We present and analyse a model for the spherical pulsations and translational motions of a pair of interacting gas bubbles in an incompressible liquid. The model is derived rigorously in the context of potential flow theory and contains all terms up to and including fourth order in the inverse separation distance between the bubbles. We use this model to study the cases of both weak and moderate applied acoustic forcing. For weak acoustic forcing, the radial pulsations of the bubbles are weakly coupled, which allows us to obtain a nonlinear time-averaged model for the relative distance between the bubbles. The two parameters of the time-averaged model classify four different dynamical regimes of relative translational motion, two of which correspond to the attraction and repulsion of classical secondary Bjerknes theory. Also predicted is a pattern in which the bubbles exhibit stable, time-periodic translational oscillations along the line connecting their centres, and another pattern in which there is an unstable separation distance such that bubble pairs can either attract or repel each other depending on whether their initial separation distance is smaller or larger than this value. Moreover, it is shown that the full governing equations possess the dynamics predicted by the time-averaged model. We also study the case of moderateamplitude acoustic forcing, in which the bubble pulsations are more strongly coupled to each other and bubble translation also affects the radial pulsations. Here, radial harmonics and nonlinear phase shifting play a significant role, as bubble pairs near resonances are observed to translate in patterns opposite to those predicted by classical secondary Bjerknes theory. In this work, dynamical systems techniques and the method of averaging are the primary mathematical methods that are employed.

We study the formation mechanism of rims created around femtosecond laser ablated craters on glass. Experimental studies of the surface morphology reveal that a thin rim is formed around the smooth craters and is raised above the undamaged surface by about 50-100 nm. To investigate the mechanism of rim formation following a single ultrafast laser pulse, we perform a one-dimensional theoretical analysis of the thermal and fluid processes involved in the ablation process. The results indicate the existence of a very thin melted zone below the surface and suggest that the rim is formed by the high pressure plasma producing a pressure-driven fluid motion of the molten material outwards from the centre of the crater. The numerical solutions of pressure-driven fluid motion of the thin melt demonstrate that the melt can flow to the crater edge and form a rim within the first nanoseconds of the ablation process. The possibility that a tall rim can be formed during the initial stages of the plasma is suggestive that the rim may tilt outwards towards the low pressure region creating a resolidified melt splash as observed in the experiments. The possibility of controlling or suppressing the rim formation is discussed also.

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Femtosecond pulsed laser ablation dynamics and ablation morphology of nickel based superalloy CMSX-4Effect of ambient air pressure on debris redeposition during laser ablation of glass J. Appl. Phys. 98, 113520 (2005); 10.1063/1.2138800Femtosecond laser ablation properties of borosilicate glass

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O( 2 ), where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ ).