Stationary Simple Waves on a Barotropic Liquid Shear Flow

Stationary threedimensional flows of a barotropic liquid in a gravity field are considered. In the shallowwater approximation, the Euler equations are transformed into a system of integrodifferential equations by the EulerLagrange change of coordinates. A system of simplewave equations is obtained, for which the theorem of existence of a solution attached to a given shear flow is proved. As an example, a particular solution analogous to the solution of the problem of a gas flow around a convex angle is given.     

Surface Water Waves and Tsunamis

Because of the enormous earthquake in Sumatra on December 26, 2004, and the devastating tsunami which followed, I have chosen the focus of my mini-course lectures at this year’s PASI to be on two topics which involve the dynamics of surface water waves. These topics are of interest to mathematicians interested in wave propagation, and particularly to Chilean scientists, I believe, because of Chile’s presence on the tectonically active Pacific Rim. My first lecture will describe the equations of fluid dynamics for the free surface above a body of fluid (the ocean surface), and the linearized equations of motion. From this, we can predict the travel time of the recent tsunami from its epicenter off of the north Sumatra coast to the coast of nearby Thailand, the easy coasts of Sri Lanka and south India, and to Africa. In fact, the signal given by ocean waves generated by the Sumatra earthquake was felt globally; within 48 h distinguishable tsunami waves were measured by wave gages in Antarctica, Chile, Rio di Janeiro, the west coast of Mexico, the east coast of the United States, and at Halifax, Nova Scotia. To describe ocean waves, we will formulate the full nonlinear fluid dynamical equations as a Hamiltonian system [19], and we will introduce the Greens function and the Dirichlet-Neumann operator for the fluid domain along with the harmonic analysis of the theory of their regularity. From an asymptotic theory of scaling transformations, we will derive the known Boussinesq-like systems and the KdV and KP equations, which govern the asymptotic behavior of tsunami waves over an idealized flat bottom. When the bottom is no longer assumed to be perfectly flat, a related theory [6, 13] gives a family of model equations taking this into account. My second lecture will describe a series of recent results in PDE, numerical results, and experimental results on the nonlinear interactions of solitary surface water waves. In contrast with the case of the KdV equations (and certain other integrable PDE), the Euler equations for a free surface do not admit clean (‘elastic’) interactions between solitary wave solutions. This has been a classical concern of oceanographers for several decades, but only recently have there been sufficiently accurate and thorough numerical simulations which quantify the degree to which solitary waves lose energy during interactions [3, 4]. It is striking that this degree of ‘inelasticity’ is remarkably small. I will describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon [14, 18].     

Resonant and Stationary Waves in Rotating Disks

The analytical conditions for resonant and stationary waves inrotating disks are presented. These conditions are derived from anonlinear plate theory pertaining to initial configurations and areapplicable to rotating disks with initial waviness and/or undergoinglarge-amplitude displacements. The rotational speeds at which theresonant and stationary waves occur for a 3.5-inch diameter computermemory disk are computed. The resonant waves for linear and nonlinear,rotating disks are simulated numerically. It is found that some diskmodes exhibit a hardening effect under which the rotational speeds forthe resonant and stationary waves increase with increasing waveamplitude, while other modes experience a softening effect with thoserotational speeds decreasing with increasing wave amplitude. Therotating-disk resonant spectrum presented in this paper is relevant tothe disk drive industry for determining the range of operationalrotation speed.     

Flows formed after the passage of a discontinuous wave over a bottom drop

The solvability of the problem of the flow formed after a discontinuous wave has passed over a bottom drop is studied within the framework of the first approximation of shallow water theory. Solutions in which the total energy of the flow is either conserved or lost at the drop are considered. Stable self-similar solutions of five qualitatively different types are derived and their domains of existence are determined in the dimensionless parameter plane.     

Stationary Flows of Shear Thickening Fluids in 2D

We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on a domain in \mathbbR²{\mathbb{R}^{2}}. We assume that the stress tensor is generated by a potential of the form H = h (|e(u)|){H = h (|\varepsilon (u)|)}, e(u){\varepsilon (u)} denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions h having the property that h′ (t)/t increases (shear thickening case).     

Transformation of Long Nonlinear Waves in a Two-Layer Viscous Fluid Between a Gently Sloping Lid and Bottom

Dynamics of three-dimensional disturbances of the interface between two fluid layers of different densities is considered analytically and numerically. An evolutionary integrodifferential equation is derived, which takes into account long-wave contributions of inertia of the layers and surface tension, small but finite amplitude of disturbances of the interface between two incompressible immiscible fluids, gentle slopes of the lid and bottom, and nonstationary shear stresses at all boundaries. Numerical solutions of this model equation for several (most typical) nonlinear problems of transformation of two- and three-dimensional waves are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 45–57, November–December, 2005.     

Nonlinear Viscous Flows with a Free Surface

An asymptotic solution of the problem of time evolution of a periodic wave on the surface of a viscous, infinitely deep fluid in the approximation quadratic in the wave amplitude is proposed.     

Surface Magnetoelastic Shear Waves in Periodic Ferrite-Dielectric Regularly Stratified Structures

Surface magnetoelastic Love waves in a regularly stratified medium with ferrite and dielectric layers are studied based on the Hamilton formalism and numerical procedures. The dispersion relations are analyzed for particular structures     

Nonlinear Waves and the Origin of Bubbles in Fluidized Beds

Early experiments in the mid-1940s established two different regimes of behavior of fluidized systems. These are broadly classified into systems that exhibit massive phase segregation, leading to particle-free regions called bubbles, and those that do not. Explaining the origin of bubbles and of these two regimes has represented both a technological and scientific challenge since then. The late 1960s through the 1970s saw a series of illuminating experiments that established many features of the flow regimes and their characteristics through both flow visualizations and quantitative measurements. Recent numerical and theoretical work has come close the resolving the problem. This paper represents the written version of the talk given at the Symposium in honor of Leen van Wijngaarden's retirement. In it, I review the history of progress on the problem in two giant 25-year steps.     

