Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary

We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a ``physical condition', related to the fact that the pressure of a fluid has to be positive. The author was supported in part by the National Science Foundation.     

激光空泡在近自由液面运动特性的实验研究

采用激光技术聚焦击穿液体产生空泡,利用高速摄像系统开展激光空泡与自由液面的相互作用的实验研究。实验研究发现,空泡与自由液面之间的无量纲距离对水下空泡的脉动特性和自由液面的水冢现象存在影响。通过大量实验总结了无量纲距离与空泡半径、空泡脉动周期、自由液面水柱的最大高度和产生水冢时间的相互关系。实验表明无量纲距离越小,空泡脉动周期越短,自由面的水冢现象越明显。同时统计出了在不同无量纲距离范围内所出现的5种不同水冢现象的规律。研究内容为空泡与自由液面相互作用的理论提供了实验依据。     

两相可压缩边界层的颗粒运动特性分析

本文采用高精度直接数值模拟方法,研究了马赫数为2的可压缩湍流边界层中,两种不同粒径颗粒的运动特性.研究发现近壁面的颗粒流向平均速度要小于流体;颗粒具有法向指向壁面的平均速度;近壁面颗粒的流向雷诺正应力要略大于流体:其中大颗粒具有更加显著的上述趋势.小颗粒的法向和展向雷诺正应力基本等于流体,而大颗粒则都要远小于流体.条件...     

Waves on the Free Surface Described by Linearized Equations of Hydrodynamics with Localized Right-Hand Sides

A linearized system of equations of hydrodynamics with time-dependent spatially localized right-hand side placed both on the free surface (and on the bottom of the basin) and also in the layer of the liquid is considered in a layer of variable depth with a given basic plane-parallel flow. A method of constructing asymptotic solutions of this problem is suggested; it consists of two stages: (1) a reduction of the three-dimensional problem to a two-dimensional inhomogeneous pseudodifferential equation on the nonperturbed free surface of the liquid, (2) a representation of the localized right-hand side in the form of a Maslov canonical operator on a special Lagrangian manifold and the subsequent application of a generalization to evolution problems of an approach, which was recently suggested in the paper [A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and M. Rouleux, Dokl. Ross. Akad. Nauk 475 (6), 624–628 (2017); Engl. transl.: Dokl. Math. 96 (1), 406–410 (2017)], to solving stationary problems with localized right-hand sides and its combination with “nonstandard” characteristics. A method of calculation (generalizing long-standing results of Dobrokhotov and Zhevandrov) of an analog of the Kelvin wedge and the wave fields inside the wedge and in its neighborhood is suggested, which uses the consideration that this method is the projection to the extended configuration space of a Lagrangian manifold formed by the trajectories of the Hamiltonian vector field issuing from the intersection of the set of zeros of the extended Hamiltonian of the problem with conormal bundle to the graph of the vector function defining the trajectory of motion of an equivalent source on the surface of the liquid.     

Boundary Singularity of Moments for the Linearized Boltzmann Equation

We study the boundary singularity for stationary solutions of the linearized Boltzmann equation with hard-sphere potential. An asymptotic estimate for the gradient of the moments is established, which shows the logarithmic singularity near the boundary. Our formula holds for the solutions of the Milne and Kramers problems obtained by Bardos-Caflish-Nicolaenko (Commun. Pure Appl. Math. 49:323–452, 1986). Our theorem requires the Hölder continuity of the boundary data. In particular, it applies to the complete condensation problem for half space.     

Two-Dimensional Wave Motion on the Charged Surface of a Viscous Liquid

The wave motion on the charged surface of a viscous Newtonian liquid is solved as an initial-value problem. Both the leaky dielectric and perfect dielectric cases are considered. The amplitude of wave is assumed to be small. The electric field induced by surface charge is shown to have a generally destabilizing effect on surface wave. The neutral stability curve is drawn in the （G, N~） plane （G： the gravitational bond number; N~： the electrical Bond number）. The Ohnesorge number, Taylor-Melcher number and permittivity ratio have little influence on the neutral stability curve. It is testified that the classical normal mode method cannot predict wave behaviour at small times.     

Algorithm for Constructing Exact Solutions of the Problem of Unsteady Plane Motion of a Fluid with a Free Surface

JETP Letters - Unsteady plane potential flows of an ideal incompressible fluid with a free surface in the absence of external forces and capillarity are studied. An algorithm to construct exact...     

Well Posed Constraint-Preserving Boundary Conditions for the Linearized Einstein Equations

In this paper we address the problem of specifying boundary conditions for Einstein's equations when linearized around Minkowski space using the generalized Einstein-Christoffel symmetric hyperbolic system of evolution equations. The boundary conditions we work out guarantee that the constraints are satisfied provided they are satisfied on the initial slice and ensures a well posed initial-boundary value formulation. We consider the case of a manifold with a non-smooth boundary, as is the usual case of the cubic boxes commonly used in numerical relativity. The techniques discussed should be applicable to more general cases, as linearizations around more complicated backgrounds, and may be used to establish well posedness in the full non-linear case.     

