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Coutinho ÍM, Miranda JA. Role of interfacial rheology on fingering instabilities in lifting Hele-Shaw flows. Phys Rev E 2023; 108:025104. [PMID: 37723719 DOI: 10.1103/physreve.108.025104] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2023] [Accepted: 08/06/2023] [Indexed: 09/20/2023]
Abstract
The lifting Hele-Shaw cell setup is a popular modification of the classic, fixed-gap, radial viscous fingering problem. In the lifting cell configuration, the upper cell plate is lifted such that a more viscous inner fluid is invaded by an inward-moving outer fluid. As the fluid-fluid interface contracts, one observes the rising of distinctive patterns in which penetrating fingers having rounded tips compete among themselves, reaching different lengths. Despite the scholarly and practical relevance of this confined lifting flow problem, the impact of interfacial rheology effects on its pattern-forming dynamics has been overlooked. Authors of recent studies on the traditional injection-induced radial Hele-Shaw flow and its centrifugally driven variant have shown that, if the fluid-fluid interface is structured (i.e., laden with surfactants, particles, proteins, or other surface-active entities), surface rheological stresses start to act, influencing the development of the viscous fingering patterns. In this paper, we investigate how interfacial rheology affects the stability as well as the shape of the emerging fingered structures in lifting Hele-Shaw flows, at linear and early nonlinear dynamic stages. We tackle the problem by utilizing the Boussinesq-Scriven model to describe the interface and by employing a perturbative mode-coupling scheme. Our linear stability results show that interfacial rheology effects destabilize the interface. Furthermore, our second-order findings indicate that interfacial rheology significantly alters intrinsically nonlinear morphological features of the shrinking interface, inducing the formation of narrow sharp-tip penetrating fingers and favoring enhanced competition among them.
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Affiliation(s)
- Írio M Coutinho
- Departamento de Física, Universidade Federal de Pernambuco, CCEN, Recife, Pernambuco 50670-901, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, CCEN, Recife, Pernambuco 50670-901, Brazil
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2
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Coutinho ÍM, Dias EO, Miranda JA. Effect of interfacial rheology on fingering patterns in rotating Hele-Shaw cells. Phys Rev E 2023; 107:025105. [PMID: 36932566 DOI: 10.1103/physreve.107.025105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2022] [Accepted: 02/13/2023] [Indexed: 06/18/2023]
Abstract
In rotating Hele-Shaw flows, centrifugal force acts, and the interface separating two viscous fluids becomes unstable, driven by the density difference between them. Complex interfacial structures develop where fingers of various shapes and sizes grow, and compete. These patterns have been well studied over the last few decades, analytically, numerically, and experimentally. However, one feature of the pattern-forming dynamics of much current interest has been underappreciated: the role of surface rheological stresses in the deformation, and time evolution of the fluid-fluid interface. In this paper, we employ a perturbative, second-order mode-coupling analysis to investigate how interfacial rheology effects influence centrifugally driven fingering phenomena, beyond the scope of linear stability theory. Describing the viscous Newtonian interface by using a Boussinesq-Scriven model, we derive a nonlinear differential equation that governs the early linear, and nonlinear time evolution of the system. In this framing, the most prevalent dynamical features of the patterns are described in terms of two dimensionless parameters: the viscosity contrast A (dimensionless viscosity difference between the fluids), and the Boussinesq number Bq which involves a ratio between interfacial and bulk viscosities. At the linear level, our results show that for a given A, surface rheological stresses dictated by Bq have a stabilizing role. Nevertheless, our weakly nonlinear findings reveal a more elaborate scenario in which the shape of the fingers, and their finger competition behavior result from the coupled influence of A and Bq. It is found that, although finger competition phenomena depend on the specific values of A and Bq, the fingers tend to widen as Bq is increased, regardless of the value of A.
