Showing papers on "Hele-Shaw flow published in 2020"

Multiphase Viscoplastic Flows in a Nonuniform Hele-Shaw Cell: A Fluidic Device to Control Interfacial Patterns

Abstract: Inspired by the problem of the removal of in situ gel-like materials via a displacement flow process, we use a fluidic device known as Hele-Shaw cell to study the flow parameters related to the dis...

Dissolution of a cylindrical disk in Hele-Shaw flow: a conformal-mapping approach

Abstract: We apply conformal mapping to find the evolving shapes of a dissolving cylinder in a potential flow. Similar equations can be used to describe melting in a flowing liquid phase. Results are compared with microfluidic experiments and numerical simulations. Shapes predicted by conformal mapping agree almost perfectly with experimental observations, after a modest (20 %) rescaling of the time. Finite-volume simulations show that the differences with experiment are connected to the underlying assumptions of the analytical model: potential flow and diffusion-limited dissolution. Approximate solutions of the equations describing the evolution of the shape of the undissolved solid can be derived from a Laurent expansion of the mapping function from the unit circle. Asymptotic expressions for the evolution of the area of the disk and the shift in its centre of mass have been derived at low and high Peclet number. Analytic approximations to the leading-order Laurent coefficients provide additional insight into the mechanisms underlying pore-scale dissolution.

Control of instability by injection rate oscillations in a radial Hele-Shaw cell

Abstract: Small spatial perturbations grow into fingers along the unstable interface of a fluid displacing a more viscous fluid in a porous medium or a Hele-Shaw cell. Mitigating this Saffman-Taylor instability increases the efficiency of fluid displacement applications (e.g., oil recovery), whereas amplifying these perturbations is desirable in, e.g., mixing applications. In this work, we investigate the Saffman-Taylor instability through analysis and experiments in which air injected with an oscillatory flow rate outwardly displaces silicone oil in a radial Hele-Shaw cell. A solution for linear instability growth that shows the competing effects of radial growth and surface tension, including wetting effects, is defined given an arbitrary reference condition. We use this solution to define a condition for stability relative to the constant flow rate case and make initial numerical predictions of instability growth by wave number for a variety of oscillations. These solutions are then modified by incorporating reference conditions from experimental data. The morphological evolution of the interface is tracked as the air bubble expands and displaces oil between the plates. Using the resulting images, we analyze and compare the linear growth of perturbations about the mean interfacial radius for constant injection rates with and without superimposed oscillations. Three distinct types of flow rate oscillations are found to modulate experimental linear growth over a constant phase-averaged rate of fluid displacement. In particular, instability growth at the interface is mildly mitigated by adding to the base flow rate provided by a peristaltic pump a second flow with low-frequency oscillations of small magnitude and, to a lesser extent, high-frequency oscillations of large amplitude. In both cases, the increased stability results from the selective suppression of the growth of large wave numbers in the linear regime. Contrarily, intermediate oscillations consistently destabilize the interface and significantly amplify the growth of the most unstable wave numbers of the constant flow rate case. Numerical predictions of low-frequency oscillations of opposite sign (initially decreasing) show promise of even greater mitigation of linear instability growth than that observed in this investigation. Looking forward, proper characterization of the unsteady, wetting, and nonlinear dynamics of instability growth will give further insight into the efficacy of oscillatory injection rates.

Dissolution phase diagram in radial geometry

Abstract: Fractures play an important role as flow paths in porous media. When an undersaturated reactive fluid is flowing in a fracture, the dissolution process will alter the flow paths locally. Dissolution patterns grow slowly, but they may lead to a dramatic reorganization of the flow. Here, we study dissolution in a radial geometry, which is relevant for a number of practical applications, e.g. the acidization of oil reservoirs. Different dissolution patterns are presented in a phase diagram with the Peclet and Damkohler numbers as parameters. We also analyze quantitatively the density of wormholes and the relation between the aperture roughness and the ramified patterns.

