Jorge F. Willemsen

Molecular Dynamics of Fluid Flow at Solid Surfaces

Molecular dynamics techniques are used to study the microscopic aspects of several slow viscous flows past a solid wall, where both fluid and wall have a molecular structure. Systems of several thousand molecules are found to exhibit reasonable continuum behavior, albeit with significant thermal fluctuations. In Couette and Poiseuille flow of liquids it is found that the no‐slip boundary condition arises naturally as a consequence of molecular roughness, and that the velocity and stress fields agree with the solutions of the Stokes equations. At lower densities slip appears, which can be incorporated into a flow‐independent slip‐length boundary condition. The trajectories of individual molecules in Poiseuille flow are examined, and it is also found that their average behavior is given by Taylor–Aris hydrodynamic dispersion. An immiscible two‐fluid system is simulated by a species‐dependent intermolecular interaction. A static meniscus is observed whose contact angle agrees with simple estimates and, when ...

Capillary displacement and percolation in porous media

We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.

Surface relaxation and pore sizes in rocks—a nuclear magnetic resonance analysis

The pulsed gradient nuclear magnetic resonance technique has been applied to the measurement of pore sizes in rocks. The measurements also yield an estimate of the strength of the magnetization relaxation at the pore‐rock interface.

Deterministic modelling of driving and dissipation for ocean surface gravity waves

Consider two questions. What qualitative features should a spectrum of wind driven wave amplitudes possess in deep water? Then, is it possible to compute such a spectrum ab initio? In answer to the first question, at the least the spectrum should exhibit a spectral peak determined by the acting wind; an asymptotic power law tail; and an angular dependence between the dominant wind and wave directions. The principal result of this paper is that a model to calculate a wind-driven sea spectrum that satisfies the first two requirements starting from a broad suite of initial conditions has been constructed and exercised. To this end, wind driving mechanisms and models for dissipation caused by wave breaking are investigated. This study does not include detailed hydrodynamic calculations, but rather an evaluation of physically plausible model interaction terms. These are appended to Hamiltons equations for a wave field in deep water. This methodology leads to deterministic ordinary differential equations for the evolution of the wave field in which three and four wave nonlinear interactions are incorporated. The deterministic form of the equations is preserved through the introduction of nonstochastic driving and dissipation terms. The time evolution results presented fulfill the qualitative expectations desired for the spectrum. The calculations also yield much more information. Full phase information is retained. The relative magnitudes of the nonlinear interaction terms may be assessed as functions of time. The same applies for the magnitudes of the driving and dissipating terms. This information will be used to improve the model to where it is ready to confront experimental data.

Enhanced Computational Methods for Nonlinear Hamiltonian Wave Dynamics

Abstract It is noted that the nonlinear Hamiltonian dynamical equations describing surface waves are of convolution form in a version derived by Krasitskii. By virtue of the convolution theorem for Fourier transforms, the dynamical equations are thus amenable to numerical calculations using FFT techniques. This, as is well known, renders the calculations much faster than direct numerical integration, of order N logN steps versus N2 per convolution integral for N discrete wavenumbers. An illustrative calculation for pure gravity waves in deep water is presented and discussed.

Enhanced Computational Methods for Nonlinear Hamiltonian Wave Dynamics. Part II: New Results

Abstract It has been noted that the nonlinear Hamiltonian dynamical equations describing surface waves are of convolution form in a version derived by Krasitskii, and thus they are amenable to numerical calculations using fast Fourier transform techniques. In this paper new results regarding the nature of the solutions are presented and numerical instabilities that may develop are discussed. Additionally various ways of displaying features of the nonlinear wave evolution are explored within the context of specific examples.

Analysis of SWADE Discus N Wind Speed and Wave Height Time Series. Part II: Quantitative Growth Rates during a Storm

Abstract In Part I, wind speed and wave height time series obtained from the Discus N buoy during two storm events recorded in the SWADE experiment were analyzed using discrete wavelet packet transforms. One result of the analysis is that distinct tightly bunched wave frequency bands exist that evolve differently from one another in response to the wind. In this paper, that result is confirmed using a more traditional windowed Fourier transform approach. Additionally, the bands in which most of the wave energy is concentrated will be shown to grow in a manner consistent with the Plant formulation during the intervals of most intense growth. That formulation is parsimoniously extended to include the presence of a growing wind, under the hypothesis that wind fluctuations are too rapid for long waves to respond. The model of Al-Zanaidi and Hui will also be considered. Neither one of the formulations is predictive above approximately 0.22 Hz. Finally, the onset and offset times of very rapid wave growth are d...

Molecular Dynamics of Slow Viscous Flows

We use molecular dynamics techniques to study the microscopic aspects of several slow viscous flows past a solid wall, where both fluid and wall have a molecular structure. Systems of several thousand molecules are found to exhibit reasonable continuum behavior, albeit with significant thermal fluctuations. In Couette and Poiseuille flow of liquids we find the no—slip boundary condition arises naturally as a consequence of molecular roughness, and that the velocity and stress fields agree with the solutions of the Stokes equations. At lower densities slip appears, which can be incorporated into a flow—independent slip—length boundary condition. An immiscible two—fluid system is simulated by a species—dependent intermolecular interaction. We observe a static meniscus whose contact angle agrees with simple estimates and, when motion occurs, velocity—dependent advancing and receding angles. The local velocity field near a moving contact line shows a breakdown of the no—slip condition.

Probability Distributions with Infinite Moments

The functional integral representation of traditional Euclidean field theory is based on the existence of Gaussian functional integrals. Perturbation theory then consists of calculating the moments of what may be thought of as a Gaussian probability distribution functional. At the Cargese Summer School, S.K. Foong discussed functional integrals based upon the Poisson distribution and how these may be applied to problems in wave propagation. This opens the door to thinking about the possible significance of other distributions to certain physical processes. Question: Can we learn something about notoriously non-renormalizable field theories such as gravity by studying distributions such as the Levy distribution, which has infinite moments? That is, can we rephrase the questions away from what we expect from the traditional moments of a Gaussian?

Analysis of a class of simplified models for nonlinear gravity wave interactions

Kolmogorov‐type cascade processes have long been postulated to explain power‐law falloff of the spectrum of gravity wave autocorrelations. In an effort to understand these processes more deeply, simplified models of the nonlinear Hasselmann resonant four‐wave interaction have been introduced. These models respect the fundamental structural aspects of that interaction, but are otherwise chosen for computational simplicity. In this paper the concept of quasilocality within the context of such models is extended to its logical limit in one spatial dimension. This results in differential rather than integral equations for the wave fields. One scaling solution to the new equations has a spectral exponent independent of the details of the model interaction, and is stable to small perturbations. In addition, under weak restrictions on the first and second derivatives of the model interactions, a further pair of two‐parameter families of scaling solutions exist. The spectral exponents in this case depend on the d...