Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics

We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is equivalent to the problem of describing all compatible Dubrovin–Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lamé equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lamé equations (a version of the inverse scattering method).     

A novel far-boundary condition for the finite element analysis of infinite reservoir

The present paper deals with the finite element analysis of the reservoir of infinite extent using a novel far-boundary condition. The equations of motion are expressed in terms of the pressure only assuming water as inviscid and incompressible. The truncation boundary condition is developed numerically from the classical wave equation. Comparative studies show that the proposed far-boundary condition is numerically efficient and accurate over the existing ones, available in the literature. The effect of the geometry of the reservoir bed and the adjacent structure on the development hydrodynamic pressure has been studied. The results show that the geometry of the reservoir bed and as well as the adjacent structure has considerable effect on the development of hydrodynamic pressure at the dam–reservoir interface.     

Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits

The present paper proves the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a quantum hydrodynamic model of semiconductors over a one-dimensional bounded domain. We also discuss on a singular limit from this model to a classical hydrodynamic model without quantum effects. Precisely, we prove that a solution for the quantum model converges to that for the hydrodynamic model as the Planck constant tends to zero. Here we adopt a non-linear boundary condition which means quantum effect vanishes on the boundary. In the previous researches, the existence and the asymptotic stability of a stationary solution are proved under the assumption that a doping profile is flat, which makes the stationary solution also flat. However, the typical doping profile in actual devices does not satisfy this assumption. Thus, we prove the above theorems without this flatness assumption. Firstly, the existence of the stationary solution is proved by the Leray-Schauder fixed-point theorem. Secondly, we show the asymptotic stability theorem by using an elementary energy method, where the equation for an energy form plays an essential role. Finally, the classical limit is considered by using the energy method again.     

Hydrodynamic Capture and Release of Passively Driven Particles by Active Particles Under Hele-Shaw Flows

The transport of active and passive particles plays central roles in diverse biological phenomena and engineering applications. In this paper, we present a theoretical investigation of a system consisting of an active particle and a passive particle in a confined micro-fluidic flow. The introduction of an external flow is found to induce the capture of the passive particle by the active particle via long-range hydrodynamic interactions among the particles. This hydrodynamic capture mechanism relies on an attracting stable equilibrium configuration formed by the particles, which occurs when the external flow intensity exceeds a certain threshold. We evaluate this threshold by studying the stability of the equilibrium configurations analytically and numerically. Furthermore, we study the dynamics of typical capture and non-capture events and characterize the basins of attraction of the equilibrium configurations. Our findings reveal a critical dependence of the hydrodynamic capture mechanism on the external flow intensity. Through adjusting the external flow intensity across the stability threshold, we demonstrate that the active particle can capture and release the passive particle in a controllable manner. Such a capture-and-release mechanism is desirable for biomedical applications such as the capture and release of therapeutic payloads by synthetic micro-swimmers in targeted drug delivery.     

Dynamik von verankerten meerestechnischen Konstruktionen

The dynamical behaviour of moored structures in ocean engineering in an irregular sea is usually investigated by means of simplified mathematical models. Nonlinear phenomena like the influence of the motion on the mooring geometry or the fluid‐structure interaction are not considered. In this paper a method is described for analyzing the dynamical behaviour of mooring systems including the hydrodynamic forces due to the fluid surrounding the structure. The model for the dynamic analysis is generated using the multibody system method.     

On the Spectrum of the Orr–Sommerfeld Equation on the Semiaxis

The Orr–Sommerfeld equation is a spectral problem which is known to play an important role in hydrodynamic stability. For an appropriate operator theoretical realization of the equation, we will determine the essential spectrum, and calculate an enclosure of the set of all eigenvalues by elementary analytical means.     

Numerical exterior algebra and the compound matrix method

Summary. The compound matrix method, which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k=2,3 dimensional subspaces of , by using compound matrices as coordinates, is reformulated in a coordinate-free way using exterior algebra spaces, . This formulation leads to a general framework for studying systems of differential equations on k-dimensional subspaces. The framework requires the development of several new ideas: the role of Hodge duality and the Hodge star operator in the construction, an efficient strategy for constructing the induced differential equations on , general formulation of induced boundary conditions, the role of geometric integrators for preserving the manifold of k-dimensional subspaces – the Grassmann manifold, , and a formulation for induced systems on an unbounded interval. The numerical exterior algebra framework is most advantageous for numerical solution of differential eigenvalue problems on unbounded domains, where there are significant difficulties in setting up matrix discretizations. The formulation is presented for k-dimensional subspaces of systems on with k and n arbitrary, and examples are given for the cases of k=2 and n=4, and k=3 and n=6, with an indication of implementation details for systems of larger dimension. The theory is illustrated by application to four differential eigenvalue problems on unbounded intervals: hydrodynamic stablity of boundary-layer flow past a compliant surface, the eigenvalue problem associated with the stability of solitary waves, the stability of Bickley jet in oceanography, and the eigenvalue problem associated with the stability of the Ekman layer in atmospheric dynamics. Received February 2, 2001 / Revised version received May 28, 2001 / Published online October 17, 2001     

Global Existence and Exponential Stability of Smooth Solutions to a Full Hydrodynamic Model to Semiconductors

 We study a full hydrodynamic semiconductor model in multi-space dimension. The global existence of smooth solutions is established and the exponential stability of the solutions as is investigated.     