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Affiliation(s)
- Írio M Coutinho
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - Eduardo O Dias
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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3
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Conrado H, Dias EO, Miranda JA. Impact of interfacial rheology on finger tip splitting. Phys Rev E 2023; 107:015103. [PMID: 36797856 DOI: 10.1103/physreve.107.015103] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2022] [Accepted: 12/22/2022] [Indexed: 06/18/2023]
Abstract
Fluid-fluid interfaces, laden with polymers, surfactants, lipid bilayers, proteins, solid particles, or other surface-active agents, often exhibit a rheologically complex response to deformations. Despite its academic and practical relevance to fluid dynamics and various other fields of research, the role of interfacial rheology in viscous fingering remains fairly underexplored. A noteworthy exception is the work by Li and Manikantan [Phys. Rev. Fluids 6, 074001 (2021)2469-990X10.1103/PhysRevFluids.6.074001], who used linear stability analysis to show that surface rheological stresses act to stabilize the development of radial viscous fingering at the linear regime. In this paper, we perform a perturbative, second-order mode-coupling analysis of the system and investigate the influence of interfacial rheology on the morphology of the fingering structures at early nonlinear stages of the dynamics. In particular, we focus on understanding how interfacial rheology impacts the emblematic finger tip-widening and finger tip-splitting phenomena that take place in radial viscous fingering in Hele-Shaw cells. We describe the viscous Newtonian fluid-fluid interface by using a Boussinesq-Scriven model, and derive a generalized Young-Laplace pressure jump condition at the fluid-fluid interface. In this framing, we go beyond the purely linear description and use Darcy's law to obtain a perturbative mode-coupling differential equation which describes the time evolution of the perturbation amplitudes, accurate to second order. Our early nonlinear mode-coupling results indicate that regardless of their stabilizing action at the linear regime, interfacial rheology effects favor finger tip widening, leading to the occurrence of enhanced finger tip-splitting events.
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Affiliation(s)
- Habakuk Conrado
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
| | - Eduardo O Dias
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
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4
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Brandão R, Carvalho GD, Miranda JA. Mode-coupling approach to near-cuspidal patterns in planar fluid flows. Phys Rev E 2021; 103:063102. [PMID: 34271760 DOI: 10.1103/physreve.103.063102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2021] [Accepted: 05/24/2021] [Indexed: 11/07/2022]
Abstract
We investigate the evolution of the interface separating two Newtonian fluids of different viscosities in two-dimensional Stokes flow driven by suction or injection. A second-order, mode-coupling theory is used to explore key morphological aspects of the emerging interfacial patterns in the stage of the flow that bridges the purely linear and fully nonlinear regimes. In the linear regime, our analysis reveals that an injection-driven expanding interface is stable, while a contracting motion driven by suction is unstable. Moreover, we find that the linear growth rate associated with this suction-driven instability is independent of the viscosity contrast between the fluids. However, second-order results tell a different story, and show that the viscosity contrast is crucial in determining the morphology of the interface. Our theoretical description is applicable to the entire range of viscosity contrasts, and provides insights on the formation of near-cusp pattern-forming structures. Reproduction of fully nonlinear, n-fold symmetric near-cuspidal shapes previously obtained through conformal mapping techniques substantiates the validity of our mode-coupling approach.
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Affiliation(s)
- Rodolfo Brandão
- Department of Mathematics, Imperial College London, London SW7 2AZ United Kingdom
| | - Gabriel D Carvalho
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
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5
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Coutinho ÍM, Miranda JA. Control of viscous fingering through variable injection rates and time-dependent viscosity fluids: Beyond the linear regime. Phys Rev E 2021; 102:063102. [PMID: 33466051 DOI: 10.1103/physreve.102.063102] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2020] [Accepted: 11/25/2020] [Indexed: 11/07/2022]
Abstract
During the past few years, researchers have been proposing time-dependent injection strategies for stabilizing or manipulating the development of viscous fingering instabilities in radial Hele-Shaw cells. Most of these studies investigate the displacement of Newtonian fluids and are entirely based on linear stability analyses. In this work, linear stability theory and variational calculus are used to determine closed-form expressions for the proper time-dependent injection rates Q(t) required to either minimize the interface disturbances or to control the number of emerging fingers. However, this is done by considering that the displacing fluid is non-Newtonian and has a time-varying viscosity. Moreover, a perturbative third-order mode-coupling approach is employed to examine the validity and effectiveness of the controlling protocols dictated by these Q(t) beyond the linear regime and at the onset of nonlinearities.
