Two-fluid spatial instability with velocity defect in the basic flow

The initial state of liquid atomization by a fast gas stream is considered by viscous linear spatial stability analysis for parallel two-fluid flow. The unbounded basic velocity profile is characterized by boundary layers near the interface and different asymptotic velocities. By computing generalized spatial branches we identify conditions for absolute instability when a velocity deficit is introduced to account for the conditions near the nozzle. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flow in the tail interaction region of a spherically symmetrical and a uniform supersonic gas stream

The flow in the tail region of two interacting supersonic streams – spherically symmetrical and planeparallel – is simulated on a supercomputer. The numerical solution is obtained by Godunov’s method. Analysis of the solutions reveals the complex structure of the flow, which includes multiple interfering shock wave structures, a near-axial circulation zone, and a near-axial forward flow zone with a velocity deficit. The detection of such a structure is an unexpected result of the simulation procedure, but it is consistent with some computational and experimental studies, where structures have been observed in supersonic jets.     

Dopant diffusion in phases with a moving interface

We consider the diffusion of a dopant through a moving interface in the suicide film-Si system during silicide layer growth. The dopant concentration distribution is derived in analytical form by the integral Fourier transform method with subsequent reduction of the dopant redistribution problem to numerical solution of two integral equations. The results are presented in the form of curves plotting the time dependence of dopant concentration on both sides of the interface for various values of diffusion coefficients and interface velocity. The effect of physical parameters on the variation of dopant concentration near the interface is demonstrated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 93–97, 1987.     

One type of waves in media with loosely bonded interface

A new type of waves that propagate in media having boundaries with slip is described. The group velocity of these waves is greater than the P — wave velocity and the less than the S-wave velocity, and the phase velocity equals the velocity of the longitudinal wave. The frequency of -waves is constant and is determined by cosntructive interference conditions. The waves under consideration are similar to Crary waves recorded in floating ice.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN USSR, Vol. 173, pp. 113–122, 1988.     

The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

Analytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier-Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen-like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall-bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of a Stokes interface problem based on formulation using the characteristic function

Numerical analysis of a model Stokes interface problem with the homogeneous Dirichlet boundary condition is considered. The interface condition is interpreted as an additional singular force field to the Stokes equations using the characteristic function. The finite element method is applied after introducing a regularization of the singular source term. Consequently, the error is divided into the regularization and discretization parts which are studied separately. As a result, error estimates of order h^1/2 in H¹ × L² norm for the velocity and pressure, and of order h in L² norm for the velocity are derived. Those theoretical results are also verified by numerical examples.     

Stability of a flame front in a divergent flow

We study the evolution of perturbations on the surface of a stationary plane flame front in a divergent flow of a combustible mixture incident on a plane wall perpendicular to the flow. The flow and its perturbations are assumed to be two-dimensional; i.e., the velocity has two Cartesian components. It is also assumed that the front velocity relative to the gas is small; therefore, the fluid can be considered incompressible on both sides of the front; in addition, it is assumed that in the presence of perturbations the front velocity relative to the gas ahead of it is a linear function of the front curvature. It is shown that due to the dependence (in the unperturbed flow) of the tangential component of the gas velocity on the combustion front on the coordinate along the front, the amplitude of the flame front perturbation does not increase infinitely with time, but the initial growth of perturbations stops and then begins to decline. We evaluate the coefficient of the maximum growth of perturbations, which may be large, depending on the problem parameters. It is taken into account that the characteristic spatial scale of the initial perturbations may be much greater than the wavelengths of the most rapidly growing perturbations, whose length is comparable with the flame front thickness. The maximum growth of perturbations is estimated as a function of the characteristic spatial scale of the initial perturbations.     

Numerical inaccuracies across the interface of a nested grid

This article addresses and discusses the inaccuracies in finite differencing across the interface of a nested grid. Explicit schemes for the advection and diffusion equations are analyzed on the fine and coarse grids and reformulated at the interface to guarantee that the evolving solution is unaffected by the abrupt change of the spatial grid resolution. The associated errors are expressed as a function of the wavelength of the initial field distribution and the ratio between the coarse and fine grid resolution. It is found that large-scale features of the coarse grid must supply energy to sustain the small-scale features of the fine grid. To not deplete the large-scale motion, a source of energy must be given at the interface in the form of a computational diffusive term with negative viscosity coefficient. On the other hand, not all the energy of the small-scale features of the fine grid has to be transferred to the large-scale motion, but some of it needs to be computationally dissipated at the interface. © 1994 John Wiley & Sons, Inc.     

