Strong convergence to global solutions for multidimensional flows of compressible,viscous fluids with polytropic equations of state and discontinuous initial data

We extend to general polytropic pressures P() = K, > 1, the existence theory of [8] for isothermal (= 1) flows of Navier-Stokes fluids in two and three space dimensions, with fairly general initial data. Specifically, we require that the initial density be close to a constant in L ² and L , and that the initial velocity be small in L ² and bounded in L ^{2 n} (in two dimensions the L ² norms must be weighted slightly). Solutions are obtained as limits of approximate solutions corresponding to mollified initial data. The key point is that the approximate densities are shown to converge strongly, so that nonlinear pressures can be accommodated, even in the absence of any uniform regularity information for the approximate densities.     

On the application of the Wiener–Hopf technique to problems in dynamic elasticity

Many problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener–Hopf functional equations defined in a strip in a complex transform plane. Apart from a few special cases, the inherent coupling between shear and compressional body motions gives rise to coupled systems of equations, and so the resulting Wiener–Hopf kernels are of matrix form. The key step in the solution of a Wiener–Hopf equation, which is to decompose the kernel into a product of two factors with particular analyticity properties, can be accomplished explicitly for scalar kernels. However, apart from special matrices which yield commutative factorizations, no procedure has yet been devised to factorize exactly general matrix kernels.

This paper shall demonstrate, by way of example, that the Wiener–Hopf approximant matrix (WHAM) procedure for obtaining approximate factors of matrix kernels (recently introduced by the author in [SIAM J. Appl. Math. 57 (2) (1997) 541]) is applicable to the class of matrix kernels found in elasticity, and in particular to problems in QNDE. First, as a motivating example, the kernel arising in the model of diffraction of skew incident elastic waves on a semi-infinite crack in an isotropic elastic space is studied. This was first examined in a seminal work by Achenbach and Gautesen [J. Acoust. Soc. Am. 61 (2) (1977) 413] and here three methods are offered for deriving distinct non-commutative factorizations of the kernel. Second, the WHAM method is employed to factorize the matrix kernel arising in the problem of radiation into an elastic half-space with mixed boundary conditions on its face. Third, brief mention is made of kernel factorization related to the problems of flexural wave diffraction by a crack in a thin (Mindlin) plate, and body wave scattering by an interfacial crack.     

On deformations of cracked anisotropic slabs

The effect of a crack on the bending and shearing of an anisotropic elastic slab is considered. The problems considered are reduced to simultaneous Fredholm equations which may be solved numerically.     

Thermal response of a periodic boundary layer near an axisymmetric stagnation point on a circular cylinder

Boundary layer solutions are provided to study the time-mean heat transfer characteristics in a laminar flow in the vicinity of an axisymmetric stagnation point. The velocity of the oncoming flow is assumed to oscillate relative to the body. Different solutions are constructed for the small and high values of the reduced frequency parameter. Numerical solutions for the temperature functions are presented, and the wall values of the thermal gradients are tabulated.     

Diffusion-Mediated Transport¶and the Flashing Ratchet

Diffusion-mediated transport is a phenomenon in which a unidirectional motion of particles is achieved as a result of two opposing tendencies: diffusion, which spreads the particles uniformly through the medium and transport, which concentrates the particles at some special sites. The flashing-ratchet version of the Brownian motor, a simple model for protein motors, where the switching between transport and diffusion is periodic, illustrates diffusion-mediated transport. In this paper we show rigorously that the flashing ratchet can be tuned in such a way that the transport of mass against the gradient of the potential takes place and the concentration of mass during the transport phase occurs at sites located at the wells of an asymmetric potential. This goal is accomplished by comparing the flashing ratchet with an approximating Markov chain. A principle achievement of this work is to establish the connection between the dynamics of the ratchet and the Markov chain in the weak^* topology.     

Stability of noisy nonlinear auto-parametric systems

The purpose of this work is to examine the stationary motion and stability properties of stationary motion of two degree-of-freedom noisy auto-parametric systems We shall use analytical techniques to extend the existing results to examine such multi-dimensional nonlinear systems with noise, and in particular additive white noise. We obtain an approximation for the top Lyapunov exponent, the exponential growth rate, of the response of the so-called single-mode stationary motion. We show analytically that the top Lyapunov exponent is positive, and for small values of noise intensity ɛ and dissipation ɛ² the exponent grows in proportion with ɛ^2/3.     

Supramolecular main-chain liquid crystalline polymers and networks with competitive hydrogen bonding

A series of supramolecular polymers and networks with variable liquid crystalline characteristics have been created. These species are formed though the benzoic acid/pyridine associations of a flexible bisacid and a mixture of a rigid bispyridyl and a non-mesogenic tetrapyridyl. The networked systems displayed liquid crystalline characteristics up to and including 22.5% netpoint inclusion. Above this concentration, only crystalline and melting behaviours were observed. This observed phenomenon would seem to be linked to the statistical correlation of hydrogen bond acceptors and donors. There was also no observed phase segregation of the species after multiple heat/cool cycles and extended periods of time in the isotropic state. This would indicate that the thermodynamically more stable mesogenic phase cannot out-compete the non-liquid crystalline network. Computational analysis indicates no significant difference in hydrogen bond strength between the two different hydrogen bond acceptors.     

Two Convenient Syntheses of 2-Aryl-1-Cyano-1-Propenes From Quaternary Ammonium Salts

Two readily available and different quaternary ammonium salts have been prepared by the alkylation of vinamidinium salt reduction products. These salts have been successfully converted to 2-aryl-1-cyano-1-propenes and this methodology represents a clean and efficient procedure for the construction of such systems.     

Determining Causal Skeletons with Information Theory

Modeling a causal association as arising from a communication process between cause and effect, simplifies the discovery of causal skeletons. The communication channels enabling these communication processes, are fully characterized by stochastic tensors, and therefore allow us to use linear algebra. This tensor-based approach reduces the dimensionality of the data needed to test for conditional independence, e.g., for systems comprising three variables, pair-wise determined tensors suffice to infer the causal skeleton. The only thing needed is a minor extension to information theory, namely the concept of path information.