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Affiliation(s)
- Írio M Coutinho
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
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6
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Computational Analysis of Interfacial Dynamics in Angled Hele-Shaw Cells: Instability Regimes. Transp Porous Media 2019. [DOI: 10.1007/s11242-019-01371-2] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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Anjos PHA, Carvalho GD, Lira SA, Miranda JA. Wrinkling and folding patterns in a confined ferrofluid droplet with an elastic interface. Phys Rev E 2019; 99:022608. [PMID: 30934336 DOI: 10.1103/physreve.99.022608] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2018] [Indexed: 11/07/2022]
Abstract
A thin elastic membrane lying on a fluid substrate deviates from its flat geometry on lateral compression. The compressed membrane folds and wrinkles into many distinct morphologies. We study a magnetoelastic variant of such a problem where a viscous ferrofluid, surrounded by a nonmagnetic fluid, is subjected to a radial magnetic field in a Hele-Shaw cell. Elasticity comes into play when the fluids are brought into contact, and due to a chemical reaction, the interface separating them becomes a gel-like elastic layer. A perturbative linear stability theory is used to investigate how the combined action of magnetic and elastic forces can lead to the development of smooth, low-amplitude, sinusoidal wrinkles at the elastic interface. In addition, a nonperturbative vortex sheet approach is employed to examine the emergence of highly nonlinear, magnetically driven, wrinkling and folding equilibrium shape structures. A connection between the magnetoelastic shape solutions induced by a radial magnetic field and those produced by nonmagnetic means through centrifugal forces is also discussed.
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Affiliation(s)
- Pedro H A Anjos
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - Gabriel D Carvalho
- Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, Rio de Janeiro 22290-180, Brazil
| | - Sérgio A Lira
- Instituto de Física, Universidade Federal de Alagoas, Maceió, Alagoas 57072-900, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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8
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Chen CY, Lin TS, Miranda JA. Rotationally induced fingering patterns in a two-dimensional heterogeneous porous medium. Phys Rev E 2016; 94:053105. [PMID: 27967138 DOI: 10.1103/physreve.94.053105] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2016] [Indexed: 11/07/2022]
Abstract
Rotating fluid flows under two-dimensional homogeneous porous media conditions (or in a rotating Hele-Shaw cell) reveal the development of complex interfacial fingering patterns. These pattern-forming structures are characterized by the occurrence of finger competition events, finger pinch-off episodes, as well as by the production of satellite droplets. In this work, we use intensive numerical simulations to investigate how these fully nonlinear pattern growth phenomena are altered by the presence of permeability heterogeneities in the rotating porous medium. This is done by employing a diffuse-interface Darcy-Cahn-Hilliard description of the problem and considering a permeability field presenting a log-Gaussian distribution, characterized by a variance s and a correlation length l. We study how the heterogeneity measures s and l couple to the governing hydrodynamic dimensionless parameters of the problem and introduce important changes on the pattern formation dynamics of the system.
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Affiliation(s)
- Ching-Yao Chen
- Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, 30010, Republic of China
| | - Ting-Shiang Lin
- Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, 30010, Republic of China
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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9
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Anjos PHA, Dias EO, Miranda JA. Kinetic undercooling in Hele-Shaw flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:043019. [PMID: 26565344 DOI: 10.1103/physreve.92.043019] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2015] [Indexed: 06/05/2023]
Abstract
A central topic in Hele-Shaw flow research is the inclusion of physical effects on the interface between fluids. In this context, the addition of surface tension restrains the emergence of high interfacial curvatures, while consideration of kinetic undercooling effects inhibits the occurrence of high interfacial velocities. By connecting kinetic undercooling to the action of the dynamic contact angle, we show in a quantitative manner that the kinetic undercooling contribution varies as a linear function of the normal velocity at the interface. A perturbative weakly nonlinear analysis is employed to extract valuable information about the influence of kinetic undercooling on the shape of the emerging fingered structures. Under radial Hele-Shaw flow, it is found that kinetic undercooling delays, but does not suppress, the development of finger tip-broadening and finger tip-splitting phenomena. In addition, our results indicate that kinetic undercooling plays a key role in determining the appearance of tip splitting in rectangular Hele-Shaw geometry.
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Affiliation(s)
- Pedro H A Anjos
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - Eduardo O Dias
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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10
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Dias EO, Lira SA, Miranda JA. Interfacial patterns in magnetorheological fluids: Azimuthal field-induced structures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:023003. [PMID: 26382499 DOI: 10.1103/physreve.92.023003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2015] [Indexed: 06/05/2023]
Abstract
Despite their practical and academic relevance, studies of interfacial pattern formation in confined magnetorheological (MR) fluids have been largely overlooked in the literature. In this work, we present a contribution to this soft matter research topic and investigate the emergence of interfacial instabilities when an inviscid, initially circular bubble of a Newtonian fluid is surrounded by a MR fluid in a Hele-Shaw cell apparatus. An externally applied, in-plane azimuthal magnetic field produced by a current-carrying wire induces interfacial disturbances at the two-fluid interface, and pattern-forming structures arise. Linear stability analysis, weakly nonlinear theory, and a vortex sheet approach are used to access early linear and intermediate nonlinear time regimes, as well as to determine stationary interfacial shapes at fully nonlinear stages.
