R. Valery Roy

A Lubrication Model of Coating Flows over a Curved Substrate in Space

Consider the three-dimensional flow of a viscous Newtonian fluid upon an arbitrarily curved substrate when the fluid film is thin as occurs in many draining, coating and biological flows. We drive the lubrication model of the dynamics of the film expressed in terms of the film thickness. The comprehensive model accurately includes the effects of the curvature of the substrate, via a physical multiple-scale approach, and gravity and inertia, via more rigorous centre manifold techniques. This new approach theoretically supports the use of the model over a wide range of parameters and provides a sound basis for further development of lubrication models. Numerical simulations exhibit some generic features of the dynamics of such thin fluid films on substrates with complex curvature: we here simulate a film thinning at a corner, the flow around a torus, and draining of a film down a cylinder. The last is more accurate than other lubrication models. The model derived here describes well thin-film dynamics over a wide range of parameter regimes.

Theoretical and numerical results for spin coating of viscous liquids

A mathematical model is developed for fluid flow in the spin coating process. Spin coating employs centrifugal force to produce coatings of uniform thickness. The long-wave or lubrication approximation is used for the flow of thin liquid layers that are exposed to the air and lie on a spinning horizontal solid substrate. For low rotation rates, steady axisymmetric drop shapes can be found analytically. The stability of these drops is investigated, using an energy method, both with and without the long-wave approximation. For industrially relevant high-speed motions, we formulate and solve a theoretical and numerical model for the three-dimensional time-dependent motion of the deforming drop. We pay particular attention to the formation of “fingers” at the expanding front. The model includes viscous, capillary, gravitational, centrifugal, Coriolis, and finite-contact-angle effects. Both homogeneous and chemically heterogeneous substrates are considered. In agreement with published experiments, the model de...

On the stability of liquid ridges

We consider the stability of a rectilinear liquid region whose boundary is composed of a solid cylindrical substrate of arbitrary shape and a free surface whose cross-section, in the absence of gravity, is a circular arc. The liquid–solid contact angle is a prescribed material property. A variational technique, using an energy functional, is developed that predicts the minimum wavelength for transverse instability under the action of capillarity. Conversely, certain configurations are absolutely stable and a simple stability criterion is derived. Stability is guaranteed if, for given substrate geometry and given contact angle, the unperturbed meniscus pressure is an increasing function of the liquid cross-sectional area. The analysis is applied to a variety of liquid/substrate configurations including (i) a liquid ridge with contact lines pinned to the sharp edges of a slot or groove, (ii) liquid ridges with free contact lines on flat and wedge-shaped substrates as well as substrates of circular or elliptical cross-section. Results are consistent with special cases previously treated including those that employ a slope-small-slope approximation.

Averaging method for strongly non-linear oscillators with periodic excitations

Abstract An averaging method is developed to predict periodic solutions of strongly non-linear and harmonically forced oscillators. The analysis is restricted to the case of period-1 orbits. The original governing equation is transformed into an autonomous set of differential equations governing the energy and resonant phase variables. The form of the transformation is given by the unperturbed conservative orbits of the system. The scheme is applied to three examples, the non-linear pendulum, the single-well Duffing oscillator, and the canonical escape oscillator. For these examples, the analysis is performed by using Jacobian elliptic functions. These examples demonstrate the ability of the averaging method to predict both transient and steady-state behavior of the system. The method has been developed in view of studying the large excursions of the response of non-linear systems induced by random perturbations.

Probabilistic analysis of flow in random porous media by stochastic boundary elements

The mathematical and numerical modeling of groundwater flows in random porous media is studied assuming that the formations hydraulic log-transmissivity is a statistically homogeneous, Gaussian, random field with given mean and covariance function. In the model, log-transmissivity may be conditioned to take exact field values measured at a few locations. Our method first assumes that the log-transmissivity may be expanded in a Fourier-type series with random coefficients, known as the Karhunen-Loeve (KL) expansion. This expansion has optimal properties and is valid for both homogeneous and nonhomogeneous fields. By combining the KL expansion with a small parameter perturbation expansion, we transform the original stochastic boundary value problem into a hierarchy of deterministic problems. To the first order of perturbation, the hydraulic head is expanded on the same set of random variables as in the KL representation of log-transmissivity. To solve for the corresponding coefficients of this expansion, we adopt a boundary integral formulation whose numerical solution is carried out by using boundary elements and dual reciprocity (DRBEM). To illustrate and validate our scheme, we solve three test problems and compare the numerical solutions against Monte Carlo simulations based on a finite difference formulation of the original flow problem. In all three cases we obtain good quantitative agreement and the present approach is shown to provide both a more efficient and accurate way of solving the problem.

Stochastic averaging of oscillators excited by colored Gaussian processes

Abstract The method of stochastic averaging has been developed and applied in the past mainly based on Stratonovich-Khasminskii theorem. We examine in this paper the application of this method in the case of arbitrary colored Gaussian excitations, which can be considered as the output of multidimensional linear filters to white Gaussian noise. The method used is based on a perturbation theoretic approach of the Fokker-Planck-Kolmogorov equation, which governs the response probability density function. First, for oscillators with linear elastic forces and non-parametric excitation, it is shown that, to leading order of perturbation, the results obtained match those derived by application of Stratonovich-Khasminskii theorem in the case of broad-band excitation. Then, more general results are derived for nearly Hamiltonian systems perturbed by parametric excitations of uncorrelated colored noises. It is shown that the state probability density function is governed by a reduced equation in the “slow” Hamiltonian variable only, which depends on a number of parameters characterizing the colored noise excitations. Several examples are given for illustration. As a preliminary to these theoretical developments, the problem of determining the eigenfunctions and eigenvalues of the Fokker-Planck operator is addressed for a general class of linear multidimensional systems.

