Asymptotic Analysis of a Bingham Fluid in a Thin Domain with Fourier and Tresca Boundary Conditions

In this paper we prove first the existence and uniqueness results for the weak solution, to the stationary equations for Bingham fluid in a three dimensional bounded domain with Fourier and Tresca boundary condition; then we study the asymptotic analysis when one dimension of the fluid domain tends to zero. The strong convergence of the velocity is proved, and a specific Reynolds limit equation and the limit of Tresca free boundary conditions are obtained.     

Global stability analysis of fluid flows using sum-of-squares

This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed optimization methods based on sum-of-squares decomposition to construct a polynomial Lyapunov function. We then show how these methods can be extended to infinite dimensional Navier-Stokes systems using robust optimization techniques. Crucially, this extension requires only the solution of infinite-dimensional linear eigenvalue problems and finite-dimensional sum-of-squares optimization problems.We further show that subject to minor technical constraints, a general polynomial Lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower-bound on the maximum Reynolds number for which a flow is globally stable, if the flow does remain globally stable for Reynolds numbers at least slightly beyond the energy stability limit. Such polynomial functions can be searched for efficiently using the SOS technique we propose.     

Global chaos synchronization of coupled parametrically excited pendula

In this paper, we study the synchronization behaviour of two linearly coupled parametrically excited chaotic pendula. The stability of the synchronized state is examined using Lyapunov stability theory and linear matrix inequality (LMI); and some sufficient criteria for global asymptotic synchronization are derived from which an estimated critical coupling is determined. Numerical solutions are presented to verify the theoretical analysis. We also examined the transition to stable synchronous state and show that this corresponds to a boundary crisis of the chaotic attractor.     

Stability of impulsively-driven natural convection with unsteady base state: implications of an adiabatic boundary

Stability conditions of a quiescent, horizontally infinite fluid layer with adiabatic bottom subject to sudden cooling from above are studied. Here, at difference from Rayleigh-Bénard convection, the temperature base state is never steady. Instability limits are studied using linear analysis while stability is analyzed using the energy method. Critical stability curves in terms of Rayleigh numbers and convection onset times were obtained for several kinematic boundary conditions. Stability curves resulting from energy and linear approaches exhibit the same temporal growth rate for large values of time, suggesting a bound for the temporal asymptotic behavior of the energy method.     

Linear Stability in an Ideal Incompressible Fluid

 We give an explicit construction of approximate eigenfunctions for a linearized Euler operator in dimensions two and three with periodic boundary conditions, and an estimate from below for its spectral bound in terms of an appropriate Lyapunov exponent. As a consequence, we prove that in dimension 2 the spectral and growth bounds for the corresponding group are equal. Therefore, the linear hydrodynamic stability of a steady state for the Euler equations in dimension 2 is equivalent to the fact that the spectrum of the linearized operator is pure imaginary. In dimension 3 we prove the estimate from below for the spectral bound that implies the same equality for every example where the relevant Lyapunov exponents could be effectively computed. For the kinematic dynamo operator describing the evolution of a magnetic field in an ideally conducting incompressible fluid we prove that the growth bound equals the spectral bound in dimensions 2 and 3. Received: 20 May 2002 / Accepted: 5 September 2002 Published online: 10 January 2003 RID="*" ID="*" The first author was partially supported by the Twinning Program of the National Academy of Sciences and National Science Foundation, and by the Research Council and Research Board of the University of Missouri. RID="**" ID="**" The second author was partially supported by the National Science Foundation grant DMS 9876947 and CRDF grant RM1-2084. Acknowledgements. The authors thank Susan Friedlander for useful discussions. Communicated by P. Constantin     

Shear flow, phase change and matched asymptotic expansions: Pattern formation in mushy layers

In many scientific and engineering problems the solidification of an alloy leads to a highly convoluted crystalline matrix modeled as a thermodynamically controlled reactive porous medium called a mushy layer. We analyze the interaction of an external shear flow with a solidifying mushy layer through a corrugated mush-liquid interface. We find that the external flow can drive forced convective motions within the mushy layer resulting in the formation of a pattern of dissolution and solidification features transverse to the overall flow. Here we seek to lay bare the underlying processes through a systematic comparison of matched asymptotic expansions and numerical solutions. The success of our modeling effort draws substantially upon understanding gleaned from the fluid mechanics of boundary layers and the theory of multi-component solidification. The results have a broad range of implications in geophysics and materials science.     

