PHYSICAL REVIEW E 91 , 023020 (2015)
Azimuthal field instability in a confined ferrofluid Eduardo O. Dias* and Jose A. Miranda* Departamento de Fi'sica, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil (Received 30 December 2014; published 27 February 2015) We report the development o f interfacial ferrohydrodynamic instabilities when an initially circular bubble of a nonmagnetic inviscid fluid is surrounded by a viscous ferrofluid in the confined geometry of a Hele-Shaw cell. The fluid-fluid interface becomes unstable due to the action of magnetic forces induced by an azimuthal field produced by a straight current-carrying wire that is normal to the cell plates. In this framework, a pattern formation process takes place through the interplay between magnetic and surface tension forces. By employing a perturbative mode-coupling approach we investigate analytically both linear and intermediate nonlinear regimes of the interface evolution. As a result, useful analytical information can be extracted regarding the destabilizing role of the azimuthal field at the linear level, as well as its influence on the interfacial pattern morphology at the onset of nonlinear effects. Finally, a vortex sheet formalism is used to access fully nonlinear stationary solutions for the two-fluid interface shapes. DOI: 10.1103/PhysRevE.91.023020
PACS number(s): 4 7 .6 5 .C b ,4 7 .1 5 .g p ,4 7 .5 4 .-r,4 7 .2 0 .M a
I. INTRODUCTION
Ferrofluids are colloidal suspensions of nanometer-sized magnetic particles suspended in a nonmagnetic carrier fluid [1,2]. This particular type of magnetic fluid behaves superparamagnetically and can easily be manipulated with ex ternal magnetic fields. The combined action of hydrodynamic and magnetic forces makes ferrofluids a remarkable material to study a variety of interfacial instabilities and pattern for mation processes [3]. One iconic example of pattern-forming phenomena in ferrofluids is the Rosensweig instability [4]. It occurs when an initially flat free surface of a ferrofluid film is subjected to a uniform, perpendicular magnetic field generated by a pair of Helmholtz coils. The competition between magnetic, gravitational, and capillary forces results in the rising of a visually striking, three-dimensional (3D) array of spiky structures (the Rosensweig’s peaks) growing from the liquid-free surface. Since then, various aspects related to the form of the peaks and their nontrivial dynamic behavior have been scrutinized by theory, simulations, and experiments (see, for instance, Refs. [5-9], and references therein). Another popular event in ferrohydrodynamics, known as the conical meniscus instability, refers to the formation of a static meniscus profile when an originally horizontal 3D layer of ferrofluid encloses a vertical current-carrying wire [10], This particular setup is quite simple and generates an azimuthal magnetic field that turns around the wire. As opposed to the perpendicular field case studied in the development of the Rosensweig’s peaks, the azimuthal field configuration produces a magnetic force directed radially inward. This force tends to attract the evolving ferrofluid toward the currentcarrying wire, making the magnetic fluid to lift it, and resulting in the establishment of a curved ferrofluid meniscus. Since the seminal work of Ref. [10] investigators have carried out theoretical and experimental studies in order to determine the precise shape of the curved meniscus in both the absence and presence of surface tension effects [11-14]. Interestingly,
*eduardodias @ df. ufpe.br tjm e@ df.ufpe.br 1539-3755/2015/91 (2)/023020(8)
the azimuthal magnetic field configuration produced by a straight current-carrying wire has also been used to examine the dynamics of solitons propagating on cylindrical ferrofluid surfaces surrounding the wire. There is a recent example in the literature in which theoretical predictions [15-17] about solitary wave propagation in ferrofluids has been realized experimentally [18]. Researchers have also analyzed how the perpendicular and azimuthal magnetic fields discussed above act on ferrofluids confined in the effectively 2D geometry of a Hele-Shaw cell. The Hele-Shaw apparatus [19,20] consists of two parallel glass plates separated by a narrow gap, where the ferrofluid can flow under the action of an applied magnetic field [2,3]. In contrast to the legitimately 3D free surface situation, the action of a magnetic field perpendicular to the plates of a Hele-Shaw cell containing an initially circular ferrofluid droplet encircled by a nonmagnetic fluid does not lead to peak formation. Instead, the applied field tends to align the tiny magnetic moments in the