Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

Abstract

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid-viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem lead to slightly fewer fingers than this linear prediction. † Email address for correspondence: scott.mccue@qut.edu.au arXiv:1901.00288v4 [physics.flu-dyn] 7 Jul 2019 , (5.3)

Modelling finger propagation in elasto-rigid channels

“…2017; Bongrand & Tsai 2018) or dynamically via the introduction of elastic walls (Pihler-Puzović et al. 2012; Al-Housseiny, Christov & Stone 2013; Lister, Peng & Neufeld 2013) or via a time-varying displacement of rigid walls (Zheng, Kim & Stone 2015; Morrow, Moroney & McCue 2019; Vaquero-Stainer et al. 2019).…”

Moving Boundary Problems for Quasi-Steady Conduction Limited Melting

“…However, an advantage of using a finite difference stencil is that it can easily be adapted to problems where the boundary integral method is not applicable. For example, we have used a similar method to the one presented in this section to study non-standard Hele-Shaw flow where pressure is not harmonic and for which the boundary integral method is much less suitable [50].…”

Computational Analysis of Interfacial Dynamics in Angled Hele-Shaw Cells: Instability Regimes

“…Since then, an explosion of works has analyzed a variety of related interfacial instability problems in "nonstandard" Hele-Shaw configurations (Morrow et al, 2019), both fixed geometries (Dias and Miranda, 2013, Al-Housseiny et al, 2012, Hu et al, 2016, Jackson et al, 2017, Grenfell-Shaw and Woods, 2017, Bongrand and Tsai, 2018 and those with flow-driven geometric changes (Pihler-Puzović et al, 2012, Pihler-Puzović et al, 2013. Further variants on the same problem also include controlling the injection flow rate (Dias and Miranda, 2010a, Dias et al, 2010, changing the permeability by adjusting the structure of the porous medium (Jackson et al, 2017, Rabbani et al, 2018, Brandão et al, 2018, applying an external force via rotation of the geometry or through a magnetic field (Carrillo et al, 1996, Alvarez-Lacalle et al, 2003, Miranda and Alvarez-Lacalle, 2005, Anjos et al, 2018b, changing the fluid properties through the viscosity ratio (Anjos et al, 2017), using non-Newtonian fluids (Vlad and Maher, 2000, Lindner et al, 2002, Tordjeman, 2007, Boronin et al, 2015 or even adding a suspended particulate phase (Xu et al, 2016, Kim et al, 2017.…”

“…We are interested in geometric controls. To this end, there are three primary ways to alter the physical geometry of an experimental Hele-Shaw apparatus: (i) creating a gradient along the flow direction by relaxing the requirement that the plates be parallel (Zhao et al, 1992, Dias and Miranda, 2010b, Al-Housseiny et al, 2012, Bongrand and Tsai, 2018, Anjos et al, 2018a, Morrow et al, 2019; (ii) using an elastic membrane (that deforms due to flow underneath it) instead of a solid top plate (Pihler-Puzović et al, 2012, Pihler-Puzović et al, 2013; and (iii) lifting one of the plates in a time dependent manner (Dias and Miranda, 2010a, Zheng et al, 2015, Díaz-Piola et al, 2017. Among these possibilities, the case of a geometric gradient in the flow direction has attracted special attention because it naturally imitates the non-uniform, fractured subsurface flow passages (Muggeridge et al, 2014, Osiptsov, 2017.…”