Warren R. Smith

Models for Solidification and Splashing in Laser Percussion Drilling

This paper studies systems of partial differential equations modelling laser percussion drilling. The particular phenomenon considered in detail is the ejection of the thin layer of molten material. This thin layer is modelled as an inviscid flow between the fluid surface and fluid/solid interface, both of which are unknown moving boundaries. Through a regular asymptotic expansion, the governing equations are reduced to a combination of the shallow water equations in the zero gravity limit and a two-phase Stefan problem; the key small parameter is the square of the aspect ratio. These leading-order problems exhibit shocks which represent a possible mechanism for the previously unexplained fluid clumping. Approximate formulas and a parameter grouping are derived to predict the rate of melt solidification during ejection. Finally, weak formulations of the convection-diffusion equation for energy conservation are presented. These weak formulations are novel because the fluid is moving across a solid surface....

Modulation equations for strongly nonlinear oscillations of an incompressible viscous drop

Large-amplitude oscillations of incompressible viscous drops are studied at small capillary number. On the long viscous time scale, a formal perturbation scheme is developed to determine original modulation equations. These two ordinary differential equations comprise the averaged condition for conservation of energy and the averaged projection of the Navier-Stokes equations onto the vorticity vector. The modulation equations are applied to the free decay of axisymmetric oblate-prolate spheroid oscillations. On the long time scale, only the modulation equation for energy is required. In this example, the results compare well with linear viscous theory, weakly nonlinear inviscid theory and experimental observations. The new results show that previous experimental observations and numerical simulations are all manifestations of a single-valued relationship between dimensionless decay rate and amplitude. Moreover, if the amplitude of the oscillations does not exceed 30 % of the drop radius, this decay rate may be approximated by a quadratic. The new results also show that, when the amplitude of the oscillations exceeds 20 % of the drop radius, fluid in the inviscid bulk of the drop is undergoing abrupt changes in its acceleration in comparison to the acceleration during small-amplitude deformations.

One-dimensional models for heat and mass transfer in pulse-tube refrigerators

This paper studies systems of partial differential equations modelling pulse-tube refrigerators. Through a regular asymptotic expansion, the pressure is found to be a known function of time. Therefore, models for heat and mass transfer are derived neglecting the equation for conservation of momentum; the cooling mechanism being highlighted in terms of this model. New approximate formulas are derived for the velocity in the tube and the non-linear thermal wave speed in the porous medium (known as the regenerator) on the short (piston oscillation) time-scale. The temperature dependence of the thermal wave speed is proposed as one measure of the efficiency of the transport of heat away from the cold heat exchanger. Progress is limited on the long time-scale, however, the importance of the difference between the gas temperature averaged over an oscillation and heat exchanger temperature is identified.

Mathematical Modelling of Electrical-Optical Effects in Semiconductor Laser Operation

A mathematical model describing the coupling of electrical and optical effects in semiconductor lasers is introduced. Through a multiple-scale asymptotic expansion, the governing equations in the active region are reduced to one parabolic and four first-order hyperbolic partial differential equations. By making further assumptions, partially and fully lumped models for the active region are deduced, which complement (and provide a more systematic derivation of) previous well-established lumped models. 1. Introduction. Numerous mathematical models have been formulated to sim- ulate electrical and optical effects in semiconductor lasers. The fully space-dependent formulation is generally accepted to comprise an electrical model, consisting of elec- tron and hole continuity equations and Poissons equation, and an optical model, made up of a wave equation and a photon rate equation (7), (19). The electrical model, employed in almost all relevant previous papers, relies on Poissons equation for the electric potential being valid across the whole semiconductor. However, this is inappropriate for the active region in which electromagnetic waves propagate, so the curl of the electric field is nonnegligible. These electromagnetic waves are normally represented by the separate optical model. The unsuitability of Poissons equation has been previously noted in the related context of the mathematical modelling of a semiconductor driven by a current of very high frequency (6). Furthermore, the high frequency currents generated by the interaction between the large electric field and the charge carriers in the active region are normally neglected in existing models and we also address this issue. Fully space-dependent models currently are solved exclusively by direct numerical approaches. However, some partially lumped models for the active region have also been reported which comprise one ordinary differential equation and two first-order wave equations (18), (9). A fully lumped model in terms of two ordinary differential equations (for electron and photon concentrations) is very well established (see, for example, (5), (14)). In this paper, we attempt to address deficiencies in the modelling of the fully space-dependent case. Using the resulting model as the starting point

