Jorge Viñals

TWO-PHASE BINARY FLUIDS AND IMMISCIBLE FLUIDS DESCRIBED BY AN ORDER PARAMETER

A unified framework for coupled Navier-Stokes/Cahn-Hilliard equations is developed using, as a basis, a balance law for microforces in conjunction with constitutive equations consistent with a mechanical version of the second law. As a numerical application of the theory, we consider the kinetics of coarsening for a binary fluid in two space dimensions.

Coarse‐grained description of thermo‐capillary flow

A mesoscopic or coarse‐grained approach is presented to study thermo‐capillary induced flows. An order parameter representation of a two‐phase binary fluid is used in which the interfacial region separating the phases naturally occupies a transition zone of small width. The order parameter satisfies the Cahn–Hilliard equation with advective transport. A modified Navier–Stokes equation that incorporates an explicit coupling to the order parameter field governs fluid flow. It reduces, in the limit of an infinitely thin interface, to the Navier–Stokes equation within the bulk phases and to two interfacial forces: a normal capillary force proportional to the surface tension and the mean curvature of the surface, and a tangential force proportional to the tangential derivative of the surface tension. The method is illustrated in two cases: thermo‐capillary migration of drops and phase separation via spinodal decomposition, both in an externally imposed temperature gradient.

Pattern formation in weakly damped parametric surface waves

We present a theoretical study of nonlinear pattern formation in parametric surface waves for fluids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing fluid motion, followed by a multiscale asymptotic expansion in the distance away from threshold. Close to onset, the asymptotic expansion yields an amplitude equation which is of gradient form, and allows the explicit calculation of the functional form of the cubic nonlinearities. In particular, we find that three-wave resonant interactions contribute significantly to the nonlinear terms, and therefore are important for pattern selection. Minimization of the associated Lyapunov functional predicts a primary bifurcation to a standing wave pattern of square symmetry for capillary-dominated surface waves, in agreement with experiments. In addition, we find that patterns of hexagonal and quasi-crystalline symmetry can be stabilized in certain mixed capillary–gravity waves, even in this case of single-frequency forcing. Quasi-crystalline patterns are predicted in a region of parameters readily accessible experimentally.

Interface and contact line motion in a two phase fluid under shear flow

We use a coarse grained description to study the steady state interfacial configuration of a two phase fluid under steady shear. Dissipative relaxation of the order parameter leads to interfacial slip at the contact line, even with no-slip boundary conditions on the fluid velocity. This relaxation occurs within a characteristic length scale l(0) = sqrt[xiD/V0], with xi the (microscopic) interfacial thickness, D an order parameter diffusivity, and V0 the boundary velocity. The steady state interfacial configuration is shown to satisfy a scaling form involving the ratio l(0)/L, where L is the width of the fluid layer, for a passive interface, and the capillary number as well for an active interface.

Pattern Selection in Faraday Waves

We present a systematic nonlinear theory of pattern selection for parametric surface waves (Farad waves), not restricted to fluids of low viscosity. A standing wave amplitude equation is derived from the Navier-Stokes equation that is of gradient form. The associated Lyapunov function is calculate for different regular patterns to determine the selected pattern near threshold as a function of a dampi parameterg. For g , 1, we show that a single wave (or stripe) pattern is selected. For g ø 1, we predict patterns of square symmetry in the capillary regime, a sequence of sixfold (hexagonal eightfold,. . . in the mixed gravity-capillary regime, and stripe patterns in the gravity dominated regime. [S0031-9007(97)04216-6]

SECONDARY INSTABILITIES AND SPATIOTEMPORAL CHAOS IN PARAMETRIC SURFACE WAVES

A two dimensional model is introduced to study pattern formation, secondary instabilities and the transition to spatiotemporal chaos (weak turbulence) in parametric surface waves. The stability of a periodic standing wave state above onset is studied against Eckhaus, zig-zag and transverse amplitude modulations (TAM) as a function of the control parameter

Grain boundary pinning and glassy dynamics in stripe phases

\varepsilon

Pattern formation in weakly damped parametric surface waves driven by two frequency components

and the detuning. A mechanism leading to a finite threshold for the TAM instability is identified. Numerical solutions of the model are in agreement with the stability diagram, and also reveal the existence of a transition to spatiotemporal chaotic states at a finite

Effect of elasticity on late stage coarsening

\varepsilon

Study of the parametric oscillator driven by narrow‐band noise to model the response of a fluid surface to time‐dependent accelerations

. Power spectra of temporal fluctuations in the chaotic state are broadband, decaying as a power law of the frequency