Session E27: Focus Session: Unanswered Questions in Viscous Fingering II

Saffman-Taylor Instability and the Inner Splitting Mechanism

The classical miscible displacement experiments of Wooding (1969) exhibit an inner splitting phenomenon that remained unexplained for over 40 years. 3D Navier-Stokes simulations presented in J. Fluid Mech. 687, 431-460 (2011), were the first ones to reproduce these experimental observations numerically, and to demonstrate that they are linked to concentrated streamwise vortices. The origin of these concentrated streamwise vortices remained a mystery, however. The current investigation, published at Phys. Rev. Lett. 118, 124502 (2017), finally resolves this long-standing issue. Towards this end, we compare 3D Navier-Stokes simulation results for neutrally buoyant, viscously unstable displacements and gravitationally unstable, constant viscosity ones. Only the former exhibit the generation of streamwise vorticity. The simulation results present conclusive evidence that it is caused by the lateral displacement of the more viscous fluid by the less viscous one, with the variable viscosity terms playing a dominant role.   

Numerical simulation of miscible viscous fingering with viscosity change in a displacing fluid by chemical reaction

Viscous fingering (VF) with viscosity changes by chemical reactions in case of miscible systems have been investigated both experimentally and theoretically in the recent years. Nagatsu \textit{et al}. investigated experimentally miscible VF in which viscosity of the displaced fluid $^{\mathrm{[1]}}_{\mathrm{\thinspace }}$or the displacing one $^{\mathrm{[2]}}$ is changed by fast chemical reaction They showed that VF was more dense by the viscosity increase whereas less dense by the viscosity increase regardless of whether the viscosity change occurs in the displaced fluid or displacing one. From a theoretical viewpoint, numerical simulation performed on the reactive VF where viscosity of the displaced fluid is changed by instantaneously fast chemical reaction $^{\mathrm{[3]}}$. The results had a good agreement with those in the corresponding experiment $^{\mathrm{[1]}}$. In this work, we have conducted numerical simulation on such reactive VF where viscosity of the displacing fluid is changed. We have found the results have a good agreement with the corresponding experimental ones $^{\mathrm{[2]}}$ [1] Y. Nagatsu \textit{et al.}, J. Fluid Mech., 571 475$-$493 (2007). [2] Y. Nagatsu \textit{et al.}, Phys. Fluids, 22 024101 (2010). [3] Y. Nagatsu and A. De Wit, Phys Fluids, 23 043103 (2011).   

Centrifugal fingering in a curved Hele-Shaw cell: A generalized Euler's elastica shape for the two-fluid interface

We study a family of generalized elastica-like equilibrium shapes that arise at the interface separating two fluids in a curved rotating Hele-Shaw cell. This family of stationary interface solutions consists of shapes that balance the competing capillary and centrifugal forces in such a curved flow environment. We investigate how the emerging interfacial patterns are impacted by changes in the geometric properties of the curved Hele-Shaw cell. A vortex-sheet formalism is used to calculate the two-fluid interface curvature, and a gallery of possible shapes is provided to highlight a number of peculiar morphological features. A linear perturbation theory is employed to show that the most prominent aspects of these complex stationary patterns can be fairly well reproduced by the interplay of just two interfacial modes. The connection of these dominant modes to the geometry of the curved cell, as well as to the fluid dynamic properties of the flow, is discussed.   

Viscous Fingering on an Immiscible Reactive Interface with Variation of Interfacial Tension

The effects of chemical reaction, in which surfactants are produced on the interface of two immiscible fluids, on viscous fingering in a radial Hele-Shaw flow are numerically investigated. The presence of surfactants reduces interfacial tension, which is an important factor to the fingering pattern formation. In the present study, influences of reaction rate and dispersion of produced surfactants, represented respectively by dimensionless parameters of Damkohler number and Peclet number, are evaluated systematically. Secondary fingering instability, e.g., tip-splitting and side-branching, is triggered by chemical reactions. Weaker surface tension generally induces tip-splitting. For the case of high Damkohler number, because of the vortex pairs generated within each finger, surfactant tends to accumulate significantly on the side of finger, so that side-branching is preferred. Nevertheless, side-branching is suppressed in the cases associated with low Peclet number, in which strong dispersion reduces the local variation of surfactant concentration. Considering the coupled effects by Damkohler number and Peclet number, the patterns obtained by the simulations qualitatively agree with the observations in the experiments.   

What selects the velocity of fingers and bubbles in a Hele-Shaw cell?

It has been widely accepted that surface tension is responsible for the selection of a single pattern out of a continuum of steady solutions for the interface dynamics. Recently, however, it was demonstrated by using time-dependent solutions that surface tension is not required for velocity selection in a Hele-Shaw cell: the velocity is selected entirely within the zero surface tension dynamics, as the selected pattern is the only attractor of the dynamics. These works changed the paradigm regarding the necessity of surface tension for selection, but were limited to a single interface. Here we show that the same selection mechanism holds for any number of interfaces. We present a new class of exact solutions for multiple time-evolving bubbles in a Hele-Shaw cell. The solution is given by a conformal mapping from a multiply connected domain and is written in closed form in terms of certain special functions (the secondary Schottky-Klein prime functions). We demonstrate that the bubbles reach an asymptotic steady velocity, $U$, which is twice greater than the velocity, $V$, of the uniform background flow, i.e., $U=2V$. The result does not depend on the number of bubbles. This confirms the prediction that contrary to common belief velocity selection does not require surface tension   

Crustal fingering: solidification on a viscously unstable interface

Motivated by the formation of gas hydrates in seafloor sediments, here we study the volumetric expansion of a less viscous gas pocket into a more viscous liquid when the gas-liquid interfaces readily solidify due to hydrate formation. We first present a high-pressure microfluidic experiment to study the depressurization-controlled expansion of a Xenon gas pocket in a water-filled Hele-Shaw cell. The evolution of the pocket is controlled by three processes: (1) volumetric expansion of the gas; (2) rupturing of existing hydrate films on the gas-liquid interface; and (3) formation of new hydrate films. These result in gas fingering leading to a complex labyrinth pattern. To reproduce these observations, we propose a phase-field model that describes the formation of hydrate shell on viscously unstable interfaces. We design the free energy of the three-phase system to rigorously account for interfacial effects, gas compressibility and phase transitions. We model the hydrate shell as a highly viscous fluid with shear-thinning rheology to reproduce shell-rupturing behavior. We present high-resolution numerical simulations of the model, which illustrate the emergence of complex crustal fingering patterns as a result of gas expansion dynamics modulated by hydrate growth at the interface.