In Hele-Shaw flows, boundaries between fluids develop unstable viscous fingers. At vanishing surface tension, the fingers further evolve to cusp-like singularities. We show that the problem admits a {\it weak solution} where shock fronts triggered by a singularity propagate together with a fluid. Shocks form a growing, branching tree of a mass deficit, and a line distribution of vorticity where pressure and velocity of the fluid have finite discontinuities. Imposing that the flow remain curl-free at macroscale determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2,3)-cusp singularity -- an elementary branching event.
In this review article, we discuss recent studies on drops and bubbles in Hele-Shaw cells, focusing on how scaling laws exhibit crossovers from the three-dimensional counterparts and focusing on topics in which viscosity plays an important role. By virtue of progresses in analytical theory and high-speed imaging, dynamics of drops and bubbles have actively been studied with the aid of scaling arguments. However, compared with three dimensional problems, studies on the corresponding problems in Hele-Shaw cells are still limited. This review demonstrates that the effect of confinement in the Hel…
Parallel flow in a Hele-Shaw cell occurs when two immiscible liquids flow with relative velocity parallel to the interface between them. The interface is unstable due to a Kelvin-Helmholtz type of instability in which fluid flow couples with inertial effects to cause an initial small perturbation to grow. Large amplitude disturbances form stable solitons. We consider the effects of applied magnetic fields when one of the two fluids is a ferrofluid. The dispersion relation governing mode growth is modified so that the magnetic field can destabilize the interface even in the absence of inertial …
We study both experimentally and theoretically the descending motion due to gravity of a fluid drop surrounded by another immiscible fluid in a confined space between two parallel plates, i.e., in the Hele-Shaw cell. As a result, we show a new scaling regime of a nonlinear drag friction in viscous liquid that replaces the well-known Stokes' drag friction through a clear collapse of experimental data thanks to the scaling law. In the novel regime, the dissipation in the liquid thin film formed between the drop and cell walls governs the dynamics. The crossover of this scaling regime to another …
Solvent exchange (also called solvent shifting or Ouzo effect) is a generally used bottom-up process to mass-produce nanoscale droplets. In this process, a good solvent for some oil is displaced by a poor one, leading to oil nanodroplet nucleation and subsequent growth. Here we perform this process on a hydrophobic substrate so that sessile droplets so-called surface nanodroplets-develop, following the work of Zhang et al. [Zhang, X.; Lu, Z.; Tan, H.; Bao, L.; He, Y.; Sun, C.; Lohse, D. Proc. Natl. Acad. Sci. U.S.A. 2015, 122, 9253-9257]. In contrast to what was done in that paper, we chose a …
In this paper we study the effects of dynamic wetting on the immiscible displacement of a high viscosity fluid subject to the radial injection of a less viscous fluid in a Hele-Shaw cell. The displaced fluid can leave behind a trailing film that coats the cell walls, dynamically affecting the pressure drop at the fluid interface. By considering the non-linear pressure drop in a boundary element formulation, we construct a Picard scheme to iteratively predict the interfacial velocity and subsequent displacement in finite mobility ratio flow regimes. Dynamic wetting delays the onset of finger bi…
We investigate the large time behavior of an agent based model describing tumor growth. The microscopic model combines short-range repulsion and cell division. As the number of cells increases exponentially in time, the microscopic model is challenging in terms of computational time. To overcome this problem, we aim at deriving the associated macroscopic dynamics leading here to a porous media type equation. As we are interested in the long time behavior of the dynamics, the macroscopic equation obtained through usual derivation method fails at providing the correct qualitative behavior (e.g. …
In this paper, the interaction between two immiscible fluids with a finite mobility ratio is investigated numerically within a Hele-Shaw cell. Fingering instabilities initiated at the interface between a low viscosity fluid and a high viscosity fluid are analysed at varying capillary numbers and mobility ratios using a finite mobility ratio model. The present work is motivated by the possible development of interfacial instabilities that can occur in porous media during the process of CO$_2$ sequestration, but does not pretend to analyse this complex problem. Instead, we present a detailed stu…
Viscous Bending Mitigates the Spontaneous Meandering of Rivulets in Hele-Shaw Cells
Matière et Systèmes Complexes UMR 7057, Université Paris Cité, CNRS（复杂物质与系统联合研究实验室，巴黎西岱大学，法国国家科学研究中心）；Laboratory of Fluid Mechanics and Instabilities, EPFL（流体力学和不稳定性实验室，洛桑联邦理工学院）
