Asymptotic and numerical solutions of three‐dimensional boundary‐layer flow past a moving wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows.     

二维可压缩油水两相渗流动边值问题的特征差分方法

可压缩可混溶油、水两相渗流动边值问题的研究,对重建盆地发育中油气资源运移、聚集的历史和评估油气资源的勘探与开发有重要的价值,其数学模型是一组非线性偶合偏微分方程组的动边值问题．本文对二维有界域的两类边值问题提出一类新的特征差分格式,应用区域变换、时间的变步长、粗细网格配套、变分形式、先验估计的理论和技巧,得到了最佳阶l2误差估计结果．将J．Douglas,Jr．提出的著名方法,成功地拓广到这一新领域,并得到实质性进展．它对这一领域的模型分析,数值方法和软件研制均有重要的价值．     

On the Geometry of Complex‐Lamellar Magnetohydrodynamics: Universal Motions

A geometric formulation is adopted for a nonlinear magnetohydrodynamic system wherein the magnetic field is aligned with the direction of the binormal to the streamlines. It is established that, for complex‐lamellar motion, if the divergence of the binormal field vanishes then the fluid streamlines are geodesics on generalized helicoids. The latter constitute the Maxwellian surfaces and the magnetic lines are helices thereon. The key geometric and physical parameters of the magneto‐hydrodynamic motion are all determined in terms of the torsion τ of the streamlines. A superposition principle is presented whereby a more general class of magnetohydrodynamic motions may be isolated with streamlines and magnetic lines no longer restricted to be geodesics or parallels on the Maxwellian surfaces. Moreover, the class so generated is not subject to the complex‐lamellar constraint. In an appendix, a particular reduction is obtained to the integrable Da Rios system.     

Dynamics of the Chaplygin sleigh on a cylinder

This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found.In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.     

Hamiltonian chaos in the interaction of moving two-level atoms with cavity vacuum

We study nonlinear dynamics of the fundamental cavity quantum-electrodynamical system consisting of a point-like collection of identical two-level atoms moving through a lossless single-mode cavity. Taking into account the interatomic and the atom-field quantum correlations of the first order, we go beyond the semiclassical model and derive a dynamical system that is able to describe the vacuum Rabi oscillations with atoms moving in a spatially inhomogeneous cavity field. A simple expression for the equilibrium points of this system provides a class of initial conditions for atoms and a cavity mode under which the atomic population and radiation may be trapped. In the strong-coupling limit and the rotating-wave approximation, the model is shown to be integrable with atoms moving through a resonant cavity with an arbitrary spatial profile of the mode along the propagation axis. The general exact solution is derived in an explicit form in terms of Jacobian elliptic functions. Numerical simulation confirms that perturbations, that are produced by a modulation of the coupling between moving atoms and a cavity mode, provide, out of resonance, a mechanism responsible for Hamiltonian chaos in the interaction of two-level atoms with cavity vacuum. These chaotic vacuum Rabi oscillations may be considered as a new kind of reversible spontaneous emission.     

A sharp estimate of the extinction time for the mean curvature flow

We establish a pointwise comparison result for a nonlinear degenerate elliptic Dirichlet problem using an isoperimetric inequality involving the total mean curvature. In particular this result provides a sharp estimate for the extinction time of a class of compact surfaces, wider than the convex one, moving by mean curvature flow. Finally we present numerical experiments to compare our estimate with those known in literature.     

Metric nonlinear connections

For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange metric. For the particular case when the system of SODE and the metric structure are Lagrangian, we prove that the canonical nonlinear connection of the Lagrange space is the only nonlinear connection which is metric and compatible with the symplectic structure of the Lagrange space. For this particular case, the metric tensor determines the symmetric part of the canonical nonlinear connection, while the symplectic structure determines the skew-symmetric part of the nonlinear connection.     

