Taper-induced control of viscous fingering in variable-gap Hele-Shaw flows

Gas Compression Systematically Delays the Onset of Viscous Fingering

Using gas to drive liquid from a Hele-Shaw cell leads to classical viscous fingering. Strategies for suppressing fingering have received substantial attention. For steady injection of an incompressible gas, the intensity of fingering is controlled by the capillary number Ca. Here, we show that gas compression leads to an unsteady injection rate controlled primarily by a dimensionless compressibility number C. Increasing C systematically delays the onset of fingering at high Ca, highlighting compressibility as an overlooked but fundamental aspect of gas-driven fingering.

Interfacial Dynamics in Dual Channels: Inspired by Cuttlebone

The cuttlebone, a chambered gas-filled structure found in cuttlefish, serves a crucial role in buoyancy control for the animal. This study investigates the motion of liquid-gas interfaces within cuttlebone-inspired artificial channels. The cuttlebone's unique microstructure, characterized by chambers divided by vertical pillars, exhibits interesting fluid dynamics at small scales while pumping water in and out. Various channels were fabricated with distinct geometries, mimicking cuttlebone features, and subjected to different pressure drops. The behavior of the liquid-gas interface was explored, revealing that channels with pronounced waviness facilitated more non-uniform air-water interfaces. Here, Lyapunov exponents were employed to characterize interface separation, and they indicated more differential motions with increased pressure drops. Channels with greater waviness and amplitude exhibited higher Lyapunov exponents, while straighter channels exhibited slower separation. This is potentially aligned with cuttlefish's natural adaptation to efficient water transport near the membrane, where more straight channels are observed in real cuttlebone.

Transformation hydrodynamic metamaterials: Rigorous arguments on form invariance and structural design with spatial variance

Transformation optics has been a powerful tool to manipulate physical fields if the governing equations in two spaces satisfy a certain form invariance. A recent interest has been applying this method to design hydrodynamic metamaterials described by the Navier-Stokes equations. However, transformation optics may not be applicable to such a general fluid model while especially the rigorous analysis is still absent. In this work, we give a definite criterion of the required form invariance, i.e., metric of one space and its affine connections that appear in curvilinear coordinates can be incorporated into material properties or interpreted by extra physical mechanisms introduced in another space. Based on this criterion, we prove that the Navier-Stokes equations, as well as its simplification in creeping flows (the Stokes equation), cannot be form invariant due to the redundant affine connections from their viscous terms. On the contrary, the creeping flows under the lubrication approximation, saying the classical Hele-Shaw model and its anisotropic counterpart, can retain the form of their governing equations for steady incompressible isothermal Newtonian fluids. Besides, we propose the design of multilayered structures with spatially varying cell depth to mimic the required anisotropic shear viscosity for modulating Hele-Shaw flows. Our results correct previous misunderstandings about the applicability of transformation optics under the Navier-Stokes equations, reveal the decisive role of lubrication approximation in maintaining form invariance (consistent with recent experiments featuring shallow configurations), and provide a feasible approach in experimental fabrication.

Stabilization of periodically forced Hele-Shaw flows by means of a nonmonotonic viscosity profile

The onset of viscous fingering in the presence of a viscosity profile is investigated theoretically for two immiscible fluids undergoing a time-dependent injection. Here, we show that the presence of a positive viscosity gradient at the interface between both fluids stabilizes the interface facilitating the spread of the perturbation. This effect is much more pronounced in the case of sinusoidal injection flows. The influence of the viscosity gradient on the dispersion relation is analyzed. Numerical simulations of the Navier-Stokes equation confirm the linear stability analysis.

Switching Modes of Mixing Due to an Adjustable Gap in a Continuous-Flow Microreactor

Microreactors are an important development in chemical engineering since the pharmaceutical industry needs flexible production rather than a large amount of product yield. The size of the microreactor may be so small that it requires the development of non-mechanical methods for reagent mixing. In this paper, we propose the design of a continuous-flow microreactor in the form of a narrow cell with a variable gap. By tuning the gap width in time and space, one can control the reaction rate and regulate the product yield. We show that the governing equation for the fluid flow can be reduced to the Darcy equation with permeability varying in space and time. As a test reaction, we consider the neutralization of nitric acid with sodium hydroxide resulting in the solutal convection in the presence of gravity. We show numerically that the prototyping spatially-distributed relief of the reactor walls can successfully separate the incoming and outgoing flows of reagents, control the mixing intensity, increase or decrease the product yield. We demonstrate also the dynamic control of the reactor efficiency via real-time local changes in the gap width.

Manipulation of viscous fingering in a radially tapered cell geometry

When a more mobile fluid displaces another immiscible one in a porous medium, viscous fingering propagates with a partial sweep, which hinders oil recovery and soil remedy. We experimentally investigate the feasibility of tuning such fingering propagation in a nonuniform narrow passage with a radial injection, which is widely used in various applications. We show that a radially converging cell can suppress the common viscous fingering observed in a uniform passage, and a full sweep of the displaced fluid is then achieved. The injection flow rate Q can be further exploited to manipulate the viscous fingering instability. For a fixed gap gradient α, our experimental results show a full sweep at a small Q but partial displacement with fingering at a sufficient Q. Finally, by varying α, we identify and characterize the variation of the critical threshold between stable and unstable displacements. Our experimental results reveal good agreement with theoretical predictions by a linear stability analysis.