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Affiliation(s)
- Eduardo O Dias
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
| | - Sérgio A Lira
- Instituto de Física, Universidade Federal de Alagoas, Maceió, Alagoas 57072-900 Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
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11
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Brandão R, Miranda JA. Viscous fluid fingering on a negatively curved surface. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:013018. [PMID: 26274280 DOI: 10.1103/physreve.92.013018] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2015] [Indexed: 06/04/2023]
Abstract
Viscous fingering formation in flat Hele-Shaw cells is a classical and widely studied fluid mechanical problem. We examine the development of viscous fluid fingering on a two-dimensional surface of constant negative Gaussian curvature, the hyperbolic plane H(2). A perturbative mode-coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system: fingertip splitting and finger competition. We also report on a time-dependent control strategy placed on the injection rate, which is able to minimize viscous fingering growth on H(2).
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Affiliation(s)
- Rodolfo Brandão
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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12
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Carvalho GD, Gadêlha H, Miranda JA. Stationary patterns in centrifugally driven interfacial elastic fingering. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:063009. [PMID: 25615189 DOI: 10.1103/physreve.90.063009] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2014] [Indexed: 06/04/2023]
Abstract
A vortex sheet formalism is used to search for equilibrium shapes in the centrifugally driven interfacial elastic fingering problem. We study the development of interfacial instabilities when a viscous fluid surrounded by another of smaller density flows in the confined environment of a rotating Hele-Shaw cell. The peculiarity of the situation is associated to the fact that, due to a chemical reaction, the two-fluid boundary becomes an elastic layer. The interplay between centrifugal and elastic forces leads to the formation of a rich variety of stationary shapes. Visually striking equilibrium morphologies are obtained from the numerical solution of a nonlinear differential equation for the interface curvature (the shape equation), determined by a zero vorticity condition. Classification of the various families of shapes is made via two dimensionless parameters: an effective bending rigidity (ratio of elastic to centrifugal effects) and a geometrical radius of gyration.
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Affiliation(s)
- Gabriel D Carvalho
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - Hermes Gadêlha
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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13
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Anjos PHA, Miranda JA. Influence of wetting on fingering patterns in lifting Hele-Shaw flows. SOFT MATTER 2014; 10:7459-7467. [PMID: 25007292 DOI: 10.1039/c4sm01047g] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We study the pattern formation dynamics related to the displacement of a viscous wetting fluid by a less viscous nonwetting fluid in a lifting Hele-Shaw cell. A perturbative weakly nonlinear analysis of the problem is presented. We focus on examining how wetting effects influence the morphology of the emerging interfacial patterns at the early nonlinear regime. Our analytical results indicate that wettability has a significant impact on the resulting nonlinear patterns. It restrains finger length variability while inducing the development of structures presenting short, blunt penetrating fingers of the nonwetting fluid, alternated by short, sharp fingers of the wetting fluid. The basic mode-coupling mechanisms leading to such behavior are discussed.
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Affiliation(s)
- Pedro H A Anjos
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil.
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14
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Carvalho GD, Gadêlha H, Miranda JA. Elastic fingering in rotating Hele-Shaw flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:053019. [PMID: 25353892 DOI: 10.1103/physreve.89.053019] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2014] [Indexed: 06/04/2023]
Abstract
The centrifugally driven viscous fingering problem arises when two immiscible fluids of different densities flow in a rotating Hele-Shaw cell. In this conventional setting an interplay between capillary and centrifugal forces makes the fluid-fluid interface unstable, leading to the formation of fingered structures that compete dynamically and reach different lengths. In this context, it is known that finger competition is very sensitive to changes in the viscosity contrast between the fluids. We study a variant of such a rotating flow problem where the fluids react and produce a gellike phase at their separating boundary. This interface is assumed to be elastic, presenting a curvature-dependent bending rigidity. A perturbative weakly nonlinear approach is used to investigate how the elastic nature of the interface affects finger competition events. Our results unveil a very different dynamic scenario, in which finger length variability is not regulated by the viscosity contrast, but rather determined by two controlling quantities: a characteristic radius and a rigidity fraction parameter. By properly tuning these quantities one can describe a whole range of finger competition behaviors even if the viscosity contrast is kept unchanged.