Clustering by mixing flows

We calculate the Lyapunov exponents for particles suspended in a random three-dimensional flow, concentrating on the limit where the viscous damping rate is small compared to the inverse correlation time. In this limit Lyapunov exponents are obtained as a power series in epsilon, a dimensionless measure of the particle inertia. Although the perturbation generates an asymptotic series, we obtain accurate results from a Padé-Borel summation. Our results prove that particles suspended in an incompressible random mixing flow can show pronounced clustering when the Stokes number is large and we characterize two distinct clustering effects which occur in that limit.     

Convective heat transfer and MHD effects on Casson nanofluid flow over a shrinking sheet

Current study examines the magnetohydrodynamic (MHD) boundary layer flow of a Casson nanofluid over an exponentially permeable shrinking sheet with convective boundary condition. Moreover, we have considered the suction/injection effects on the wall. By applying the appropriate transformations, system of non-linear partial differential equation along with the boundary conditions are transformed to couple non-linear ordinary differential equations. The resulting systems of non-linear ordinary differential equations are solved numerically using Runge-Kutta method. Numerical results for velocity, temperature and nanoparticle volume concentration are presented through graphs for various values of dimensionless parameters. Effects of parameters for heat transfer at wall and nanoparticle volume concentration are also presented through graphs and tables. At the end, fluid flow behavior is examined through stream lines. Concluding remarks are provided for the whole analysis.     

Exact solution for the fractional cable equation with nonlocal boundary conditions

The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.     

Stability and vibration of a helical rod with circular cross section in a viscous medium

The stability and vibration of a thin elastic helical rod with circular cross section in a viscous medium are discussed. The dynamical equations of the rod in the viscous medium are established in the Frenet coordinates of the centreline with the Euler angles describing the attitudes of the cross section as variables. We have proved that the Lyapunov and Euler conditions of stability of a helical rod in the space domain are the necessary conditions for the asymptotic stability of the rod in the time domain. The free frequencies and damping coefficients of torsional and flexural vibrations of the helical rod in the viscous medium are calculated.     

Existence and Asymptotic Stability of Periodic Solutions of the Reaction–Diffusion Equations in the Case of a Rapid Reaction

A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.     

Invariant slow manifold approach for exact dynamic inversion of singularly perturbed linear mechanical systems with admissible output constraints

We propose an approach for the exact dynamic inversion of singularly perturbed second-order linear systems through asymptotic expansion in a singular parameter. We show that the inversion solution, corresponding to the invariant slow manifold, can be expressed as a converging infinite series under desired output constraints composed of exponential support functions in the complex domain. We provide systematic mathematical procedures to obtain the closed-form invariant slow manifold, along with required admissible boundary conditions. Numerical examples are given to validate the proposed approach.     

On the global asymptotic stability of the chemical rate equations for irradiation-produced point defects

We prove that a Lyapunov function, for the chemical rate equations for irradiation-produced point defects, exists which ensures their global asymptotic stability.     

Brouwer?s problem on a heavy particle in a rotating vessel: Wave propagation, ion traps, and rotor dynamics

In 1918 Brouwer considered stability of a heavy particle in a rotating vessel. This was the first demonstration of a rotating saddle trap which is a mechanical analogue for quadrupole particle traps of Penning and Paul. We revisit this pioneering work in order to uncover its intriguing connections with classical rotor dynamics and fluid dynamics, stability theory of Hamiltonian and non-conservative systems as well as with the modern works on crystal optics and atomic physics. In particular, we find that the boundary of the stability domain of the undamped Brouwer?s problem possesses the Swallowtail singularity corresponding to the quadruple zero eigenvalue. In the presence of dissipative and non-conservative positional forces there is a couple of Whitney umbrellas on the boundary of the asymptotic stability domain. The handles of the umbrellas form a set where all eigenvalues of the system are pure imaginary despite the presence of dissipative and non-conservative positional forces.     

A Cahn-Hilliard model in a domain with non-permeable walls

We consider in this article a Cahn-Hilliard model in a bounded domain with non-permeable walls, characterized by dynamic-type boundary conditions. Dynamic boundary conditions for the Cahn-Hilliard system have recently been proposed by physicists in order to account for the interactions with the walls in confined systems and are obtained by writing that the total bulk mass is conserved and that there is a relaxation dynamics on the boundary. However, in the case of non-permeable walls, one should also expect some mass on the boundary. It thus seems more realistic to assume that the total mass, in the bulk and on the boundary, is conserved, which leads to boundary conditions of a different type. For the resulting mathematical model, we prove the existence and uniqueness of weak solutions and study their asymptotic behavior as time goes to infinity.     