ferrofluid in a direction perpendicular to the plates. Consequently, these magnetic moments repel each other within the plane of the Hele-Shaw cell, and the two-fluid interface starts to distort. On the other hand, the surface tension between the fluids tends to stabilize the interface. The interplay between these two forces ultimately leads to the emergence of mazelike, multiply bifurcated structures, where a labyrinth-type pattern is formed. This characterizes the celebrated labyrinthine ferrofluid instability [21-24]. A considerably different scenario arises when a circular ferrofluid droplet, surrounded by a nonmagnetic fluid, is placed in a Hele-Shaw cell and subjected to an azimuthal magnetic field produced by current-carrying wire. In this context, the wire is normal to the cell plates and passes through the center of the ferrofluid droplet. As noted earlier, the azimuthal magnetic field generates a radial magnetic body force pointing inward, attracting the ferrofluid droplet toward the wire. Under such circumstances both surface tension and magnetic forces tend to stabilize the fluidfluid interface. This azimuthal field stabilizing strategy has been proven effective to control interfacial instabilities in ferrofluids under centrifugally induced fingering in rotating Hele-Shaw cells [25-27], as well as during the stretch
023020-1
©2015 American Physical Society
EDUARDO O. DIAS AND JOSE A. MIRANDA
PHYSICAL REVIEW E 91 , 023020 (2015)
flow of thin ferrofluid films in the lifting Hele-Shaw flow setup [28,29]. In this work we study a still unexplored aspect of the azimuthal magnetic field configuration in Hele-Shaw geom etry. More specifically, we consider the influence of the azimuthal field on the fluid-fluid interface, now assuming that a viscous ferrofluid is the outer fluid, while the inner fluid is inviscid and nonmagnetic. The interesting facet of the problem resides on the fact that in this situation the azimuthal field acts to destabilize the interface separating the fluids, offering the possibility of investigating innovative dynamical and pattern-forming behaviors. It should be noted that, due its simplicity, the azimuthal magnetic field offers plenty of room to analytical treatment of the rising interfacial phenomena. This is in contrast to what happens in the perpendicular magnetic field case, where destabilization relies on complicated demagnetizing effects that usually defy the analytical treatment of the system, mainly during nonlinear stages of the dynamics. Here we tackle the ferrohydrodynamic problem analytically: first, a perturbative mode-coupling approach [30] is employed to access relevant features of the time-evolving interface up to quadratic nonlinearities. This allows one to examine the linear stability of the problem, as well as the most salient morphological characteristics of the two-fluid interface at the onset of nonlinear effects. Fully nonlinear features of the interface are also studied, via a vortex sheet approach [31], leading to the determination of stationary solutions for the interface shape.
of viscosity /?. Both fluids are incompressible and are located between two narrowly spaced flat plates of a Hele-Shaw cell of thickness b. The surface tension at the fluid-fluid interface is denoted by a . We consider the action of an azimuthal magnetic field produced by a long, straight, current-carrying wire that is perpendicular to (coaxial with) the plates of the Hele-Shaw cell,
The electric current is represented by I . r is the radial distance from the origin of the coordinate system (located at the center of the cell), and is a unit vector in the azimuthal direction. The azimuthal angle in the plane of the cell is denoted by ip. A magnetic body force ~ V / / , where H = |H|, acts on the ferrofluid pointing in the inward radial direction [1,25]. While surface tension tends to keep the two-fluid inter face circular, the magnetic force induced by the azimuthal magnetic field (1) acts to deform it. In the framework of our second-order perturbative approach, we describe the deformed interface shape as TZ(ip,t) — R + f( 0 and bn = 0. In this approach, finger tip broadening, and finger tip splitting are described by considering the influence of a fundamental mode n on the growth of its first harmonic 2n [30].
023020-3
EDUARDO O. DIAS AND JOSE A. MIRANDA
PHYSICAL REVIEW E 91 , 023020 (2015)
Writing the equations of motion for the harmonic mode we have
closed form solution can be written as £n(0 = ?„(0)exp{A(n)l}
a2„ = A.(2n)a2„ + ^T(2n,n)aft,
(11) +
b2n=X(2n)b2n,
( 12)
n '^ 0
where the finger tip function is defined as
x
T(2n,n) = F(2n,n) + X(n)G(2n,n) - \n\[xblB( x n 2 - 4 ) + 6«2 - 3].
