Luis G. Reyna

Smallest scale estimates for the Navier-Stokes equations for incompressible fluids

We consider solutions of the Navier-Stokes equations for incompressible fluids in two and three space dimensions. We obtain improved estimates, in the limit of vanishing viscosity, for the Fourier coefficients. The coefficients decay exponentially fast for wave numbers larger than the square root of the maximum of the velocity gradients divided by the square root of the viscosity. This defines the minimum scale, the size of the smallest feature in the flow.

A fourth-order-accurate difference approximation for the incompressible Navier-Stokes equations☆

Abstract We discuss fourth-order-accurate difference approximations for parabolic systems and for the incompressible Navier—Stokes equations. A general principle for deriving numerical boundary conditions for higher-order-accurate difference schemes is described. Some difference approximations for parabolic systems are analyzed for stability and accuracy. The principle is used to derive stable and accurate numerical boundary conditions for the incompressible Navier—Stokes equations. Numerical results are given from a fourth-order-accurate scheme for the incompressible Navier—Stokes equations on overlapping grids in two- and three-space dimensions.

On the smallest scale for the incompressible Navier-Stokes equations

We prove that, for solutions to the two- and three-dimensional incompressible Navier-Stokes equations, the minimum scale is inversely proportional to the square root of the Reynolds number based on the kinematic viscosity and the maximum of the velocity gradients. The bounds on the velocity gradients can be obtained for two-dimensional flows, but have to be assumed in three dimensions. Numerical results in two dimensions are given which illustrate and substantiate the features of the proof. Implications of the minimum scale result, to the decay rate of the energy spectrum are discussed.

High electric field approximation to charge transport in semiconductor devices

Abstract We consider the Boltzmann equation describing the charge transport in semiconductor devices, including the scattering of electrons with acoustic and optical phonons. Taking into account that at a high electric field the scattering events are nearly elastic, we develop a formal expansion for the distribution function valid for general band structures.

Internal layers, small eigenvalues, and the sensitivity of metastable motion

On a semi-infinite domain, an analytical characterization of exponentially slow internal layer motion for the Allen–Cahn equation and for a singularly perturbed viscous shock problem is given. The results extend some previous results that were restricted to a finite geometry. For these slow motion problems, we show that the slow dynamics associated with the semi-infinite domain are not preserved, even qualitatively, by imposing a commonly used form of artificial boundary condition to truncate the semi-infinite domain to a finite domain. This extreme sensitivity to boundary conditions and domain truncation is a direct result of the exponential ill-conditioning of the underlying linearized problem. For Burgers equation, many of the analytical results are verified by calculating certain explicit solutions. Some related ill-conditioned internal layer problems are examined.

LASER ABLATION OF MULTILAYER POLYMER FILMS

We study the efficiency of using multilayer structures as an etch‐stop mechanism in the ablation of polyimide films by ultraviolet lasers. The study is done using a photothermal model that includes the light absorption by the decomposed fragments, which shield the polymer from the laser beam, an intermediate zone in which the polymer is suffering a phase transition and the underlying unburned material. The layers are differentiated from each other through their optical properties. Variation in the optical properties of polyimide has been achieved by a proper selection of impurities. From our modeling work, we conclude that optically thin foils may be used as etch stop in the ablation process when the penetration depth of the middle layer is around three times larger than the penetration depth of the surrounding layers, this for fluences below 200 mJ/cm2. We also present some experimental results.

Conservation of expected momentum and energy in Monte Carlo particle simulation

Rarefied gas dynamics can be simulated numerically by Monte Carlo particle methods in which energy and momentum are conserved in expectation, but not exactly, through each collision. The conservation of the expected values of these moments does not imply the conservation of other expected second moments. For example, in Nanbu’s method [J. Phys. Soc. Jpn. 49, 2042 (1980)] (which has been proved to converge), the expected value of the temperature decreases through each collision step, and the relaxation of a gas calculated by this scheme leads to the zero temperature state. The decrease in expected temperature is of order O(1/N), where N is the number of simulated particles.

Numerical prediction flow in slender vortices

Abstract We study the slender vortex approximation with emphasis on the high Reynolds number behavior. It is shown that the breakdown of the approximation coincides with the critically condition as introduced by Benjamin [1]. We study free vortices with and without an adverse pressure gradient for viscous and inviscid flows. Finally, we compare with the experimental results from Faler and Leibovich [2].

On exponential ill-conditioning and internal layer behavior

In the limit of small diffusivity, the internal layer behavior associated with the initial-boundary value problems for a viscous shock equation and a reaction diffusion equation is analyzed. As a result of the occurrence of exponentially small eigenvalues for the linearized problems, the steady state internal layer solutions are shown to very sensitive to small perturbations. For the time dependent problems, the small eigenvalues give rise to exponentially slow internal layer motion. Accurate numerical methods are used to compute the steady state internal layer solutions and the slow internal layer motion. The relationship between the viscous shock problem and some exponentially ill-conditioned linear singular perturbation problems is discussed.

Repetition rate effect on the laser ablation of polymer structures

Multilayer polyimide structures may be used as an etch stop mechanism in laser ablation processing. By changing the optical properties of the polymer it is possible to better control the depth of the patterns produced by the lasers. We present the effect of the repetition rate in the ablation of multilayer structures. The results are obtained from a photothermal model that includes the light absorption by the decomposed fragments, which shield the polymer from the incoming laser beam. The model also includes an intermediate zone in which the polymer suffers a phase transition. The evolution of the temperature profiles is simulated during each pulse taking into account the pulse shape; however, a simple diffusion model is used between pulses. The results of the simulations indicate the range of values for which the multilayer structure may be used as an effective etch stop mechanism. We found that the effectiveness of the multilayer structure deteriorates for increasing repetition rates.