untere Lösungszweig endet, wenn der Auftriebsparameter gegen Null strebt. Für die zweite Problemstellung zeigen wir, daß eine eindeutige Lösung existiert für alle positiven Werte des Auftriebsparameters und daß für negative Werte eine singuläre Lösung mit algebraischem Zerfall erreicht wird.

     

Doppler effect in the motion of a source and a refractive surface

The space-time ray-tracing method is used to derive equations for the instantaneous frequency and instantaneous wave vector of a wave radiated by a moving source in a time-dependent inhomogeneous medium and reflected from a moving surface.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 52–58, 1987.The author is indebted to V. S. Buldyrev for useful discussions.     

Perturbations of the Haar wavelet by convolution

In this note we show that the standard convolution regularization of the Haar system generates Riesz bases of smooth functions for , providing in this way an alternative to the approach given by Govil and Zalik [Proc. Amer. Math. Soc. 125 (1997), 3363-3370].

     

Typical Changes in Spectral Properties under Perturbations by a Rank-One Operator

We show that under perturbations by a rank-one typical operator, a fixed eigenvalue loses one Jordan block of maximal order, while the orders of the other Jordan blocks remain unchanged. We construct the first-order perturbation theory for the new eigenvalues and the zero-order approximations to the corresponding eigenvectors.     

Scattering of oblique waves by bottom undulations in a two-layer fluid

Scattering of waves obliquely incident on small cylindrical undulations at the bottom of a two-layer fluid wherein the upper layer has a free surface and the lower layer has an undulating bottom, is investigated here assuming linear theory. There exists two modes of time-harmonic waves propagating at each of the free surface and the interface. Due to an obliquely incident wave of a particular mode, reflected and transmitted waves of both the modes are created in general by the bottom undulations. For small undulations, a simplified perturbation analysis is used to obtain first-order reflection and transmission coefficients of both the modes due to oblique incidence of waves of again both modes, in terms of integrals involving the shape function describing the bottom. For sinusoidal undulations, these coefficients are plotted graphically to illustrate the energy transfer between the waves of different modes induced by the bottom undulations.     

On the free convection from a horizontal plate

Zusammenfassung Die freie Konvektion an einer beheizten horizontalen Platte wird nach den üblichen Methoden der Grenzschichttheorie untersucht. Weist die wärmeübertragende Plattenseite nach oben, so ist das Problem auf dieser Basis unlösbar, wohl aber findet man eine widerspruchsfreie Lösung, wenn diese Plattenseite nach unten weist. In diesem ist die Nusselt Zahl, die den Wärmeübergang kennzeichnet, proportional der 1/5-ten Potenz der Rayleigh Zahl, welche die Beheizung charakterisiert. Die Übereinstimmung mit dem Experiment ist befriedigend.