Rotating Hele-Shaw Cell Research Articles

Convection induced by centrifugal and Coriolis buoyancy in a rotating Hele-Shaw reactor

The study of heat and mass transfer in a Hele-Shaw cell rotating around a perpendicular axis has various advanced technological applications. These include the design of microfluidic devices and continuous-flow chemical microreactors, to name a couple. In this setup configuration, the quasi-two-dimensional design allows for recording the density field using optical methods, and the rotation enables control of this field through spatially distributed inertial forces. As is known, in the limit of an infinitely thin layer, the Coriolis force vanishes within a standard mathematical model. However, experimental observations of fluid flow in a rotating Hele-Shaw cell indicate the opposite. In this paper, we show that the correct derivation of the equation of motion under the Hele-Shaw approximation leads to the appearance of a Boussinesq-type term for the Coriolis force. To study the effect of the Coriolis buoyancy, we consider the problem of fluid stability during the internal generation of a transfer component, which can be either the concentration of the dissolved substance or the temperature of the medium. The careful study of system dynamics involves linear stability analysis, weakly nonlinear analysis, and direct numerical simulation. The general properties of the disturbance spectrum are analyzed. The branching of solutions near the first bifurcation is studied using the technique of multiple time scales. A stationary convection is replaced by an oscillatory one under the action of the Coriolis force, as demonstrated by weakly nonlinear analysis. Finally, we investigate the nonlinear dynamics using direct numerical simulation.

Elastic fingering in a rotating Hele-Shaw cell.

We consider the steady-state fingering instability of an elastic membrane separating two fluids of different density under external pressure in a rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are considered, and the role of membrane tension is detailed in each case. Both systems exhibit a centrifugally driven Rayleigh-Taylor-like instability when the density of the inner fluid exceeds that of the outer one, and this instability competes with the restoring forces arising from curvature and tension, thereby setting the finger scale. Numerical continuation is used to compute not only strongly nonlinear primary finger states up to the point of self-contact, but also secondary branches of mixed modes and circumferentially localized folds as a function of the rotation rate and the externally imposed pressure. Both reflection-symmetric and symmetry-broken chiral states are computed. The results are presented in the form of bifurcation diagrams. The ratio of system scale to the natural length scale is found to determine the ordering of the primary bifurcations from the unperturbed circle state as well as the solution profiles and onset of secondary bifurcations.

Dynamics of liquids with highviscosity contrast in unevenly rotating Hele-Shaw cell.

The paper is devoted to the experimental study of the dynamics of an interface between two liquids with highviscosity contrast filling a vertical, circular narrow gap rotating about a horizontal axis at a speed modulated by librations. The equilibrium shape and stability of the interphase boundary are considered. In the absence of librations, under the action of the centrifugal force the boundary has an axisymmetric shape. At librations, at certain frequency ratios, the interface loses its axisymmetric position: in the cavity reference frame, it is displaced in the radial direction. It is theoretically shown that the discovered phenomenon is explained by the average action of gravity; the results of experiments and theory are consistent. The experiments reveal that with an increase in the modulation amplitude the circular interface loses stability in a threshold manner: an azimuthally periodic relief, quasi-stationary in the cavity frame, emerges. This is associated with the Kelvin-Helmholtz instability due to tangential oscillations of the less viscous liquid near the interface. A dimensionless parameter that determines the stability of the interface is obtained in the limit of high dimensionless libration frequency. Thestability threshold increases with the decrease in the dimensionless frequency. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Effect of interfacial rheology on fingering patterns in rotating Hele-Shaw cells.

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 equationthat 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.

A boundary integral method to investigate pattern formation in a rotating Hele-Shaw cell with time dependent gap

In this article we present a spectrally accurate boundary integral method to investigate the evolution of a Hele Shaw blob in a rotating, radial cell with time dependent gap. The blob of the viscous fluid is affected by the shearing forces induced by a changing gap distance, however the rotation also has significant influence on the shape of the blob. Using the method of complex moments, we characterize the solution of the moving boundary problem that models the evolution of the blob and investigate the morphology using linear analysis. The results of the linear analysis indicate the de-stabilizing influence of the rotation. In the numerical methods, we develop a spectrally accurate boundary integral technique to explore the complete nonlinear dynamics of the problem. Using the boundary integral simulations we demonstrate that, for a given strength of surface tension, the rotational effect causes a more complex pattern on the fluid interface that involves the formation of narrow fingers and more splitting of the interface. The effect is more pronounced if the angular speed increases. We also prescribe a method for controlling the viscous fingering using insights from linear analysis. Our numerical simulations validate the proposed controlling mechanism.

