Santiago Madruga

Decomposition driven interface evolution for layers of binary mixtures. I. Model derivation and stratified base states

A dynamical model is proposed to describe the coupled decomposition and profile evolution of a free surface film of a binary mixture. An example is a thin film of a polymer blend on a solid substrate undergoing simultaneous phase separation and dewetting. The model is based on model-H describing the coupled transport of the mass of one component (convective Cahn-Hilliard equation) and momentum (Navier-Stokes-Korteweg equations) supplemented by appropriate boundary conditions at the solid substrate and the free surface. General transport equations are derived using phenomenological nonequilibrium thermodynamics for a general nonisothermal setting taking into account Soret and Dufour effects and interfacial viscosity for the internal diffuse interface between the two components. Focusing on an isothermal setting the resulting model is compared to literature results and its base states corresponding to homogeneous or vertically stratified flat layers are analyzed.

Homology and symmetry breaking in Rayleigh-Bénard convection: Experiments and simulations

Algebraic topology (homology) is used to analyze the state of spiral defect chaos in both laboratory experiments and numerical simulations of Rayleigh-Benard convection. The analysis reveals topological asymmetries that arise when non-Boussinesq effects are present. The asymmetries are found in different flow fields in the simulations and are robust to substantial alterations to flow visualization conditions in the experiment. However, the asymmetries are not observable using conventional statistical measures. These results suggest homology may provide a new and general approach for connecting spatiotemporal observations of chaotic or turbulent patterns to theoretical models.

Re-entrant hexagons in non-Boussinesq convection

The stability and dynamics of hexagonal patterns is analyzed numerically in strongly nonlinear non-Boussinesq convection. The uid parameters correspond to those of water, with emphasis on realistic experimental conditions. The stability analysis of the spatially periodic solution is performed with a Galerkin approach allowing arbitrary perturbations. The temporal evolution of the patterns is studied for periodic boundary conditions and for conditions corresponding to a circular container. While hexagons are known to be stable close to threshold (i.e. for Rayleigh numbers R Rc) it has been commonly assumed that quite generally they become unstable to rolls already for slightly higher Rayleigh numbers and remain unstable as the Rayleigh number is increased. In contrast we nd that depending on the strength of the non-Boussinesq eects the hexagons can restabilize again for larger Rayleigh numbers. For instance, hexagons that become unstable at (R Rc)=Rc = 0:15 can become stable again at = 0:2. Direct simulations for circular containers show that these reentrant hexagons can prevail even for side-wall conditions that favor convection in the form of the competing stable rolls. For sucien tly strong non-Boussinesq eects hexagons are stable even over the whole R range considered, 0 1:5. The reentrance of hexagons for larger R is attributed to the increase of the non-Boussinesq eects with Rayleigh number and the existence of stable hexagons in standard Boussinesq convection at larger Rayleigh numbers. These results provide insight into recent convection experiments using SF6 in which reentrant hexagons were observed.

Geometric diagnostics of complex patterns: Spiral defect chaos

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems, we present an automated approach that aims at characterizing quantitatively spiral-like elements in complex stripe-like patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arc length and their winding number. In addition, it yields the number of pattern components (Betti number of order 1), as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven Rayleigh-Benard convection and find that the arc length of spirals decreases monotonically with decreasing Prandtl number of the fluid and increasing heating. By contrast, the winding number of the spirals is nonmonotonic in the heating. The distribution function for the number of spirals is significantly narrower than a Poisson distribution. The distribution function for the winding number shows approximately an exponential decay. It depends only weakly on the heating, but strongly on the Prandtl number. Large spirals arise only for larger Prandtl numbers (Pr approximately > 1). In this regime the joint distribution for the spiral length and the winding number exhibits a three-peak structure, indicating the dominance of Archimedean spirals of opposite sign and relatively straight sections. For small Prandtl numbers the distribution function reveals a large number of small compact pattern components.

Free surface liquid films of binary mixtures. Two-dimensional steady structures at off-critical compositions

We present steady non-linear solutions of films of confined polymer blends deposited on a solid substrate at off-critical concentrations with a free deformable surface. The solutions are obtained numerically using a variational form of the Cahn-Hilliard equation in the static limit, which allows for internal diffuse interfaces between the two components of the mixture. Existence of most of the branches of non-linear solutions at off-critical concentrations can be predicted from the knowledge of the branching points obtained with a linear stability analysis plus the non-linear solutions at critical concentrations. However, some families of solutions are found not to have correspondence at critical compositions. We take a value for surface tension that allows strong deformations at the sharp free upper surface. Varying the average composition and the length and thickness of the films we find a rich morphology of static films in the form of laterally structure films, layered films, droplets on the substrate,...

Reentrant and whirling hexagons in non-Boussinesq convection

Abstract.We review recent computational results for hexagon patterns in non-Boussinesq convection. For sufficiently strong dependence of the fluid parameters on the temperature we find reentrance of steady hexagons, i.e. while near onset the hexagon patterns become unstable to rolls as usually, they become again stable in the strongly nonlinear regime. If the convection apparatus is rotated about a vertical axis the transition from hexagons to rolls is replaced by a Hopf bifurcation to whirling hexagons. For weak non-Boussinesq effects they display defect chaos of the type described by the two-dimensional (2D) complex Ginzburg–Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and localized bursting of the whirling amplitude is found. In this regime the coupling of the whirling amplitude to (small) deformations of the hexagon lattice becomes important. For yet stronger non-Boussinesq effects this coupling breaks up the hexagon lattice and strongly disordered states characterized by whirling and lattice defects are obtained.