Control of mechanical systems on Lie groups and ideal hydrodynamics

In contrast to the Euler–Poincaré reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is the velocity field for a stationary flow of an ideal fluid. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. Now we consider a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant with respect to left translations on this group, and assume that the mass geometry f the system may change under the action of internal control forces. Such a system can also be reduced to a Lie group. Without controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and, therefore, its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. We show that under these conditions, by changing the mass geometry, one can also bring one vortex manifold to any other preassigned vortex manifold. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

On upper bounds for the growth rate in the extended Taylor-Goldstein problem of hydrodynamic stability

For the extended Taylor-Goldstein problem of hydrodynamic stability governing the stability of shear flows of an inviscid, incompressible but density stratified fluid in sea straits of arbitrary cross-section a new estimate for the growth rate of an arbitrary unstable normal mode is given for a class of basic flows. Furthermore the Howard’s conjecture, namely, the growth rate kc _i → 0 as the wave number k → ∞ is proved for two classes of basic flows.     

张量封闭模型与三维取向分布对纤维悬浮槽流稳定性的影响

应用3种不同的纤维方向张量封闭模型,数值模拟了纤维悬浮槽流的流动稳定性问题,从而研究封闭模型和纤维的三维取向分布对稳定性分析的影响．结果发现,采用3种不同封闭模型所得到的流动稳定特性与纤维参数之间的关系是相同的,但采用三维混合封闭模型时,由于纤维的取向与流向的偏差程度较大,所以纤维对流动的不稳定性具有最强的抑制作用．而采用二维混合封闭模型时,由于纤维在平面取向条件下,其轴线整体上趋于呈流向排列,使得对流体的作用削弱,导致纤维对流动不稳定性抑制的作用最弱．     

Linearized hydrodynamic stability of a viscous liquid in an open container

Linearized hydrodynamic stability of an incompressible viscous liquid in an open container, subject to a constant external acceleration, is studied. A normal mode analysis is employed to show that initial perturbations of the flow and the free surface tend to zero in time.     

血液锥形分离杯内血液流动稳定性的边界摄动解

本文首次用边界摄动方法求解了二相同角速度转动的圆锥面间(窄间隙)均质流体的流动,和锥面间流动稳定性,证实了在小轴向Re数、窄间隙血液分离杯中血液流动的稳定性.并进而采用一种新的数学技巧证实了二相同角速度转动的圆柱面间(窄间隙)流动流体动力稳定性. 理论分析得到上海医疗器械研究所实验的证实.     

A Diffusion Equation Illustrating Spectral Theory for Boundary Layer Stability

The stability of a special solution of a nonlinear diffusion equation is examined. It is shown that the spectral resolution of the linearized stability equation is best accomplished by requiring sufficiently strong decay of the eigenfunctions at infinity. The techniques employed illustrate what is involved in determining the spectral resolution for the equations of hydrodynamic stability when the boundary layer has a transverse component at infinity.     

Continuous dependence on the time and spatial geometry for the equations of thermoelasticity

In this paper we establish the stability of the solution of the standard initial-boundary value problem of linear anisotropic thermoelasticity under perturbations of the initial time geometry and of the spatial geometry. This is done by deriving appropriate explicit a priori inequalities which permit us to bound in particular the L₂ integral of the perturbation in terms of some well defined measure of the perturbation in the geometry.     

On the Whitham system for the (2+1)-dimensional nonlinear Schrödinger equation

Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)-dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann-type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.     

The geometry and number of the root invariant regions for linear systems

The stability domain is a feasible set for numerous optimization problems. D-decomposition technique is targeted to describe the stability domain in the parameter space for linear parameter-dependent systems. This technique is very simple and efficient for robust stability analysis and design of low-order controllers. However, the geometry of the arising parameter space decomposition into root invariant regions has not been studied in detail; it is an objective of the present paper. We estimate the number of root invariant regions and provide examples, where this number is attained.     

Classification of the Associativity Equations with A First-Order Hamiltonian Operator

We study the Hamiltonian geometry of systems of hydrodynamic type that are equivalent to the associativity equations in the case of three primary fields and obtain the complete classification of the associativity equations with respect to the existence of a first-order Dubrovin–Novikov Hamiltonian structure.     

Global Existence and Exponential Stability of Smooth Solutions to a Full Hydrodynamic Model to Semiconductors

 We study a full hydrodynamic semiconductor model in multi-space dimension. The global existence of smooth solutions is established and the exponential stability of the solutions as is investigated. Received November 14, 2000; in revised form March 25, 2002 Published online August 5, 2002     

Linear Stability of Pipe Poiseuille Flow at High Reynolds Number Regime

In this paper, we prove the linear stability of the pipe Poiseuille flow for general perturbations at high Reynolds number regime. This has been a long-standing problem since the experiments of Reynolds in 1883. Our work lays a foundation for the theoretical analysis of hydrodynamic stability of pipe flow, which is one of the oldest yet unsolved problems in fundamental fluid dynamics. © 2022 Wiley Periodicals LLC.