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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: 0] [Impact Index Per Article: 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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Yu Z, Christov IC. Delayed Hopf bifurcation and control of a ferrofluid interface via a time-dependent magnetic field. Phys Rev E 2023; 107:055102. [PMID: 37329044 DOI: 10.1103/physreve.107.055102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2022] [Accepted: 04/13/2023] [Indexed: 06/18/2023]
Abstract
A ferrofluid droplet confined in a Hele-Shaw cell can be deformed into a stably spinning "gear," using crossed magnetic fields. Previously, fully nonlinear simulation revealed that the spinning gear emerges as a stable traveling wave along the droplet's interface bifurcates from the trivial (equilibrium) shape. In this work, a center manifold reduction is applied to show the geometrical equivalence between a two-harmonic-mode coupled system of ordinary differential equations arising from a weakly nonlinear analysis of the interface shape and a Hopf bifurcation. The rotating complex amplitude of the fundamental mode saturates to a limit cycle as the periodic traveling wave solution is obtained. An amplitude equation is derived from a multiple-time-scale expansion as a reduced model of the dynamics. Then, inspired by the well-known delay behavior of time-dependent Hopf bifurcations, we design a slowly time-varying magnetic field such that the timing and emergence of the interfacial traveling wave can be controlled. The proposed theory allows us to determine the time-dependent saturated state resulting from the dynamic bifurcation and delayed onset of instability. The amplitude equation also reveals hysteresislike behavior upon time reversal of the magnetic field. The state obtained upon time reversal differs from the state obtained during the initial (forward-time) period, yet it can still be predicted by the proposed reduced-order theory.
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Affiliation(s)
- Zongxin Yu
- School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA
| | - Ivan C Christov
- School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA
- Department of Computer Science, University of Nicosia, 46 Makedonitissas Avenue, CY-2417 Nicosia, Cyprus
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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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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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Coutinho ÍM, Miranda JA. Field-controlled flow and shape of a magnetorheological fluid annulus. Phys Rev E 2022; 106:025105. [PMID: 36109920 DOI: 10.1103/physreve.106.025105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2022] [Accepted: 08/05/2022] [Indexed: 06/15/2023]
Abstract
We investigate the behavior of a magnetorheological (MR) fluid annulus, bounded by a nonmagnetic fluid and confined in a Hele-Shaw cell, under the simultaneous effect of in-plane, external radial and azimuthal magnetic fields. A second-order mode-coupling theory is used to study the early nonlinear stage of the pattern-forming dynamics. We examine changes in the morphology of the MR fluid annular structure as a function of its magnetic-field-tunable rheological properties, as well as the combined magnetic field's intensities, and thickness of the ring. Our weakly nonlinear perturbative results show that, depending on the system control parameters, the MR fluid annulus adopts various stationary shapes. These equilibrium annular structures present slightly bent, asymmetric fingered protrusions which may emerge on the inner, outer, or even on both boundaries of the magnetic fluid ring. On top of these morphological changes, we find that the resulting permanent shape patterns rotate with a well defined angular velocity. We focus on analyzing how the overall shape of the fingered patterns, in particular their sharpness and asymmetric form, as well as the number of resulting fingers are impacted by the magnetic-field-dependent yield stress of the MR fluid annulus. The influence of the magnetically controlled rheological properties of the MR fluid on the angular velocity of the rotating annulus is also scrutinized.
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Affiliation(s)
- Írio M Coutinho
- Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, Pernambuco, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, Pernambuco, Brazil
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Livera POS, Anjos PHA, Miranda JA. Ferrofluid annulus in crossed magnetic fields. Phys Rev E 2022; 105:045106. [PMID: 35590587 DOI: 10.1103/physreve.105.045106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2022] [Accepted: 04/06/2022] [Indexed: 06/15/2023]
Abstract
We study the dynamics and pattern formation of a ferrofluid annulus enveloped by two nonmagnetic fluids in a Hele-Shaw cell, subjected to an in-plane crossed magnetic field configuration involving the combination of radial and azimuthal magnetic fields. A perturbative, second-order mode-coupling analysis is employed to investigate how the ferrofluid annulus responds to variations in the relative strength of the radial and azimuthal magnetic field components, as well as in the thickness of magnetic fluid ring. By tuning the magnetic field components and the annulus' thickness, we have found the development of several stationary annular shapes, presenting polygon-shaped structures typically having skewed, peaked fingers. Such fingered structures may vary their skewness, sharpness, and number and arise on the inner, outer, or even both boundaries of the annulus. In addition to controlling the morphologies of the ferrofluid annuli, the external field can be used to put the annulus into a rotational motion, with an angular velocity having prescribed magnitude, and direction. Our second-order theory is utilized to obtain a correction to the linear stability analysis prediction of such angular velocity, usually resulting in a decreased weakly nonlinear value as compared with the magnitude predicted by purely linear theory. These theoretical results suggest the use of magnetic-field-controlled ferrofluid annuli in Hele-Shaw cells as a potential laboratory for microscale applications related to the manipulation of shape-programmable magnetic fluid objects and tunable fluidic-mixing devices in confined environments.
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Affiliation(s)
- Pedro O S Livera
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
| | - Pedro H A Anjos
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil
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