Conformal mapping methods for interfacial dynamics

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.Comment: 37 pages, 12 (mostly color) figure

Occurrence of coexisting dendrite morphologies: immiscible fluid displacement in an anisotropic radial hele-shaw cell under a high flow rate regime

Viscous fingering morphologies during the displacement of a high viscosity fluid by a low viscosity immiscible fluid in a radial fourfold anisotropic Hele-Shaw cell are examined. By using the kerosene-glycerin system for which the µ/T ratio (µ being the relative viscosity and T the interfacial tension between the fluids) is about ten times higher than that for the commonly used air-glycerin system, we have been able to access the hitherto unexplored Nca 1 regime (capillary number Nca=Uµ/T, U being the advancing fingertip velocity). Within the anisotropy-dominated regime, and when flow rates are significantly high (capillary number well beyond Nca=1), a new phase is seen to evolve wherein the dendrites grow simultaneously along the channels and along the directions making an angle of 45° with the channels, both being kinetically driven. This new phase resembles the one observed in a miscible fluid system at all flow rates of the displacing fluid

Novel Experimentally Observed Phenomena in Soft Matter

Soft materials such as colloidal suspensions, polymer solutions and liquid crystals are constituted by mesoscopic entities held together by weak forces. Their mechanical moduli are several orders of magnitude lower than those of atomic solids. The application of small to moderate stresses to these materials results in the disruption of their microstructures. The resulting flow is non-Newtonian and is characterised by features such as shear rate-dependent viscosities and non-zero normal stresses. This article begins with an introduction to some unusual flow properties displayed by soft matter. Experiments that report a spectrum of novel phenomena exhibited by these materials, such as turbulent drag reduction, elastic turbulence, the formation of shear bands and the existence of rheological chaos, flow-induced birefringence and the unusual rheology of soft glassy materials, are reviewed. The focus then shifts to observations of the liquid-like response of granular media that have been subjected to external forces. The article concludes with examples of the patterns that emerge when certain soft materials are vibrated, or when they are displaced with Newtonian fluids of lower viscosities.Comment: 30 pages, 11 figures, invited review article, supplementary videos may be obtained from the journal websit

Viscous fingering in liquid crystals: Anisotropy and morphological transitions

We show that a minimal model for viscous fingering with a nematic liquid crystal in which anisotropy is considered to enter through two different viscosities in two perpendicular directions can be mapped to a two-fold anisotropy in the surface tension. We numerically integrate the dynamics of the resulting problem with the phase-field approach to find and characterize a transition between tip-splitting and side-branching as a function of both anisotropy and dimensionless surface tension. This anisotropy dependence could explain the experimentally observed (reentrant) transition as temperature and applied pressure are varied. Our observations are also consistent with previous experimental evidence in viscous fingering within an etched cell and simulations of solidification.Comment: 12 pages, 3 figures. Submitted to PR

Viscous fingering of miscible fluids in an anisotropic radial hele-shaw cell: coexistence of kinetic and surface-tension dendrite morphology types and an exploration of small-scale influences

The evolution of viscous fingering morphology is examined for the case of a system of miscible fluids in an anisotropic radial Hele-Shaw cell. It is shown that dendritic morphologies similar to the kinetic and surface-tension morphology types coexist for this case. The critical role of the means of introducing anisotropy in the Hele-Shaw cell is established, and an explanation of the pattern behavior is offered on the basis of shape discontinuities of the individual elements of the lattice used to induce anisotropy. The ramifications of such an explanation are experimentally verified by demonstrating a clear difference in the morphology evolution in two halves of a single Hele-Shaw cell, one half of which contains square lattice elements, and the other half of which contains circular lattice elements