Obstructed viscoplastic flow in a Hele-Shaw cell

Abstract: Masoud Daneshi,1 Jordan MacKenzie,1 Neil J. Balmforth,2 D. Mark Martinez,1 and Duncan R. Hewitt 3 1Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, Canada V6T 1Z4 2Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z4 3Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom

Flow-Driven Precipitation Patterns with Microemulsions in a Confined Geometry

Abstract: Precipitation reactions occurring under fluid flow can selforganize a variety of complex spatiotemporal patterns. Herein, we investigate the structures formed when a CoCl2-containing water-in-oil microemulsion is injected into an aqueous NaOH solution. The reacting liquids are confined to the thin gap of a horizontal Hele-Shaw cell and injection is carried out from a central port. Dark-field detection of the patterns reveals borders that can deviate from smooth circles by either jagged or sinusoidal deformations. The interior of the patterns shows speckle-like features that can arrange along either spike-like tracks or concentric circular segments. The area ratio of light scattering regions increases with increasing flow rates but decreases sharply with increasing NaOH concentration. The latter transition is interpreted as a shift from patterns dominated by the physical breakdown of the microemulsion to patterns strongly affected by the precipitation of α-Co(OH)2.

Numerical Analysis of Density-Driven Reactive Flows in Hele-Shaw Cell Geometry

Abstract: In this paper, a two-dimensional numerical simulation of the unsteady state of a two non-isothermal immiscible liquids layer system filling a reactor formed by two closely spaced parallel glass sheets, which is called an Hele-Shaw cell, vertically oriented, with an expected neutralization reaction between an acid and a base in the lower layer, under the action of gravity, is studied. Attention is given on the general behavior of the complete temporal pattern evolution (velocity, temperature, and concentration profiles) and the identification of the exothermic reaction’s role in giving birth to chemo-hydrodynamic patterns that occur because of concentration gradients. The effects of gravity and changes in initial acid and base concentrations on the formed patterns were studied. The mathematical model governing the phenomenon was solved numerically by the CFD software package COMSOL Multiphysics, with the finite element method and a comparison with the experimental data was made. The results show that this numerical tool is promising for the understanding of the reactive instabilities happening when two immiscible fluids come into contact.

Mixing layer and turbulent jet flow in a Hele–Shaw cell

Abstract: A plane turbulent mixing in a shear flow of an ideal homogeneous fluid confined between two relatively close rigid walls is considered. The character of the flow is determined by interaction of vortices arising at the non-linear stage of the Kelvin–Helmholtz instability development and by turbulent friction. In the framework of the shallow water theory and a three-layer representation of the flow, one-dimensional models of a mixing layer are proposed. The obtained equations allow one to determine averaged boundaries of the region of intense fluid mixing. Stationary solutions of the governing equations are constructed and analysed. Using the averaged flow characteristics obtained by one-dimensional equations, a hyperbolic system for determining the velocity profile and Reynolds shear stress across the mixing layer is derived. Comparison with the experimental results of the evolution of turbulent jet flows in a Hele–Shaw cell shows that the proposed models provide a fairly accurate description of the average boundaries of the region of intense mixing, as well as the velocity profile and Reynolds shear stress across the mixing layer.

An ill-posed problem in hydrodynamic stability of multi-layer Hele-Shaw flow

Abstract: An useful approximation for the displacement of two immiscible fluids in a porous medium is the Hele-Shaw model. We consider several liquids with different constant viscosities, inserted between the displacing fluids. The linear stability analysis of this model leads us to an ill-posed problem. The growth rates (in time) of the perturbations exist iff some compatibility conditions on the interfaces are verified. We prove that these conditions cannot be fulfilled.

Control of Instability by Injection Rate Oscillations in a Radial Hele-Shaw Cell

Abstract: Small spatial perturbations grow into fingers along the unstable interface of a fluid displacing a more viscous fluid in a porous medium or a Hele-Shaw cell. Mitigating this Saffman-Taylor instability increases the efficiency of fluid displacement applications (e.g., oil recovery), whereas amplifying these perturbations is desirable in, e.g., mixing applications. In this work, we investigate the Saffman-Taylor instability through analysis and experiments in which air injected with an oscillatory flow rate outwardly displaces silicone oil in a radial Hele-Shaw cell. A solution for linear instability growth that shows the competing effects of radial growth and surface tension, including wetting effects, is defined given an arbitrary reference condition. We use this solution to define a condition for stability relative to the constant flow rate case and make initial numerical predictions of instability growth by wave number for a variety of oscillations. These solutions are then modified by incorporating reference conditions from experimental data. The morphological evolution of the interface is tracked as the air bubble expands and displaces oil between the plates. Using the resulting images, we analyze and compare the linear growth of perturbations about the mean interfacial radius for constant injection rates with and without superimposed oscillations. Three distinct types of flow rate oscillations are found to modulate experimental linear growth over a constant phase-averaged rate of fluid displacement. In particular, instability growth at the interface is mildly mitigated by adding to the base flow rate provided by a peristaltic pump a second flow with low-frequency oscillations of small magnitude and, to a lesser extent, high-frequency oscillations of large amplitude. In both cases, the increased stability results from the selective suppression of the growth of large wave numbers in the linear regime. Contrarily, intermediate oscillations consistently destabilize the interface and significantly amplify the growth of the most unstable wave numbers of the constant flow rate case. Numerical predictions of low-frequency oscillations of opposite sign (initially decreasing) show promise of even greater mitigation of linear instability growth than that observed in this investigation. Looking forward, proper characterization of the unsteady, wetting, and nonlinear dynamics of instability growth will give further insight into the efficacy of oscillatory injection rates.

