On some smooth approximations on thick periodic multi-structures

In this paper we study the approximation properties of measurable and square-integrable functions. In particular we show that any $L^2$-bounded function can be approximated in $L^2$-norm by smooth functions defined on a highly oscillating boundary of thick multi-structures in ${\mathbb{R}}^n$. We derive the norm estimates for the approximating functions and study their asymptotic behaviour.

Observations of Nearbed Turbulence over Mobile Bedforms in Combined, Collinear Wave-Current Flows

Collinear wave-current shear interactions are often assumed to be the same for currents following or opposing the direction of regular wave propagation; with momentum and mass exchanges restricted to the thin oscillating boundary layer (zero-flux condition) and enhanced but equal wave-averaged bed shear stresses. To examine these assumptions, a prototype-scale experiment investigated the nature of turbulent exchanges in flows with currents aligned to, and opposing, wave propagation over a mobile sandy bed. Estimated mean and maximum stresses from measurements above the bed exceeded predictions by models of bed shear stress subscribing to the assumptions above, suggesting the combined boundary layer is larger than predicted by theory. The core flow experiences upward turbulent fluxes in aligned flows, coupled with sediment entrainment by vortex shedding at flow reversal, whilst downward fluxes of eddies generated by the core flow, and strong adverse shear can enhance near-bed mass transport, in opposing currents. Current-aligned coherent structures contribute significantly to the stress and energy dissipation, and display characteristics of wall-attached eddies formed by the pairing of counter-rotating vortices. These preliminary findings suggest a notable difference in wave-following and wave-opposing wave-current interactions, and highlight the need to account for intermittent momentum-exchanges in predicting stress, boundary layer thickness and sediment transport.

Damped Vibration Response of an Axially Moving Wire Subject to an Oscillating Boundary Condition and the Application to Slurry Wiresaws

This paper presents the analysis of a new class of differential continuum system with a solution of traveling waves containing coupled spatial and temporal variables. Herein, we derive the analytical solution of the damped vibration response of a longitudinally moving wire with damping, subject to an oscillating boundary condition. The vibration response is the outcome of combining four traveling waves, induced by a wave initiating from the oscillating boundary, and traveling between the two boundaries. The four different traveling waves are the independent bases of the vibration responses that span the solution space of vibration of such continuum system. The combination, or the interference, of these traveling waves in the undamped condition produces nodal points in the vibration response, which can be formulated through the analytical solution. The impacts of wire speed, oscillating frequency at the boundary and damping factors on the vibration response are investigated. Furthermore, the vibration induced by the oscillating motion of the boundary has a profound impact on the effectiveness of slicing ingots with rocking motion of oscillating wire guides in wiresaw manufacturing processes.

Experimental Study of the Impact of a Vibrating Wire on Free Abrasive Machining as Correlated to the Modeling of Vibration Subject to an Oscillating Boundary Condition

In this paper, we study experimentally the impact of a vibrating wire on the free abrasive machining (FAM) process in removing material from the surface of brittle materials, such as silicon. An experimental setup was designed to study the FAM process on silicon substrate surface by using a slurry-fed wire with a periodic excitation. An analytical solution of a wire moving axially, subject to an oscillating boundary condition with damping from abrasive slurry, was derived based on the partial differential equation of motion. Experiments were conducted on the apparatus using a wire with an oscillating boundary. It was found that the amplitudes of vibration were larger at the side of the oscillatory boundary, which caused more FAM interaction near the edge of the oscillatory boundary with larger material removal that was measured and validated. Furthermore, experiments were conducted to elucidate the effectiveness of brittle material removal using FAM with abrasive grits: (i) under dry condition, (ii) with water, and (iii) with abrasive slurry. Experimental results showed that the vibration of wire resulted in plastic deformation on the surface of silicon wafer. The abrasive grits in slurry driven by a vibrating wire generated material removal through observable grooves and fractures on the surface of silicon due to FAM in just a few minutes. The grooves from FAM process is an outcome of brittle machining through fracture formation and concatenation, generated by the indentation of abrasive grits on the silicon surface.

Transport of Temperature Fluctuations Across a Two-Phased Laminate Conductor

In the periodic composite materials temperature or displacement fluctuations suppressed in directions perpendicular to the periodicity surfaces should expect a damping reaction from the composite. This phenomenon, known as the boundary effect behavior has been investigated only in the framework of approximated models. In this paper extended tolerance model of heat transfer in periodic composites is used as a tool allows analytical investigations of highly oscillating boundary thermal loadings. It has been shown that mentioned reaction is dual – different for even and odd fluctuations.

Oscillating PDE in a rough domain with a curved interface: homogenization of an optimal control problem

Homogenization of an elliptic PDE with periodic oscillating coefficients and an associated optimal control problems with energy type cost functional is considered. The domain is a 3-dimensional region (method applies to any $n$ dimensional region) with oscillating boundary, where the base of the oscillation is curved and it is given by a Lipschitz function. Further, we consider a general elliptic PDE with oscillating coefficients. We also include very general type cost functional of Dirichlet type given with oscillating coefficients which can be different from the coefficient matrix of the equation. We introduce appropriate unfolding operators and approximate unfolded domain to study the limiting analysis. The present article is new in this generality.