(14)
into the second-order terms on the right-hand side of Eq. (5), we obtain
tn = UnKn + ^ [T («,n O + A(nOG(«,nO]
Azimuthal field instability in a confined ferrofluid Eduardo O. Dias* and Jose A. Miranda* Departamento de Fi'sica, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901 Brazil (Received 30 December 2014; published 27 February 2015) We report the development o f interfacial ferrohydrodynamic instabilities when an initially circular bubble of a nonmagnetic inviscid fluid is surrounded by a viscous ferrofluid in the confined geometry of a Hele-Shaw cell. The fluid-fluid interface becomes unstable due to the action of magnetic forces induced by an azimuthal field produced by a straight current-carrying wire that is normal to the cell plates. In this framework, a pattern formation process takes place through the interplay between magnetic and surface tension forces. By employing a perturbative mode-coupling approach we investigate analytically both linear and intermediate nonlinear regimes of the interface evolution. As a result, useful analytical information can be extracted regarding the destabilizing role of the azimuthal field at the linear level, as well as its influence on the interfacial pattern morphology at the onset of nonlinear effects. Finally, a vortex sheet formalism is used to access fully nonlinear stationary solutions for the two-fluid interface shapes. DOI: 10.1103/PhysRevE.91.023020
PACS number(s): 4 7 .6 5 .C b ,4 7 .1 5 .g p ,4 7 .5 4 .-r,4 7 .2 0 .M a
I. INTRODUCTION
Ferrofluids are colloidal suspensions of nanometer-sized magnetic particles suspended in a nonmagnetic carrier fluid [1,2]. This particular type of magnetic fluid behaves superparamagnetically and can easily be manipulated with ex ternal magnetic fields. The combined action of hydrodynamic and magnetic forces makes ferrofluids a remarkable material to study a variety of interfacial instabilities and pattern for mation processes [3]. One iconic example of pattern-forming phenomena in ferrofluids is the Rosensweig instability [4]. It occurs when an initially flat free surface of a ferrofluid film is subjected to a uniform, perpendicular magnetic field generated by a pair of Helmholtz coils. The competition between magnetic, gravitational, and capillary forces results in the rising of a visually striking, three-dimensional (3D) array of spiky structures (the Rosensweig’s peaks) growing from the liquid-free surface. Since then, various aspects related to the form of the peaks and their nontrivial dynamic behavior have been scrutinized by theory, simulations, and experiments (see, for instance, Refs. [5-9], and references therein). Another popular event in ferrohydrodynamics, known as the conical meniscus instability, refers to the formation of a static meniscus profile when an originally horizontal 3D layer of ferrofluid encloses a vertical current-carrying wire [10], This particular setup is quite simple and generates an azimuthal magnetic field that turns around the wire. As opposed to the perpendicular field case studied in the development of the Rosensweig’s peaks, the azimuthal field configuration produces a magnetic force directed radially inward. This force tends to attract the evolving ferrofluid toward the currentcarrying wire, making the magnetic fluid to lift it, and resulting in the establishment of a curved ferrofluid meniscus. Since the seminal work of Ref. [10] investigators have carried out theoretical and experimental studies in order to determine the precise shape of the curved meniscus in both the absence and presence of surface tension effects [11-14]. Interestingly,
*eduardodias @ df. ufpe.br tjm e@ df.ufpe.br 1539-3755/2015/91 (2)/023020(8)
the azimuthal magnetic field configuration produced by a straight current-carrying wire has also been used to examine the dynamics of solitons propagating on cylindrical ferrofluid surfaces surrounding the wire. There is a recent example in the literature in which theoretical predictions [15-17] about solitary wave propagation in ferrofluids has been realized experimentally [18]. Researchers have also analyzed how the perpendicular and azimuthal magnetic fields discussed above act on ferrofluids confined in the effectively 2D geometry of a Hele-Shaw cell. The Hele-Shaw apparatus [19,20] consists of two parallel glass plates separated by a narrow gap, where the ferrofluid can flow under the action of an applied magnetic field [2,3]. In contrast to the legitimately 3D free surface situation, the action of a magnetic field perpendicular to the plates of a Hele-Shaw cell containing an initially circular ferrofluid droplet encircled by a nonmagnetic fluid does not lead to peak formation. Instead, the applied field tends to align the tiny magnetic moments in the ferrofluid in