Direct numerical simulation of turbulent wall flows at constant power input

Abstract: We propose a new flow condition for conducting DNS of internal flows: Instead of keeping either a flow rate or a pressure gradient constant the new simulation approach is to maintain the total power input, at a constant value. This new condition is closer to a practical situation where the power input, i.e. the product of flow rate and pressure drop (and not flow rate or pressure drop independently), is given by a pump for example. From the flow control point of view, the constant power input approach is the most suitable approach for an energetic analyses of flows with/without control. BACKGROUND & OBJECTIVES Typically direct numerical simulations (DNS) of turbulent wall flows are carried out while keeping constant in time either the flow rate (CFR) or, less often, the pressure gradient (CPG), under the generally acknowledged assumption that the statistics of the flow are insensitive to this arbitrary choice. However, this apparently uninfluential choice becomes critical when drag-redution techniques are applied to turbulent channel or pipe flows. Under the CFR condition, a successful drag reducing technique reduces friction drag, which immediately translates into a reduction of the pumping power. One important drawback of the CFR constraint is that the wall shear stress, which is a dominant factor in near-wall turbulence dynamics, is changed due to the applied control, so that it is difficult to extract the essential effects of a control input itself owing to superimposed Reynolds number effects. On the other hand, when the CPG condition is used, friction drag is unchanged by design, and 'drag reduction' manifests itself through an increase of the flow rate, which implies an increase in the power required to drive the flow. Recently, we introduced a novel way of assessing the energetic performance of flow control techniques by explicitly ac- counting for the total power spent for pumping and control (Frohnapfel et al. (2012)). The presented energy-convenience plane allows the designer to weight the energetic cost of the applied control against the increased performance, that is observed through an increased flow rate. This evaluation plane graphically emphasizes that besides CFR and CPG, many different routes link an uncontrolled turbulent state to the laminar flow state, which is the ultimate goal in drag reduction control. Among these various routes, the one of constant power input (CPI) is an alternative that puts the energetic advan- tage of laminarizing the flow into an interesting perspective: at a Reynolds number of 75000 relaminarisation corresponds to the total energy savings of 86% for CPI while stunning 50 times larger flow rate will be reached for CPG. In addition to this novel evaluation perspective that might be useful in judging the potential economical benefit of (active) flow control techniques, the CPI approach provides an interesting tool for the fundamental analysis of turbulent flows and their control. If we assume that an analysis of the energy flow through the system and its modifications in flow control can teach us something about the physics of turbulence itself and possibly allow generalized statements on how to efficiently control a flow, we need to be able to compare different flow control scenarios at the same energy flux and dissipation rate. This is basically impossible to realize with CFR or CPG but straightforward to implement with the CPI approach. The aim of the present study is thus to introduce CPI as a physically sound strategy to carry out numerical experiments. We show how the CPI approach naturally leads to identifying a characteristic velocity scale, based on the power consumption, and thus a related Reynolds number. In this way it becomes natural and easy to carry out DNS under the CPI condition. First results for CPI in controlled flows are presented.

Development of magnetoelastic fingering patterns in a rectangular Hele-Shaw cell

Abstract: A ferrofluid and a nonmagnetic fluid flow in a rectangular Hele-Shaw cell subjected to an in-plane magnetic field. A chemical reaction occurs, and the two-fluid interface becomes elastic. The produced patterns range from straight front to sharp tip ones, and they differ from usual Saffman-Taylor fingers.