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Affiliation(s)
- Gabriel D Carvalho
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - Hermes Gadêlha
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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15
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Rocha FM, Miranda JA. Control of centrifugal fingering via a variable interfacial tension approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:033008. [PMID: 24125345 DOI: 10.1103/physreve.88.033008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/20/2013] [Indexed: 06/02/2023]
Abstract
We study the centrifugal fingering instabilities that occur in rotating Hele-Shaw cells containing two different fluids. A weakly nonlinear analysis of the problem is performed, considering that the surface tension between the rotating fluids changes with the local curvature of the interface. It is shown that the coupling between the contact angle and the variable interfacial tension permits the control of the interface disturbances. As a result, linear perturbations and nonlinear finger competition phenomena can be properly controlled and suppressed.
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Affiliation(s)
- Francisco M Rocha
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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16
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Dias EO, Miranda JA. Control of centrifugally driven fingering in a tapered Hele-Shaw cell. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:053014. [PMID: 23767627 DOI: 10.1103/physreve.87.053014] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2013] [Indexed: 06/02/2023]
Abstract
Conventional studies of the centrifugally driven fingering instability are performed in rotating Hele-Shaw cells presenting perfectly parallel plates. In this setup, the fluid-fluid interface can become unstable due to the density difference between the fluids, forming a variety of complex patterns. We study a modified, tapered version of this rotating flow problem where the cell plates are not exactly parallel, but present a constant gap gradient in the radial direction. Our analytical results indicate that the stability of the interface is significantly sensitive to the presence of the depth gradient. This allows a proper monitoring of pattern-forming features just by slightly changing the cell's taper angle. In this context, we have verified that by manipulating the sign and magnitude of the gap gradient one can favor, restrain, or even suppress the development of centrifugally driven instabilities. A taper-induced mechanism for the selection of the number of emerging fingers is also discussed.
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Affiliation(s)
- Eduardo O Dias
- Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, Pernambuco, Brazil
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17
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Chen CY, Huang YS, Miranda JA. Diffuse-interface approach to rotating Hele-Shaw flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:046302. [PMID: 22181256 DOI: 10.1103/physreve.84.046302] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2011] [Indexed: 05/31/2023]
Abstract
When two fluids of different densities move in a rotating Hele-Shaw cell, the interface between them becomes centrifugally unstable and deforms. Depending on the viscosity contrast of the system, distinct types of complex patterns arise at the fluid-fluid boundary. Deformations can also induce the emergence of interfacial singularities and topological changes such as droplet pinch-off and self-intersection. We present numerical simulations based on a diffuse-interface model for this particular two-phase displacement that capture a variety of pattern-forming behaviors. This is implemented by employing a Boussinesq Hele-Shaw-Cahn-Hilliard approach, considering the whole range of possible values for the viscosity contrast, and by including inertial effects due to the Coriolis force. The role played by these two physical contributions on the development of interface singularities is illustrated and discussed.
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Affiliation(s)
- Ching-Yao Chen
- Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan 30010, Republic of China.
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18
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Dias EO, Miranda JA. Inertial effects on rotating Hele-Shaw flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:046311. [PMID: 21599299 DOI: 10.1103/physreve.83.046311] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2011] [Indexed: 05/30/2023]
Abstract
We examine the finite surface tension radial viscous fingering problem in a rotating Hele-Shaw cell, and focus on the action of inertial effects on the stability and morphology of the emerging patterns. To study such a flow we use an alternative version of the usual Darcy's law derived by gap-averaging the Navier-Stokes equation, but retaining its inertial terms. The importance of inertial forces is pertinently characterized by a rotational Reynolds number. Linear and weakly nonlinear stages of the dynamics are described analytically through a mode coupling approach. The linear stability results indicate that inertia has a stabilizing role. However, the characteristic number of fingers and the width of the band of linearly unstable modes are not altered by inertia. In the early nonlinear regime we find that inertia acts to favor the development of fingers with narrower tips than those obtained in its absence. We have also verified that finger competition events are affected by inertial effects which tend to restrain finger length variability among inward-moving fingers.