Incompressible Limit of the Compressible Magnetohydrodynamic Equations with Periodic Boundary Conditions

This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. It is rigorously shown that the weak solutions of the compressible magnetohydrodynamic equations converge to the strong solution of the viscous or inviscid incompressible magnetohydrodynamic equations as long as the latter exists both for the well-prepared initial data and general initial data. Furthermore, the convergence rates are also obtained in the case of the well-prepared initial data.     

Excitation of Vortex Flows on the Free Surface of a Liquid by a Vibrating Plate

Technical Physics - The surface flows of a liquid in the presence of a partially submerged vibrating plate are investigated. In the case of low-frequency plate vibrations, two vortices are formed...     

Free Surface and Interface Thermodynamics of Liquid Nickel in Contact with Alumina

Sessile drop experiments of Ni and Ni(2at.%Al) were conducted under controlled working conditions, at 1500°C, P(O₂) 10^–9 Torr. It is shown that Al and oxygen atoms engaged in the capillary driven mass transport at the interface have a significant impact on the surface/interface thermodynamics. The surface energy of liquid Ni determined from experiments in which Ni comes into contact with Al₂O₃ is significantly lower than that of high purity Ni, due to the segregation of Al. The free energy of segregation of Al to the free surface of Ni ( G _S) was found to range from –164 to –152 kJ/mol, indicating a relatively strong tendency for segregation of Al to the free surface of Ni(Al). It is proposed that an Al(O)-rich liquid layer forms adjacent to the Ni-Al₂O₃ interface, which improves interfacial adhesion. In the Ni(Al)-Al₂O₃ system, an increase in the Al content of the alloy leads to the improvement of both wetting and adhesion of the alloy on the ceramic, correlating with the improvement in the interface strength after solidification.     

Exact Solutions to the Problem of Dynamics of a Liquid with a Free Surface between Two Approaching Vertical Walls

Doklady Physics - Exact solutions of the classical problem of a plane unsteady potential flow of an ideal incompressible fluid with a free boundary are presented. The fluid occupies a semi-infinite...     

Flow Characteristics of Flapping Motion of a Plane Water Jet Impinging onto Free Surface

A submerged turbulent plane jet in shallow water impinging vertically onto the free surface will produce a large-scale flapping motion when the jet exit velocity is larger than a critical one. The flapping phenomenon is verified in this paper through a large eddy simulation where the free surface is modeled by volume of fluid approach. The quantitative results for flapping jet are found to be in good agreement with available experimental data in terms of mean velocity, flapping-induced velocity and turbulence intensity. Results show that the flapping motion is a new flow pattern with characteristic flapping frequency for submerged turbulent plane jets, the mean centerline velocity decay is considerably faster than that of the stable impinging jet without flapping motion, and the flapping-induced velocities are as important as the turbulent fluctuations.     

Generation Mechanism of Liquid Column during the Burst of a Rising Bubble near a Free Surface

In the frame of inviscid and incompressible flow and by neglecting surface tension effects, the dynamical mechanism of the generation and the development of the liquid column during the burst of a rising bubble near a free surface is studied theoretically and numerically by the volume-of-fluid method.     

Boundary Layers for the Navier–Stokes Equations¶of Compressible Fluids

The global unique solvability is proved for the Navier–Stokes equations of compressible fluids for the one-dimensional spiral flows between two circular cylinders. The zero shear viscosity limit μ→ 0 is justified. The value O(μ^α), 0 < α < 1/2, is established for the boundary layer thickness. Received: 11 December 1998 / Accepted: 9 June 1999     

Experimental Study on Liquid Free Surface in Buoyant-Thermocapillary Convection

We investigate the surface deformations of buoyant-thermocapillary convection in a rectangular cavity due to gravity and temperature gradient between the two sidewalls. The cavity is 52mm×42 mm in horizontal cross section, the thickness of liquid layer h is changed from 2.5 mm to 6.5 mm. Surface deformations of h = 3.5 mm and 6.0mm are discussed and compared. Temperature difference is increased gradually, and the flow in the liquid layer will change from stable convection to unstable convection. Two kinds of optical diagnostic system with image processor are developed for study of the kinetics of buoyant-thermocapillary convection, they give out the information of liquid free surface. The quantitative results are calculated by Fourier transform and correlation analysis, respectively. With the increasing temperature gradient, surface deformations calculated are more declining. It is interesting phenomenon that the inclining directions of the convections in thin and thick liquid layers are different. For a thin layer, the convection is mainly controlled by thermocapillary effect. However, for a thick layer, the convection is mainly controlled by buoyancy effect. The surface deformation theoretically analysed is consistent with our experimental results. The present experiment proves that surface deformation is related to temperature gradient and thickness of the liquid layer. In other words, surface deformation lies on capillary convection and buoyancy convection.