Liouville Theorems for Stationary Flows of Shear Thickening Fluids in the Plane

We consider entire solutions of the equations for stationary flows of shear thickening fluids in 2D and prove Liouville results under conditions like global boundedness of the velocity field or finiteness of the energy.     

Exact Analytic Solutions of the Nonlinear Long-Wave Equations in the Case of Axisymmetric Fluid Vibrations in a Parabolic Rotating Vessel

A class of exact analytic solutions of the system of nonlinear long-wave equations is found. This class corresponds to the axisymmetric vibrations of an ideal incompressible homogeneous fluid in a rotating vessel in the shape of a paraboloid of revolution. The radial velocity of these motions is a linear function, and the azimuthal velocity and free surface displacements are polynomials in the radial coordinate with time-dependent coefficients. The nonlinear vibration frequency is equal to the frequency of the lowest mode of linear axisymmetric standing waves in the parabolic vessel.     

Linear Dynamics of Perturbations in Flows with Constant Horizontal Shear

The dynamics of perturbations in shallow water and incompressible stratified fluid flows with constant horizontal shear are described using the nonmodal analysis. It is shown that the shear flow perturbations can be divided into two classes on the basis of the potential vorticity: rapidly oscillating wave perturbations with zero potential vorticity and slow vortex perturbations with nonzero potential vorticity. In the cases of weak and strong shear the main features of the dynamics of wave and vortex perturbations are studied analytically (using the WKBG method) and numerically. It is shown that for large times the wave perturbation energy increases linearly, i.e., the shear flow is algebraically unstable due to the growth of rapid wave perturbations. This instability can be of importance in processes of turbulence development and surface and internal wave generation.     

Localization of Shear Waves Near Layers Separating Two Regularly Laminated Half-Spaces

The existence conditions for surface and normal shear waves in finite and infinite periodically laminated structures with broken translational symmetry are studied theoretically and numerically. The problems posed are reduced to systems of linear algebraic equations. The existence condition for their nontrivial solutions yield dispersion relations. Conditions for the existence of surface and normal shear waves are established. Some results are plotted. The dependence of the dispersion spectrum on the physical and geometrical properties of the symmetry breaker is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 128–134, January 2005.     

Dam-Break Flows Over a Bottom Drop

The single-layer shallow-water model is used to study flows generated by dam break over a bed level discontinuity in the form of a drop from which water flows. Emphasis is given to submerged regimes in which downstream wave processes affect the upstream flow. The paper considers solutions in which the total flow energy is conserved on the drop and solutions in which the energy is lost on the drop.     

Dam-Break Flows Over a Bottom Step

A single-layer shallow-water model is used to study the solvability of the problem of flows generated by dam break over a bed level discontinuity in the form of a step onto which water flows. Solutions in which the total flow energy is conserved on the step and solutions in which the energy is lost on the step are considered.     

Nonlinear traveling waves in a compressible Mooney-Rivlin rod I. Long finite-amplitude waves

In literature, nonlinear traveling waves in elastic circular rods have only been studied based on single partial differential equation (pde) models, and here we consider such a problem by using a more accurate coupled-pde model. We derive the Hamiltonian from the model equations for the long finite-amplitude wave approximation, analyze how the number of singular points of the system changes with the parameters, and study the features of these singular points qualitatively. Various physically acceptable nonlinear traveling waves are also discussed, and corresponding examples are given. In particular, we find that certain waves, which cannot be counted by the single-equation model, can arise. The project supported by the Research Grants Council of the HKSAR, China (City U 1107/99P) and the National Natural Science Foundation of China (10372054 and 10171061)     

Hamiltonian formulation of nonlinear water waves in a two-fluid system

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Propagation of Magnetoelastic Shear Waves Through a Regularly Laminated Medium with Metallized Interfaces

An exact solution is found for magnetoelastic shear waves in an infinite structure consisting of three metallized layers. The core layer is ferrite and the face layers are nonmagnetic dielectrics. The wave process in the layers is described by a linearized system of magnetoelastic equations. The problem posed is reduced to a system of linear algebraic equations. The existence conditions for an undamped solution to this system yield the existence conditions for magnetoelastic bulk waves. The dispersion relations derived are analyzed in detail     

A Note on Liouville Theorem for Stationary Flows of Shear Thickening Fluids in the Plane

In this paper we consider the entire weak solutions of the equations for stationary flows of shear thickening fluids in the plane and prove Liouville theorem under the global boundedness condition of velocity fields.     

开放剪切流空间全局动力学与空间相位斑图间的关系

开放流动空间动力学可基于两类全局能量关系式进行研究;而空间相位斑图则可通过互谱空间演化加以测定。全局能量关系式以时间Fourier系数的形式建立流场任意两点问速度脉动能量间的关系,籍此可定义全局意义上的线性、非线性和线性一非线性机制。基于轴对称剪切流、变密度轴对称圆射流以及平面对称剪切流的实验发现：轴对称旋涡结构的配对由线性、线性一非线性机制表征,对应有序空间相位斑图;并且能量可通过线性一非线性机制在具有相同相速度的扰动间传递。螺旋结构由线性机制表征,对应有序相位斑图。全局自激励振荡由非线性的能量共振表征,对应无序相位斑图。籍此,有序空间相位斑图对应线性和线性一非线性机制;而混沌相位斑图则对应非线性机制。