The acoustic field of a high-frequency source moving in a waveguide

The acoustic field of a source moving at a subsonic velocity in a regular waveguide with perfectly reflecting boundaries is considered. The acceleration of the source is assumed to be small. In a moving coordinate system, the asymptotics of the wave field is obtained. This asymptotics is inapplicable near the critical cross sections, for which the Doppler frequency of the source coincides with the frequency of the waveguide mode under consideration. It is demonstrated that, in this case, the wave field can be represented locally by a special type of integral, which is analyzed by the saddle-point method.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 71–78.This work was supported in part by the Russian Foundation for Basic Research under grant 95-01-01285a.     

On the existence of leapfrogging pair of circular vortex filaments

We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. As a preliminary analysis, we first consider the case when the filaments are arranged as straight and parallel lines, and explicitly solve the system to show that the motion of the lines resemble that of point vortices moving in a plane. Then, we consider the motion of a pair of coaxial circular vortex filaments. We show that in this case, the system can be reduced to a two‐dimensional Hamiltonian system. Based on this formulation, we give a condition for the initial configuration and parameters of the filaments for leapfrogging to occur, and prove that the condition is in fact necessary and sufficient.     

Buoyant Poiseuille–Couette flow with viscous dissipation in a vertical channel

Steady combined forced and free convection is investigated in a vertical channel having a wall at rest and a moving wall subjected to a prescribed shear stress. The moving wall is thermally insulated, while the wall at rest is kept at a uniform temperature. The analysis deals with the fully–developed parallel flow regime. The governing equations yield a boundary value problem, that is solved analytically by employing a power series expansion of the velocity field with respect to the transverse coordinate. It is shown that the nonlinear interplay between buoyancy and viscous dissipation may determine the existence of dual solutions of the boundary value problem corresponding to fixed values of the applied shear stress on the moving wall and of the hydrodynamic pressure gradient. It is shown that a nontrivial fully separated flow may occur such that the hydrodynamic pressure gradient is zero and the shear stress vanishes on both walls. E. Magyari: On leave from Institute of Building Technology, ETH – Zürich     

Low Dimensional Born-Infeld Equations Coupled with a Collisionless Matter Model

We consider the Born-Infeld nonlinear electromagnetic field equations and study its Cauchy problem in the case that the Vlasov equation is considered as a matter model. In the present paper, the Vlasov equation is considered on the so-called one and one-half dimensional phase space, and in consequence the Born-Infeld equations are reduced to a quasilinear hyperbolic system with two unknowns. A transformation is introduced in order to make the field equations easy to handle, and suitable assumptions are made on initial data so that the nonlinearity of the field is controlled.     

Bifurcation analysis of a viscoelastic fluid heated from below

Convection of a viscoelastic fluid in a square domain heated from below is investigated for the case of nondeformable free surfaces. To describe the rheological behavior of the fluid the generalized Oldroyd model is used. A weakly nonlinear analysis is performed in order to determine the character of branching for both the monotonic and oscillatory modes. We also perform a reduction of the boundary value-problem to the set of nonlinear amplitude equations. The analysis of this dynamic system demonstrates the onset and competition of five convection modes.     

Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases

We consider problems for the nonlinear Boltzmann equation in the framework of two models: a new nonlinear model and the Bhatnagar-Gross-Krook model. The corresponding transformations reduce these problems to nonlinear systems of integral equations. In the framework of the new nonlinear model, we prove the existence of a positive bounded solution of the nonlinear system of integral equations and present examples of functions describing the nonlinearity in this model. The obtained form of the Boltzmann equation in the framework of the Bhatnagar-Gross-Krook model allows analyzing the problem and indicates a method for solving it. We show that there is a qualitative difference between the solutions in the linear and nonlinear cases: the temperature is a bounded function in the nonlinear case, while it increases linearly at infinity in the linear approximation. We establish that in the framework of the new nonlinear model, equations describing the distributions of temperature, concentration, and mean-mass velocity are mutually consistent, which cannot be asserted in the case of the Bhatnagar-Gross-Krook model.     