Asymptotic analysis of first-passage problems

Abstract The response of stochastically-forced dynamical systems is analyzed in the limit of vanishing noise strength e. We predict asymptotic expressions for the stationary response probability density function (p.d.f.) and for the probability of first-passage of the response to the boundary of a domain in state space. The analysis is limited to Gaussian white noise type perturbations and to domains D in the phase plane “attracted” to an equilibrium point O of the system: all unperturbed trajectories enter D and converge asymptotically to O . In the first stage, the p.d.f. is expressed in terms of an asymptotic WKB form wexp( −Ψ e ) where the “quasi-potential” Ψ can be readily determined numerically by a method of “rays”. A domain of reliability D may then be defined as one bounded by a given contour of quasi-potential, since the latter is a Lyapunov function of the deterministic system. In a second stage, the probability of first-passage is determined in terms of the mean first-passage time to the boundary ∂ D . The latter is found in a singular perturbation solution devised by Matkowsky and Schuss [ SIAM. Appl. Math. 33 , 365 (1977)] in terms of the values reached on ∂ D by Ψ, w and by the deterministic force vector. Several examples demonstrate the validity and usefulness of this approach.

Noise perturbations of a non-linear system with multiple steady states

Abstract We examine the effects of small white Gaussian noise perturbations on a harmonically forced Duffing oscillator. For specific values of the parameters, the noise-free system admits two coexisting steady-state attractors. The presence of noise induces transitions from one attractor to the other, however small the noise intensity may be. As a first step, the equation of motion is transformed into a system of stochastic differential equations for the slowly varying van der Pol variables, by assuming that the predominant frequency of response is that of the forcing term. The condition for bistability, the stable fixed points and the separatrix defining the domains of attraction are then examined. In the second step, the effects of noise perturbations on the system are analyzed by determining the steady-state probability density of the fluctuations of the response, as well as the probabilities of escape from one attractor to the other. The latter are found by determination of the expected times of first-passage to the separatrix starting from a point in the domain of attraction of each stable state. This analysis is done in the limit of small damping and small noise intensity by use of an averaging scheme which reduces the dimensionality of the problem from two to one; this yields valuable information about the relative stability of the stable states. The obtained theoretical results are supported by digital simulation data. The analytical theory gives good agreement even for large noise.

Probabilistic analysis of a nonlinear pendulum

SummaryWe investigate the reliability of a nonlinear pendulum forced by a resonant harmonic excitation and interacting in a random environment. Two types of random perturbations are considered: additive weak noise, and random phase fluctuations of the harmonic resonant forcing. Our goal is to predict the probability of a transition of the response from oscillatory regime to rotatory regime. In the first stage, the noise-free system is analyzed by an averaging method in view of predicting period-1 resonant orbits. By averaging the fast oscillations of the response, these orbits are mapped into equilibrium points in the space of the energy and resonant phase variables. In the second stage, the random fluctuating terms exciting the averaged system are evaluated, leading to a Fokker-Planck-Kolmogorov equation governing the probability density function of the energy and phase variables. This equation is solved asymptotically in the form of a WKB approximationp∼exp(−Q/ε) as the parameter ε characterizing the smallness of the random perturbations tends to zero. The quasipotentialQ is solution of a Hamilton-Jacobi equation, and can be obtained numerically by a method of characteristics. Of critical importance is the evaluation of the minimum difference of quasipotential between the equilibrium point and the boundary across which the transitions occur. We show that this minimum difference determines to logarithmic accuracy the mean first-passage time to the critical boundary and hence the probability of failure of the oscillatory regime. The effects of the two types of random perturbations are analyzed separately.We investigate the reliability of a nonlinear pendulum forced by a resonant harmonic excitation and interacting in a random environment. Two types of random perturbations are considered: additive weak noise, and random phase fluctuations of the harmonic resonant forcing. Our goal is to predict the probability of a transition of the response from oscillatory regime to rotatory regime. In the first stage, the noise-free system is analyzed by an averaging method in view of predicting period-1 resonant orbits. By averaging the fast oscillations of the response, these orbits are mapped into equilibrium points in the space of the energy and resonant phase variables. In the second stage, the random fluctuating terms exciting the averaged system are evaluated, leading to a Fokker-Planck-Kolmogorov equation governing the probability density function of the energy and phase variables. This equation is solved asymptotically in the form of a WKB approximationp∼exp(−Q/e) as the parameter e characterizing the smallness of the random perturbations tends to zero. The quasipotentialQ is solution of a Hamilton-Jacobi equation, and can be obtained numerically by a method of characteristics. Of critical importance is the evaluation of the minimum difference of quasipotential between the equilibrium point and the boundary across which the transitions occur. We show that this minimum difference determines to logarithmic accuracy the mean first-passage time to the critical boundary and hence the probability of failure of the oscillatory regime. The effects of the two types of random perturbations are analyzed separately.