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Affiliation(s)
- Eduardo O Dias
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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Mishra M, Trevelyan PMJ, Almarcha C, De Wit A. Influence of double diffusive effects on miscible viscous fingering. PHYSICAL REVIEW LETTERS 2010; 105:204501. [PMID: 21231238 DOI: 10.1103/physrevlett.105.204501] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2010] [Indexed: 05/30/2023]
Abstract
Miscible viscous fingering classically occurs when a less viscous fluid displaces a miscible more viscous one in a porous medium. We analyze here how double diffusive effects between a slow diffusing S and a fast diffusing F component, both influencing the viscosity of the fluids at hand, affect such fingering, and, most importantly, can destabilize the classically stable situation of a more viscous fluid displacing a less viscous one. Various instability scenarios are classified in a parameter space spanned by the log-mobility ratios R(s) and R(f) of the slow and fast component, respectively, and parametrized by the ratio of diffusion coefficients δ. Numerical simulations of the full nonlinear problem confirm the existence of the predicted instability scenarios and highlight the influence of differential diffusion effects on the nonlinear fingering dynamics.
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Affiliation(s)
- M Mishra
- Nonlinear Physical Chemistry Unit and Center for Nonlinear Phenomena and Complex Systems, Faculté des Sciences, Université Libre de Bruxelles (ULB), CP231, 1050 Brussels, Belgium
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20
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Lira SA, Miranda JA, Oliveira RM. Stationary shapes of confined rotating magnetic liquid droplets. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:036318. [PMID: 21230182 DOI: 10.1103/physreve.82.036318] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2010] [Indexed: 05/30/2023]
Abstract
We study the family of steady shapes which arise when a magnetic liquid droplet is confined in a rotating Hele-Shaw cell and subjected to an azimuthal magnetic field. Two different scenarios are considered: first, the magnetic fluid is assumed to be a Newtonian ferrofluid, and then it is taken as a viscoelastic magnetorheological fluid. The influence of the distinct material properties of the fluids on the ultimate morphology of the emerging stationary patterns is investigated by using a vortex-sheet formalism. Some of these exact steady structures are similar to the advanced time patterns obtained by existing time-evolving numerical simulations of the problem. A weakly nonlinear approach is employed to examine this fact and to gain analytical insight about relevant aspects related to the stability of such exact stationary solutions.
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Affiliation(s)
- Sérgio A Lira
- Departamento de Física, LFTC, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil
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Dias EO, Miranda JA. Control of radial fingering patterns: a weakly nonlinear approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:016312. [PMID: 20365465 DOI: 10.1103/physreve.81.016312] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2009] [Indexed: 05/29/2023]
Abstract
It is well known that the constant injection rate flow in radial Hele-Shaw cells leads to the formation of highly branched patterns, where finger tip-splitting events are plentiful. Different kinds of patterns arise in the lifting Hele-Shaw flow problem, where the cell's gap width grows linearly with time. In this case, the morphology of the emerging structures is characterized by the strong competition among inward moving fingers. By employing a mode-coupling theory we find that both finger tip-splitting and finger competition can be restrained by properly adjusting the injection rate and the time-dependent gap width, respectively. Our theoretical model approaches the problem analytically and is capable of capturing these important controlling mechanisms already at weakly nonlinear stages of the dynamics.
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Affiliation(s)
- Eduardo O Dias
- Departamento de Física, LFTC, Universidade Federal de Pernambuco, Recife, PE, Brazil
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Folch R, Alvarez-Lacalle E, Ortín J, Casademunt J. Pattern formation and interface pinch-off in rotating Hele-Shaw flows: a phase-field approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:056305. [PMID: 20365071 DOI: 10.1103/physreve.80.056305] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2009] [Revised: 07/29/2009] [Indexed: 05/29/2023]
Abstract
Viscous fingering dynamics driven by centrifugal forcing is studied for arbitrary viscosity contrast. Theoretical methods, including exact solutions, and numerics based on a phase-field approach are used. Both confirm that pinch-off singularities in patterns originated from the centrifugally driven instability may occur spontaneously and be inherent to the two-dimensional Hele-Shaw dynamics. They are systematically more frequent for lower viscosity contrasts consistently with experimental evidence. The analytical insights provide an interpretation of this fact in terms of the asymptotic matching of the different regions of the fingering patterns. The phase-field numerical scheme is shown to be particularly adequate to elucidate the existence of finite-time singularities through the dependence of the singularity time on the interface thickness, in particular for varying viscosity contrast.