a direction perpendicular to the plates. Consequently, these magnetic moments repel each other within the plane of the Hele-Shaw cell, and the two-fluid interface starts to distort. On the other hand, the surface tension between the fluids tends to stabilize the interface. The interplay between these two forces ultimately leads to the emergence of mazelike, multiply bifurcated structures, where a labyrinth-type pattern is formed. This characterizes the celebrated labyrinthine ferrofluid instability [21-24]. A considerably different scenario arises when a circular ferrofluid droplet, surrounded by a nonmagnetic fluid, is placed in a Hele-Shaw cell and subjected to an azimuthal magnetic field produced by current-carrying wire. In this context, the wire is normal to the cell plates and passes through the center of the ferrofluid droplet. As noted earlier, the azimuthal magnetic field generates a radial magnetic body force pointing inward, attracting the ferrofluid droplet toward the wire. Under such circumstances both surface tension and magnetic forces tend to stabilize the fluidfluid interface. This azimuthal field stabilizing strategy has been proven effective to control interfacial instabilities in ferrofluids under centrifugally induced fingering in rotating Hele-Shaw cells [25-27], as well as during the stretch
023020-1
©2015 American Physical Society
EDUARDO O. DIAS AND JOSE A. MIRANDA
PHYSICAL REVIEW E 91 , 023020 (2015)
flow of thin ferrofluid films in the lifting Hele-Shaw flow setup [28,29]. In this work we study a still unexplored aspect of the azimuthal magnetic field configuration in Hele-Shaw geom etry. More specifically, we consider the influence of the azimuthal field on the fluid-fluid interface, now assuming that a viscous ferrofluid is the outer fluid, while the inner fluid is inviscid and nonmagnetic. The interesting facet of the problem resides on the fact that in this situation the azimuthal field acts to destabilize the interface separating the fluids, offering the possibility of investigating innovative dynamical and pattern-forming behaviors. It should be noted that, due its simplicity, the azimuthal magnetic field offers plenty of room to analytical treatment of the rising interfacial phenomena. This is in contrast to what happens in the perpendicular magnetic field case, where destabilization relies on complicated demagnetizing effects that usually defy the analytical treatment of the system, mainly during nonlinear stages of the dynamics. Here we tackle the ferrohydrodynamic problem analytically: first, a perturbative mode-coupling approach [30] is employed to access relevant features of the time-evolving interface up to quadratic nonlinearities. This allows one to examine the linear stability of the problem, as well as the most salient morphological characteristics of the two-fluid interface at the onset of nonlinear effects. Fully nonlinear features of the interface are also studied, via a vortex sheet approach [31], leading to the determination of stationary solutions for the interface shape.
of viscosity /?. Both fluids are incompressible and are located between two narrowly spaced flat plates of a Hele-Shaw cell of thickness b. The surface tension at the fluid-fluid interface is denoted by a . We consider the action of an azimuthal magnetic field produced by a long, straight, current-carrying wire that is perpendicular to (coaxial with) the plates of the Hele-Shaw cell,
The electric current is represented by I . r is the radial distance from the origin of the coordinate system (located at the center of the cell), and is a unit vector in the azimuthal direction. The azimuthal angle in the plane of the cell is denoted by ip. A magnetic body force ~ V / / , where H = |H|, acts on the ferrofluid pointing in the inward radial direction [1,25]. While surface tension tends to keep the two-fluid inter face circular, the magnetic force induced by the azimuthal magnetic field (1) acts to deform it. In the framework of our second-order perturbative approach, we describe the deformed interface shape as TZ(ip,t) — R + f( 0 and bn = 0. In this approach, finger tip broadening, and finger tip splitting are described by considering the influence of a fundamental mode n on the growth of its first harmonic 2n [30].
023020-3
EDUARDO O. DIAS AND JOSE A. MIRANDA
PHYSICAL REVIEW E 91 , 023020 (2015)
Writing the equations of motion for the harmonic mode we have
closed form solution can be written as £n(0 = ?„(0)exp{A(n)l}
a2„ = A.(2n)a2„ + ^T(2n,n)aft,
(11) +
b2n=X(2n)b2n,
( 12)
n '^ 0
where the finger tip function is defined as
x
T(2n,n) = F(2n,n) + X(n)G(2n,n) - \n\[xblB( x n 2 - 4 ) + 6«2 - 3].
(14)
into the second-order terms on the right-hand side of Eq. (5), we obtain
tn = UnKn + ^ [T («,n O + A(nOG(«,nO]