Rotational friction coefficient of a permeable cylinder in a viscous fluid

An exact expression is derived for the rotational friction coefficient of a cylinder of infinite length and constant permeability immersed in an incompressible viscous fluid. An asymptotic expression for the translational friction coefficient of a permeable cylinder moving in a sheet of viscous fluid embedded on both sides in a fluid of much lower viscosity is also given.     

Stationary Waves for the Discrete Boltzmann Equation in the Half Space with Reflective Boundaries

The present paper is concerned with stationary solutions for discrete velocity models of the Boltzmann equation with reflective boundary condition in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of stationary solutions satisfying the reflective boundary condition as well as the spatially asymptotic condition given by a Maxwellian state. First, the sufficient condition is obtained for the linearized system. Then, this result is applied to prove the existence theorem for the nonlinear equation through the contraction mapping principle. Also, it is shown that the stationary solution approaches the asymptotic Maxwellian state exponentially as the spatial variable tends to infinity. Moreover, we show the time asymptotic stability of the stationary solutions. In the proof, we employ the standard energy method to obtain a priori estimates for nonstationary solutions. The exponential convergence at the spatial asymptotic state of the stationary solutions gives essential information to handle some error terms. Then we discuss some concrete models of the Boltzmann type as an application of our general theory. Received: 7 July 1999 / Accepted: 3 November 1999     

On the Global Existence and Stability of a Multi-Dimensional Supersonic Conic Shock Wave

We establish the global existence and stability of a three-dimensional supersonic conic shock wave for a compactly perturbed steady supersonic flow past an infinitely long circular cone with a sharp angle. The flow is described by a 3-D steady potential equation, which is multi-dimensional, quasilinear, and hyperbolic with respect to the supersonic direction. Making use of the geometric properties of the pointed shock surface together with the Rankine–Hugoniot conditions on the conic shock surface and the boundary condition on the surface of the cone, we obtain a global uniform weighted energy estimate for the nonlinear problem by finding an appropriate multiplier and establishing a new Hardy-type inequality on the shock surface. Based on this, we prove that a multi-dimensional conic shock attached to the vertex of the cone exists globally when the Mach number of the incoming supersonic flow is sufficiently large. Moreover, the asymptotic behavior of the 3-D supersonic conic shock solution, which is shown to approach the corresponding background shock solution in the downstream domain for the uniform supersonic constant flow past the sharp cone, is also explicitly given.     

Two-dimensional Euler flows in slowly deforming domains

We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow compared to the fluid motion. The Eulerian flow is found to remain approximately steady throughout the evolution. At leading order, the velocity field depends instantaneously on the shape of the domain boundary, and it is determined by the steadiness and vorticity-preservation conditions. This is made explicit by reformulating the problem in terms of an area-preserving diffeomorphism g_Λ which transports the vorticity. The first-order correction to the velocity field is linear in the boundary velocity, and we show how it can be computed from the time derivative of g_Λ.The evolution of the Lagrangian position of fluid particles is also examined. Thanks to vorticity conservation, this position can be specified by an angle-like coordinate along vorticity contours. An evolution equation for this angle is derived, and the net change in angle resulting from a cyclic deformation of the domain boundary is calculated. This includes a geometric contribution which can be expressed as the integral of a certain curvature over the interior of the circuit that is traced by the parameters defining the deforming boundary.A perturbation approach using Lie series is developed for the computation of both the Eulerian flow and geometric angle for small deformations of the boundary. Explicit results are presented for the evolution of nearly axisymmetric flows in slightly deformed discs.     

SPH方法进出口数值边界条件研究

基于虚粒子概念,提出进出口数值边界条件处理方法.在进口边界外设置进口区域,并赋予进口虚粒子相应的物理量;出口边界外设置出口和缓冲区域,出口虚粒子的物理量由计算获得,缓冲区的虚粒子初始物理量是指定的;根据每个时间步的流动情况,改变粒子的区域属性,添加/删除相应的粒子.利用拉格朗日形式的SPH方法,通过管内流动问题验证数值进出口边界方法的适用性,研究进出口边界条件在激波管、绕流问题中的应用.提出的进出口边界处理方法,避免了边界附近流体粒子积分截断问题,保证流体粒子能够流出边界,激波能够透射边界.