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Affiliation(s)
- R Folch
- Departament d'Enginyeria Química, Universitat Rovira i Virgili, Av. dels Països Catalans 26, E-43007 Tarragona, Spain
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Dias EO, Miranda JA. Interfacial instabilities in periodically driven Hele-Shaw flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:026303. [PMID: 19792245 DOI: 10.1103/physreve.80.026303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2009] [Revised: 06/15/2009] [Indexed: 05/28/2023]
Abstract
We consider flow in a Hele-Shaw cell for which the upper plate moves up and down, making the fluid-fluid interface be driven periodically. To study such a flow we employ a mode-coupling approach, which allows the analytical assessment of important aspects about the stability and morphology of the evolving interface. At the linear level, it is shown that both the amplitude and the frequency of the drive have a significant role in determining the ultimate number of fingers formed. The influence of these factors on the mechanisms of finger competition and finger tip behavior at the onset of nonlinear effects is also studied.
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Affiliation(s)
- Eduardo O Dias
- Departamento de Física, LFTC, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, PE, Brazil
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Gadêlha H, Miranda JA. Effects of normal viscous stresses on radial viscous fingering. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:066312. [PMID: 19658599 DOI: 10.1103/physreve.79.066312] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2009] [Indexed: 05/28/2023]
Abstract
We revisit the radial viscous fingering problem in a Hele-Shaw cell, and consider the action of viscous stresses originated from velocity gradients normal to the fluid-fluid interface. The evolution of the interface during linear and weakly nonlinear stages is described analytically through a mode-coupling approach. We find that the introduction of normal stresses influences the stability and the ultimate morphology of the emerging patterns. Although at early stages normal stresses tend to stabilize the interface, they act to favor the development of tip-splitting phenomena at the weakly nonlinear regime. We have also verified that finger competition events are only significantly affected by normal stresses for circumstances involving the development of a large number of interfacial fingers.
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Affiliation(s)
- Hermes Gadêlha
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK.
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Alvarez-Lacalle E, Gadêlha H, Miranda JA. Coriolis effects on fingering patterns under rotation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:026305. [PMID: 18850934 DOI: 10.1103/physreve.78.026305] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2008] [Revised: 07/01/2008] [Indexed: 05/26/2023]
Abstract
The development of immiscible viscous fingering patterns in a rotating Hele-Shaw cell is investigated. We focus on understanding how the time evolution and the resulting morphologies are affected by the action of the Coriolis force. The problem is approached analytically and numerically by employing a vortex sheet formalism. The vortex sheet strength and a linear dispersion relation are derived analytically, revealing that the most relevant Coriolis force contribution comes from the normal component of the averaged interfacial velocity. It is shown that this normal velocity, uniquely due to the presence of the Coriolis force, is responsible for the complex-valued nature of the linear dispersion relation making the linear phases vary with time. Fully nonlinear stages are studied through intensive numerical simulations. A suggestive interplay between inertial and viscous effects is found, which modifies the dynamics, leading to different pattern-forming structures. The inertial Coriolis contribution plays a characteristic role: it generates a phase drift by deviating the fingers in the sense opposite to the actual rotation of the cell. However, the direction and intensity of finger bending is predominantly determined by viscous effects, being sensitive to changes in the magnitude and sign of the viscosity contrast. The finger competition behavior at advanced time stages is also discussed.
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Affiliation(s)
- Enrique Alvarez-Lacalle
- Departament de Fisica Aplicada, Universitat Politecnica de Catalunya (UPC), EPSEB, Avenue Doctor Marañón, 44-50, Barcelona 08028, Spain
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Chen CY, Huang CW, Gadêlha H, Miranda JA. Radial viscous fingering in miscible Hele-Shaw flows: a numerical study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:016306. [PMID: 18764049 DOI: 10.1103/physreve.78.016306] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2008] [Revised: 04/24/2008] [Indexed: 05/26/2023]
Abstract
A modified version of the usual viscous fingering problem in a radial Hele-Shaw cell with immiscible fluids is studied by intensive numerical simulations. We consider the situation in which the fluids involved are miscible, so that the diffusing interface separating them can be driven unstable through the injection or suction of the inner fluid. The system is allowed to rotate in such a way that centrifugal and Coriolis forces come into play, imposing important changes on the morphology of the arising patterns. In order to bridge from miscible to immiscible pattern forming structures, we add the surface tensionlike effects due to Korteweg stresses. Our numerical experiments reveal a variety of interesting fingering behaviors, which depend on the interplay between injection (or suction), diffusive, rotational, and Korteweg stress effects. Whenever possible the features of the simulated miscible fronts are contrasted to existing experiments and other theoretical or numerical studies, usually resulting in close agreements. A number of additional complex morphologies, whose experimental realization is still not available, are predicted and discussed.