A Host of Traveling Waves in a Model of Three-Dimensional Water-Wave Dynamics

Summary. We describe traveling waves in a basic model for three-dimensional water-wave dynamics in the weakly nonlinear long-wave regime. Small solutions that are periodic in the direction of translation (or orthogonal to it) form an infinite-dimensional family. We characterize these solutions through spatial dynamics, by reducing a linearly ill-posed mixed-type initial-value problem to a center manifold of infinite dimension and codimension. A unique global solution exists for arbitrary small initial data for the two-component bottom velocity, specified along a single line in the direction of translation (or orthogonal to it). A dispersive, nonlocal, nonlinear wave equation governs the spatial evolution of bottom velocity. Received July 20, 2001; accepted November 5, 2001     

Lq$L^q$-weak solutions to the time-dependent 3D Oseen system: Decay estimates

This article deals with $L^q$-weak solutions to the 3D time-dependent Oseen system. This type of solution is defined in terms of the velocity only. It is shown that the velocity may be represented by a sum of integrals none of which involves the pressure and without a surface integral of the spatial gradient of the velocity. On the basis of this representation formula, an estimate of the spatial decay of the velocity and its spatial gradient is derived. No boundary conditions have to be imposed for these results.     

Scattering of antiplane shear wave by a propagating crack at the interface of two dissimilar elastic media

An analysis of the scattering of horizontally polarized shear wave by a semi-infinite crack running with uniform velocity along the interface of two dissimilar semi-infinite elastic media has been carried out. The mixed boundary value problem has been solved completely by the Wiener-Hopf technique. The effect of different values of the material parameter, the angle of incidence of incident wave and the crack propagation velocity on the stress intensity factor have been illustrated graphically.     

On the Temporal and the Spatial Kelvin-Helmholtz Instability of a Hypersonic Jet Confined by a Longitudinal Magnetic Field

Both the temporal and the spatial Kelvin-Helmholtz instability of a hypersonic jet confined by a longitudinal magnetic field are investigated. The reflection modes are calculated for a set of typical jet parameters. Analytical treatments for both temporal and spatial resonant modes, i.e., the most important reflection modes with the local maximum growth rates, are given. Various properties of the resonant modes are plotted against the jet temperature. The growth rates of the temporal and the spatial resonant modes are connected by the group velocity. The astrophysical implications of the present investigation are discussed.     

Optimal elliptic regularity at the crossing of a material interface and a Neumann boundary edge

We investigate optimal elliptic regularity of anisotropic div–grad operators in three dimensions at the crossing of a material interface and an edge of the spatial domain on the Neumann boundary part within the scale of Sobolev spaces. Bibliography: 39 titles. Illustration: 8 figures.     

Torsional wave dispersion relations in a pre-stressed bi-material compounded cylinder with an imperfect interface

This paper studies the influence of the imperfectness of the contact condition on the torsional wave propagation in the initially stressed (stretched) bi-material compounded circular cylinder. The investigation is carried out within the scope of the piecewise homogeneous body model with the use of the Three-dimensional Linearized Theory of Elastic Waves in Initially Stresses Bodies. The mathematical formulation of the corresponding eigen-value problem is formulated and the solution method for that is developed. The two cases considered are the bi-material compounded cylinder consists of the solid inner and surrounding hollow outer cylinders (Case 1); the bi-material compounded cylinder consists of the hollow inner and surrounding hollow outer cylinders (Case 2). The mechanical relations of the cylinders’ materials are written through the Murnaghan potential. It is proven that the imperfectness of the contact condition does not influence the asymptotic-limit values of the wave propagation velocity. Moreover, the numerical results on the effects of the imperfectness of the boundary condition on the influence of the initial stresses on the wave propagation velocity are presented and discussed.     

Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows

We consider a phase field model for the flow of two partly miscible incompressible, viscous fluids of non-Newtonian (power law) type. In the model it is assumed that the densities of the fluids are equal. We prove the existence of weak solutions for general initial data and arbitrarily large times with the aid of a parabolic Lipschitz truncation method, which preserves solenoidal velocity fields and was recently developed by Breit, Diening, and Schwarzacher.     