Traveling Waves in Two-Dimensional Plane Poiseuille Flow

The asymptotic structure of laminar modulated traveling waves in two-dimensional high-Reynolds-number plane Poiseuille flow is investigated on the upper-energy branch. A finite set of independent slowly varying parameters are identified which parameterize the solution of the Navier--Stokes equations in this subset of the phase space. Our parameterization of the weakly stable modes describes an attracting manifold of maximum-entropy configurations. The complementary modes, which have been neglected in this parameterization, are strongly damped. In order to seek a closure, a countably infinite number of modulation equations are derived on the long viscous time scale: a single equation for averaged kinetic energy and momentum; and the remaining equations for averaged powers of vorticity. Only a finite number of these vorticity modulation equations are required to determine the finite number of unknowns. The new results show that the evolution of the slowly varying amplitude parameters is determined by the vo...

Solidification of a two-dimensional high-Reynolds-number flow and its application to laser percussion drilling

The competition between inertia and solidification for the high-Reynolds-number flow of molten aluminium across a cool solid aluminium surface is investigated. A two-dimensional molten aluminium droplet is of finite extent and is surrounded by a passive gas. The droplet initially freezes due to rapid thermal conduction into the solid. Depending on the initial velocity of the molten aluminium, one of two situations may develop: (i) If the molten aluminium has a non-decreasing initial velocity profile, solidification continues until the passing of the trailing edge of the liquid/gas interface or the flow is engulfed; (ii) If the molten aluminium has a decreasing initial velocity profile, the droplet narrows and thickens resulting in a reduction in the heat flux and in the rate of solidification; this will eventually lead to fluid clumping and shock formation. The rate of solidification may also be reduced by increasing the ambient temperature. The results are interpreted in terms of the recast observed during the solidification phase of laser percussion drilling.

Predictions of thermoelastic stress in a broad-area semiconductor laser

The coupling of electrical, optical, and thermal effects in broad-area semiconductor lasers has been investigated using a multilateral mode mathematical model. Numerical solutions for the active layer temperature rise are input into the thermal source terms of elasticity equations, leading to the prediction of the thermoelastic stresses which occur in regions of high defect concentration. The magnitude of this prediction is compared with the size of other stresses reported elsewhere in experimental observations of the degraded facet of broad-area devices.

Asymptotic analysis of the attractors in two-dimensional Kolmogorov flow

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Wave–structure interactions for the distensible tube wave energy converter

A comprehensive linear mathematical model is constructed to address the open problem of the radiated wave for the distensible tube wave energy converter. This device, full of sea water and located just below the surface of the sea, undergoes a complex interaction with the waves running along its length. The result is a bulge wave in the tube which, providing certain criteria are met, grows in amplitude and captures the wave energy through the power take-off mechanism. Successful optimization of the device means capturing the energy from a much larger width of the sea waves (capture width). To achieve this, the complex interaction between the incident gravity waves, radiated waves and bulge waves is investigated. The new results establish the dependence of the capture width on absorption of the incident wave, energy loss owing to work done on the tube, imperfect tuning and the radiated wave. The new results reveal also that the wave–structure interactions govern the amplitude, phase, attenuation and wavenumber of the transient bulge wave. These predictions compare well with experimental observations.

Necessary conditions for breathers on continuous media to approximate breathers on discrete lattices

We start by considering the sine-Gordon partial differential equation (PDE with an arbitrary perturbation. Using the method of Kuzmak-Luke, we investigate those conditions the perturbation must satisfy in order for a breather solution to be a valid leading-order asymptotic approximation to the perturbed problem. We analyse the cases of both stationary and moving breathers. As examples, we consider perturbing terms which include typical linear damping, periodic sinusoidal driving, and dispersion caused by higher order spatial derivatives. The motivation for this study is that the mathematical modelling of physical systems, often leads to the discrete sine-Gordon system of ODEs which are then approximated in the long wavelength limit by the continuous sine-Gordon PDE. Such limits typically produce fourth-order spatial derivatives as higher order correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasi-continuum SG equation and the forced/damped SG system. However, the moving breather is only a consistent solution of the quasi- continuum SG equation and not the damped SG system.