Vector acoustic solitons from the coupling of long and short waves in a paramagnetic crystal

We investigate the propagation of a longitudinal-transverse elastic pulse in a statically deformed crystal containing paramagnetic impurities and placed in an external magnetic field. We derive a system of three nonlinear wave equations describing the interaction of the pulse with the paramagnetic impurities in the quasiresonance approximation in the Faraday geometry. We assume that the transverse components of the pulse, which cause quantum transitions, have carrier frequencies and are short-wave (acoustic), while the longitudinal component has no carrier frequency and is long-wave. We show that in the case of an equilibrium initial distribution of populations of quantum levels of paramagnetic impurities, the coupling between the longitudinal and transverse components is weak, the pulse is therefore strictly transverse, and its dynamics are described by the Manakov system. With a nonequilibrium initial distribution of populations, conditions of effective interaction between all components of the elastic pulse can be reached, and their nonlinear dynamics are described by a vector generalization of the Zakharov equations. In the case of a unidirectional propagation of the pulse, these equations reduce to the Yajima-Oikawa vector system. We show that the obtained system of equations and its version with an arbitrary number of short-wave components can be integrated using the inverse scattering transform. We construct infinite hierarchies of solutions of the Yajima-Oikawa vector system (including a solution on a nontrivial background). We consider stationary (complex-valued Garnier system) and self-similar reductions of that system, also admitting a representation in the form of compatibility conditions.     

Invariants and Bonnet-type theorem for surfaces in ℝ4

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.     

Vector breathers in the Manakov system

We study theoretically the nonlinear interactions of vector breathers propagating on an unstable wavefield background. As a model, we use the two-component extension of the one-dimensional focusing nonlinear Schrödinger equation—the Manakov system. With the dressing method, we generate the multibreather solutions to the Manakov model. As shown previously in [D. Kraus, G. Biondini, and G. Kovačič, Nonlinearity 28(9), 3101, (2015)], the class of vector breathers is presented by three fundamental types I, II, and III. Their interactions produce a broad family of the two-component (polarized) nonlinear wave patterns. First, we demonstrate that the type I and the types II and III correspond to two different branches of the dispersion law of the Manakov system in the presence of the unstable background. Then, we investigate the key interaction scenarios, including collisions of standing and moving breathers and resonance breather transformations. Analysis of the two-breather solution allows us to derive general formulas describing phase and space shifts acquired by breathers in mutual collisions. The found expressions enable us to describe the asymptotic states of the breather interactions and interpret the resonance fusion and decay of breathers as a limiting case of infinite space shift in the case of merging breather eigenvalues. Finally, we demonstrate that only type I breathers participate in the development of modulation instability from small-amplitude perturbations withing the superregular scenario, while the breathers of types II and III, belonging to the stable branch of the dispersion law, are not involved in this process.     

动边值问题的混合元方法和数值分析

油气运移、聚集史软件的功能是重建盆地发育中油气资源运移、聚集的历史。其数学模型是一组非线性耦合偏微分方程组的动边值问题。本文对二维、有界区域的一般情况，提出一类特征混合元逼近并得到了逼近解的最佳Ｌ＾２模误差估计。     

The geometrically nonlinear dynamic responses of simply supported beams under moving loads

This paper presents a method for determining the nonlinear dynamic responses of structures under moving loads. The load is considered as a four degrees-of-freedom system with linear suspensions and tires flexibility, and the structure is modeled as an Euler–Bernoulli beam with simply supported at both ends. The nonlinear dynamic interaction of the load–structure system is discussed, and Kelvin−Voigt material model is employed for the beam. The nonlinear partial differential equations of the dynamic interaction are derived by using the von Kármán nonlinear theory and D'Alembert's principle. Based on the Galerkin method, the partial differential equations of the system are transformed into nonlinear ordinary equations, which can be solved by using the Newmark method and Newton−Raphson iteration method. To validate the approach proposed in this paper, the comparison are performed using a moving mass and a moving oscillator as the excitation sources, and the investigations demonstrate good reliability.     

Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions

In this paper we consider quasilinear Keller–Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite time of solutions) finite-time blowup result. In the case of a cross-diffusion system we give results which are optimal provided one assumes some proper non-decay of a nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross-diffusion terms. To this end we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller–Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.     

Interaction of Two Charges in a Uniform Magnetic Field: II. Spatial Problem

The interaction of two charges moving in ℝ³ in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry, we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either “bounce-back”, “pass-through”, or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into “free” and “trapped” parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories that come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low-order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies, an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.