Spatial chaotic vibrations when there is a periodic change in the position of the centre of mass of a body in the atmosphere

The spatial chaotic motion of a blunt body in the atmosphere when there is a periodic change in the position of the centre of mass is considered. A restoring moment, described by a biharmonic dependence on the spatial angle of attack, a small perturbing moment, due to the periodic change in the position of the centre of mass, and also a small damping moment, acts on the body. The motion when the velocity head remains constant is investigated. When there are no small perturbations, the phase portrait of the system can have points of stable and unstable equilibrium. The behaviour of the system in the neighbourhood of the separatrice is investigated using Mel’nikov's method. An analytic solution of the equation of the body motion along the separatrice is obtained. The criteria for the occurrence of chaos are obtained and the results of numerical modelling, which confirm the correctness of the solutions obtained, are presented.     

The quickest transfer of a spatial dynamic object into a circular domain

The spatial problem of the time-optimal transfer of a point mass by a limited force onto a terminal set in the form of a circle without fixing the final velocity is investigated. The optimal modes of motion are constructed and investigated for arbitrary initial values of the three-dimensional position and velocity vectors using the maximum principle. The governing relations are obtained in the form of fourth-order and eighth-order algebraic equations for the minimum time of motion, which enable the dependence on the initial data to be investigated constructively. The qualitative features of the solution due to a jump discontinuity in the minimum time of motion, which lead to jumps in the control vector, are established. The problem is solved approximately by perturbation methods for the cases of motion close to singular ones. A complete investigation of the control problem for the motion of an object in the plane of a circle and close to it is presented using an original numerical-analytical approach.     

Transfer of a Passive Vector Admixture by a Two-Dimensional Turbulent Flow

We consider a model of a passive vector field transfer by a random two-dimensional transverse velocity field that is uncorrelated in time and has Gaussian spatial statistics given by a powerlike correlator. We use the renormalization group and the operator product expansion techniques to show that the asymptotic approximation of the structure functions of a vector field in the inertial range is determined by the energy dissipation fluctuations. The dependence of the asymptotic approximation on the external scale of turbulence is essential and has a powerlike form (the case of an anomalous scaling). The corresponding exponents are calculated in the one-loop approximation for structure functions of an arbitrary order.     

Influence of the gravity on interface shape during crystal growth of LICAF

A Galerkin finite element method, together with the boundary conformal mapping technique, is used to investigate the change of melWcrystal interface under low gravity during the growth of LEAF system. Results have shown that strong convection can cause a deeply concave interface toward the crystal, and significantly increase radial thermal gradients near the interface. The flow intensity and the change of the gravity have a linear relationship under low gravity (g _o u = 10⁻²-10⁻⁶). At smallMa number, the maximum acceleration for keeping a planar growth interface is g_max = 1 × 10⁻³ g under our given conditions. In addition, the growth velocity may have some influence on the growth interface shape even atpg gravity level, indicating that the growth velocity cannot be too fast even when convection is very weak.     

Rapid tearing along an interface

A half-space composed of two joined elastic quarter spaces of different material properties is subjected to anti-plane surface tractions parallel to the interface. We investigate the case that the application of the surface tractions instantaneously gives rise to rupture at the surface trace of the interface. A zone of rupture, or delamination, subsequently propagates at a relatively large velocity as a stress-free crack along the interface. Wave motions in the two quarter spaces are analyzed, and the elastodynamic stress intensity factor of the interface stress is computed. An expression is derived for the rate of energy flux into the propagating crack tip.

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Zusammenfassung In dieser Arbeit wird die Ausbreitung eines Risses untersucht, der bei hoher Geschwindigkeit in der Verbindungsfläche zweier Viertelräume erzeugt wird. Die durch den sich ausbreitenden Riss verursachte Wellenbewegung wird analysiert und die elastodynamischen Intensitätsfaktoren für die Schubspannung in der Verbindungsfläche werden erhalten. Der Energiefluss in das Ende des Risses wird berechnet.

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Dedicated to Professor Heinz Parkus on the occasion of his sixty-fifth birthday

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The efforts of one of the authors (J.D.A.) were sponsored by Grant 71-G155 from the U.S. Army Research Office-Durham to Northwestern University.