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Affiliation(s)
- Ching-Yao Chen
- Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China.
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Gadêlha H, Brito N, Miranda JA. Dynamics of viscous fingers in rotating Hele-Shaw cells with Coriolis effects. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:016305. [PMID: 17358251 DOI: 10.1103/physreve.75.016305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2006] [Indexed: 05/14/2023]
Abstract
A growing number of experimental and theoretical works have been addressing various aspects of the viscous fingering formation in rotating Hele-Shaw cells. However, only a few of them consider the influence of Coriolis forces. The studies including Coriolis effects are mostly restricted to the high-viscosity-contrast limit and rely on either purely linear stability analyses or intensive numerical simulations. We approach the problem analytically and use a modified Darcy's law including the exact form of the Coriolis effects to execute a mode-coupling analysis of the system. By imposing no restrictions on the viscosity contrast A (dimensionless viscosity difference) we go beyond linear stages and examine the onset of nonlinearities. Our results indicate that when Coriolis effects are taken into account, an interesting interplay between the Reynolds number Re and A arises. This leads to important changes in the stability and morphological features of the emerging interfacial patterns. We contrast our mode-coupling approach with previous theoretical models proposed in the literature.
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Affiliation(s)
- Hermes Gadêlha
- Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
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Chen CY, Chen CH, Miranda JA. Numerical study of pattern formation in miscible rotating Hele-Shaw flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:046306. [PMID: 16711928 DOI: 10.1103/physreve.73.046306] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2005] [Indexed: 05/09/2023]
Abstract
The dynamics of the diffusing interface separating two miscible fluids in a rotating Hele-Shaw cell is studied by intensive and highly accurate numerical simulations. We perform numerical experiments in a wide range of parameters, focusing on the influence of viscosity contrast and Korteweg stresses on the shape of the interfacial patterns. A great variety of morphological behaviors is systematically introduced, and a wealth of interesting phenomena related to finger competition dynamics, filament stretching, and interface pinch off are reveal. Our simulations exhibit miscible patterns that bear a strong resemblance to their immiscible counterparts for larger Korteweg stresses. The quantitative equivalence between such stresses and the usual immiscible surface tension is studied. The concept of an effective interfacial tension is considered, allowing the direct and precise calculation of the important fingering properties under miscible circumstances. Our results show excellent agreement with existing experiments and simulations for corresponding immiscible displacements. This agreement refers to a striking similarity between miscible and immiscible pattern morphologies, and also to an accurate prediction for the typical number of miscible fingering structures formed. Our findings suggest that the effective interfacial tension is both qualitatively and quantitatively equivalent to its immiscible counterpart.
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Affiliation(s)
- Ching-Yao Chen
- Department of Mechanical Engineering, National Yunlin University of Science & Technology, Yunlin, Taiwan, Republic of China.
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Oliveira RM, Miranda JA. Stretching of a confined ferrofluid: influence of viscous stresses and magnetic field. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:036309. [PMID: 16605653 DOI: 10.1103/physreve.73.036309] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2005] [Revised: 02/06/2006] [Indexed: 05/08/2023]
Abstract
An analytical investigation is presented for the stretch flow of a viscous Newtonian ferrofluid highly confined between parallel plates. We focus on the development of interfacial instabilities when the upper plate is lifted at a described rate, under the action of an applied magnetic field. We derive the mode-coupling differential equation for the interface perturbation amplitudes and study both linear and nonlinear flow regimes. In contrast to the great majority of works in stretch flow we take into account stresses originated from velocity gradients normal to the ferrofluid interface. The impact of such normal stresses is accounted for through a modified Young-Laplace pressure jump interfacial boundary condition, which also includes the contribution from magnetic normal traction. We study how the stability properties of the interface and the shape of the emerging patterns respond to the combined action of normal stresses and magnetic field, both in the presence and absence of surface tension. We show that the inclusion of normal viscous stresses introduces a pertinent dependence on the initial aspect ratio, indicating that the number of fingers formed would be overestimated if such stresses are not taken into account. At early linear stages it is found that such stresses regularize the system, acting as an effective interfacial tension. At weakly nonlinear stages we verified that normal stresses reduce finger competition, which can be completely suppressed with the assistance of an azimuthal magnetic field. We have also found that the magnetic normal traction introduces a purely nonlinear contribution to the problem, revealing the key role played by the magnetic susceptibility in the control of finger competition.
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Affiliation(s)
- Rafael M Oliveira
- Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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