Propagation of Waves in an Elastic Half-Space Interacting with a Viscous Liquid Layer

International Applied Mechanics - The problem of the propagation of acoustic waves in a layer of a compressible viscous fluid that interacts with an elastic half-space is solved using the...     

Propagation of Quasi-Lamb Waves in an Elastic Layer Interacting with a Viscous Liquid Half-Space

International Applied Mechanics - The propagating of quasi-Lamb waves in an elastic layer that interacts with a half-space of a viscous compressible fluid is studied. The three-dimensional...     

Surface Waves for a Compressible Viscous Fluid

In this paper, we are concerned with free boundary problem for compressible viscous isotropic Newtonian fluid. Our problem is to find the three-dimensional domain occupied by the fluid which is bounded below by the fixed bottom and above by the free surface together with the density, the velocity vector field and the absolute temperature of the fluid satisfying the system of Navier-Stokes equations and the initial-boundary conditions. The Navier-Stokes equations consist of the conservations of mass, momentum under the gravitational field in a downward direction and energy. The effect of the surface tension on the free surface is taken into account. The purpose of this paper is to establish two existence theorems to the problem mentioned above: the first concerns with the temporary local solvability in anisotropic Sobolev-Slobodetskiĭ spaces and the second the global solvability near the equilibrium rest state. Here the equilibrium rest state (heat conductive state) means that the temperature distribution is a linear function with respect to a vertical direction and the density is determined by an ordinary differential equation which involves equation of state. For the proof, we rely on the methods due to Solonnikov in the case of incompressible fluid with some modifications, since our problem is hyperbolic-parabolic coupled system. Dedicated to Professors Takaaki Nishida and Masayasu Mimura on their sixtieth birthdays     

Flow Asymptotics of a Viscous Compressible Fluid with Discontinuous Initial Data

Asymptotics of a continuous solution to a plane problem on the motion of a viscous incompressible fluid with discontinuous initial velocity and pressure fields is studied by the boundarylayer method with simultaneous stretching of space and time coordinates.     

Elastic Waves in Bodies with Initial (Residual) Stresses

An analysis is made of the results obtained in the three-dimensional linearized theory of elastic waves propagating in initially stressed solids. Consideration is given to surface waves along planar and curvilinear boundaries and interfaces, waves in layers and cylinders, waves in composite materials, waves in hydroelastic systems, and dynamic problems for moving loads. The results were obtained in exact formulations.     

Nonstationary Axisymmetric Problem for a Half-Space of a Compressible Fluid*

International Applied Mechanics - The exact analytical solution to the axisymmetric problem for a half-space of an ideal compressible fluid under nonstationary pressure is found. The method of...     

Contact Problem for an Elastic Ring Punch and a Half-Space with Initial (Residual) Stresses*

International Applied Mechanics - The problem of contact interaction without friction between an elastic cylindrical ring punch and an elastic half-space with initial (residual) stresses under...     

Instabilities in Dynamic Anti-plane Sliding of an Elastic Layer on a Dissimilar Elastic Half-Space

The stability of dynamic anti-plane sliding at an interface between an elastic layer and an elastic half-space with dissimilar elastic properties is studied. Friction at the interface is assumed to follow a rate- and state-dependent law, with a positive instantaneous dependence on slip velocity and a rate weakening behavior in the steady state. The perturbations at the interface are of the form exp(ikx ₁+pt), where k is the wavenumber, x ₁ is the coordinate along the interface, p is the time response to the perturbation and t is time. A key feature of the problem is that interfacial waves both in freely slipping contact as well as in bonded contact exist for the problem. Attention is focused on the role of the interfacial waves on slip stability. Instabilities are plotted in the $\operatorname{Re} (pL/V_{o})$ versus $\operatorname{Im} (p/|k|c_{s})$ plane, where L is a length scale in the friction law, V _o is the unperturbed slip velocity and c _s is the shear wave speed of the layer. Stability of both rapid and slow slip is studied. The results show one mechanism by which instabilities occur is the destabilization by friction of the interfacial waves in freely slipping contact/bonded contact. This occurs even in slow sliding, thus confirming that the quasi-static approximation is not valid for slow sliding. The effect of material properties and layer thickness on the stability results is studied.     

On the Motion of Rigid Bodies in a Viscous Compressible Fluid

We prove the existence of global-in-time weak solutions to a model describing the motion of several rigid bodies in a viscous compressible fluid. Unlike most recent results of similar type, there is no restriction on the existence time, regardless of possible collisions of two or more rigid bodies and/or a contact of the bodies with the boundary. (Accepted September 23, 2002) Published online February 4, 2003 Communicated by Y. Brenier     

On the Exponential Stability of the Rest State of a Viscous Compressible Fluid

We prove that the rest state of a viscous isothermal gas filling a bounded rigid vessel, is exponentially stable with respect a large class of "weak" perturbations that, in particular, allow for supersonic flow and discontinuous densities. In the inviscid limit, marginal stability is recovered.     

Hopf Bifurcation of Viscous Shock Waves in Compressible Gas Dynamics and MHD

Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier–Stokes and magnetohydrodynamics equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in H ^s for data in H ^s . We accomplish this by a novel energy estimate carried out in Lagrangian coordinates; interestingly, this estimate is false in Eulerian coordinates. At the same time, we greatly sharpen and simplify the analysis of the previous work. Research of B.T. was partially supported under NSF grant number DMS-0505780. Research of K.Z. was partially supported under NSF grant number DMS-0300487.