Session M9: Instability: Interfacial and Thin-Film VI - Fingering

Stability Results on Multi-Layer Hele-Shaw Flows

Saffman-Taylor instability, which occurs when a less viscous fluid drives a more viscous fluid, has been studied for many years and has a wide range of applications. In particular, an understanding of this phenomenon is helpful in the attempt to maximize the effectiveness of chemically enhanced oil recovery techniques. We study this instability through linear stability analysis of multi-layer radial Hele-Shaw flows of immiscible fluids. We take classic results on the instability of flows consisting of two fluids and extend them to flows with an arbitrary number of fluid phases. Using upper bound results on the growth rate of instabilities obtained in this general setting, we are able to give conditions under which this regime is less unstable than the single interface case.   

A Solutal Fingering Instability during Capillary Imbibition in Porous Media

We report the existence of a solute-driven fingering instability that occurs during capillary imbibition into cellulosic porous media. Contacting a piece of paper with an aqueous solution containing hydrophobic solutes causes the liquid to move forward into the paper. For sufficiently low solute concentrations and sufficiently high ambient humidities, the imbibition front moves forward smoothly as expected. For higher concentrations and lower humidities, however, the imbibition front develops spatially periodic oscillations that grow with time, i.e., a fingering instability occurs. Surprisingly, under these conditions the solute concentration becomes larger at the imbibition front compared to the bulk, contrary to the behaviour expected based on chromatographic separation. We present a stability analysis predicated on solutal changes in the interfacial tension driven by water imbibition into a precursor film ahead of the macroscopically observable air/water interface, and we derive a critical P\'{e}clet number above which the interface is unstable.   

Magnetically induced solitons in a Hele-Shaw cell

We study the development of propagating solitons on the interface separating two viscous fluids flowing in parallel in a vertical Hele-Shaw cell. One of the fluids is a ferrofluid and a uniform magnetic field is applied in the plane of the cell, making an angle with the initially undisturbed interface. We derive a Korteweg-de Vries equation for long waves at this confined geometry which predicts the possibility of controlling the speed of the solitons by magnetic means. The influence of the tilted magnetic field on the velocity and on the shape of the solitary waves is investigated.   

Effect of transient interfacial tension on miscible viscous fingering

The pressure-driven displacement flow of a more viscous fluid by a less viscous one is an unstable configuration in the context of miscible viscous fingering in porous media. Steep concentration, density or temperature gradient at the interface of the underlying fluids gives rise to a nonconventional stress in the system, which causes an effective or transient interfacial tension. Such tension has been incorporated using Korteweg stresses in the momentum equation. The system has been modeled by coupling the continuity and Darcy-Korteweg equations with the convection-diffusion equation for the evolution of the solvent concentration. We have shown by a numerical simulation based on Fourier-spectral method that such system can remain stable for a comparatively longer initial transient period. This delay on the onset of instability is due to transient interfacial tension acting at the miscible diffusive interface. The results show that increasing the strength of the stress the onset of instability can be delayed significantly. However, the system may or may not become completely stable depending upon the configuration of the displaced fluid. Linear stability analysis of such system having two semi-infinite fluids and with a finite slice of fluid has been also investigated.   

Strong sample solvent and viscous fingering effects on the dynamics of adsorbed solutes

The pressure driven displacement flow in a porous medium with viscosity increasing in the direction of the flow leads to viscous fingering of the rear interface of finite samples. Sample solvent effects exist if the adsorption constant of solutes on the porous matrix depends on the solvent composition. A sample solvent stronger than the displacing fluid then leads to spatially variable retention of the solute initially dissolved in the sample. We investigate here the influence of these two effects, variable retention and viscosity contrast, on the dynamics of the solute. The continuity equation and Darcy's law coupled to convection-diffusion equations for the evolution of the sample and solute concentration are solved numerically to analyze the above phenomena. The sample viscosity and solute retention are assumed to depend exponentially on the concentration of a solute initially contained in the sample. The results demonstrate the development of two solute concentration zones, one of them being affected by the viscous fingering pattern. The effect of the fingering instability on the retained solute zone increases with an increase in the strength of the sample solvent. This, in turn, increases the spreading zone of the solute and delays the disengagement of the solute from the sample zone.   

Wavelength selection in injection-driven Hele-Shaw flows: A maximum amplitude criterion

As in most interfacial flow problems, the standard theoretical procedure to establish wavelength selection in the viscous fingering instability is to maximize the linear growth rate. However, there are important discrepancies between previous theoretical predictions and existing experimental data [T. Maxworthy, Phys. Rev. A {\bf 39}, 5863 (1989)]. In this work we perform a linear stability analysis of the radial Hele-Shaw flow system that takes into account the combined action of viscous normal stresses and wetting effects. Most importantly, we introduce an alternative selection criterion for which the selected wavelength is determined by the maximum of the interfacial perturbation amplitude. The effectiveness of such a criterion is substantiated by the significantly improved agreement between theory and experiments.   

Determining the number of fingers in the lifting Hele-Shaw problem

The lifting Hele-Shaw cell flow is a variation of the celebrated radial viscous fingering problem for which the upper cell plate is lifted uniformly at a specified rate. This procedure causes the formation of intricate interfacial patterns. Most theoretical studies determine the total number of emerging fingers by maximizing the linear growth rate, but this generates discrepancies between theory and experiments. In this work, we tackle the number of fingers selection problem in the lifting Hele-Shaw cell by employing the recently proposed maximum-amplitude criterion [Dias and Miranda, Phys. Rev. E {\bf 88}, 013016 (2013)]. Our linear stability analysis accounts for the action of capillary, viscous normal stresses, and wetting effects, as well as the cell confinement. The comparison of our results with very precise laboratory measurements for the total number of fingers shows a significantly improved agreement between theoretical predictions and experimental data.