Rotating Hele-Shaw cell with a time-dependent angular velocity

Most rotating Hele-Shaw cell studies assume a constant cell angular velocity and neglect inertial effects. This work considers a time-dependent angular velocity and includes inertia. These two new elements introduce important changes in the linear and weakly nonlinear dynamics of the problem.

Generalized elastica patterns in a curved rotating Hele-Shaw cell.

We study a family of generalized elasticalike equilibrium shapes that arise at the interface separating two fluids in a curved rotating Hele-Shaw cell. This family of stationary interface solutions consists of shapes that balance the competing capillary and centrifugal forces in such a curved flow environment. We investigate how the emerging interfacial patterns are impacted by changes in the geometric properties of the curved Hele-Shaw cell. A vortex-sheet formalism is used to calculate the two-fluid interface curvature, and a gallery of possible shapes is provided to highlight a number of peculiar morphological features. A linear perturbation theory is employed to show that the most prominent aspects of these complex stationary patterns can be fairly well reproduced by the interplay of just two interfacial modes. The connection of these dominant modes to the geometry of the curved cell, as well as to the fluid dynamic properties of the flow, is discussed.

Labyrinthine instabilities of miscible magnetic fluids in a rotating Hele-Shaw cell

This study presents the first experimental results of confining miscible magnetic fluids in a rotating Hele-Shaw cell. Variations in the prominence of labyrinthine instabilities are observed under a range of experimental conditions, with different magnetic field strengths, gap depths, and rotation speeds. These instabilities are characterized by two modified Péclect numbers, namely, Pem (the ratio of the characteristic magnetic advection rate and the diffusion rate) and Pec (the ratio of characteristic rotation advection and the diffusion rate). The magnetic effect is characterized by dipolar repulsion, which triggers a distinctive fingering pattern differing from the progressive diffusion pattern that occurs without magnetic fields or rotation. Under the same rotation speed, the magnetoviscous effect will hinder the growth rate of the magnetic drops at the later stage. However, both the rotation effect and the gap depth greatly enhance the growth rate of the magnetic drops, as these conditions help to intensify the labyrinthine instabilities. In contrast, the countering pressure gradient produces an opposite force that constrains the trend toward expansion. Two major phases in the growth of instabilities are defined: a magnetization phase and a rotation phase, which are dominated by the magnetic and the rotation effect, respectively. The significance of the rotation effect is confirmed by the linear regression between the rotation growth rate and Pec. Finally, main fingering structures that evolve from the secondary waves are verified as having a wavelength λ to gap depth h relation of λ≈(7±1)h.

Rotationally induced fingering patterns in a two-dimensional heterogeneous porous medium.

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 howthesefully 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.

Ferrofluid patterns in Hele-Shaw cells: Exact, stable, stationary shape solutions.

We investigate a quasi-two-dimensional system composed of an initially circular ferrofluid droplet surrounded by a nonmagnetic fluid of higher density. These immiscible fluids flow in a rotating Hele-Shaw cell, under the influence of an in-plane radial magnetic field. We focus on the situation in which destabilizing bulk magnetic field effects are balanced by stabilizing centrifugal forces. In this framing, we consider the interplay of capillary and magnetic normal traction effects in determining the fluid-fluid interface morphology. By employing a vortex-sheet formalism, we have been able to find a family of exact stationary N-fold polygonal shape solutions for the interface. A weakly nonlinear theory is then used to verify that such exact interfacial solutions are in fact stable.

Stationary patterns in centrifugally driven interfacial elastic fingering.

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.

Elastic fingering in rotating Hele-Shaw flows.

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.

Interfacial instability in a time-periodic rotating Hele-Shaw Cell

The effect of time-periodic angular velocity on the interfacial instability of two immiscible, viscous fluids of different densities and confined in an annular Hele-Shaw cell is investigated. An inviscid linear stability analysis of the viscous and time dependent basic flow leads to a periodic Mathieu oscillator describing the evolution of the interfacial amplitude. We show that the relevant parameters that control the interface are the Bond number, viscosity ratio, Atwood number and the frequency number.

Two-phase flow in rotating Hele-Shaw cells with Coriolis effects

The free boundary problem of a two phase flow in a rotating Hele-Shaw cell with Coriolis effects is studied. Existence and uniqueness of solutions near spheres is established, and the asymptotic stability and instability of the trivial solution is characterized in dependence on the fluid densities.

Control of centrifugal fingering via a variable interfacial tension approach

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.

Control of centrifugally driven fingering in a tapered Hele-Shaw cell

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.

Diffuse-interface approach to rotating Hele-Shaw flows

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.

Inertial effects on rotating Hele-Shaw flows

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.

Stationary shapes of confined rotating magnetic liquid droplets

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.

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele–Shaw Cell

We study the radial movement of an incompressible fluid located in a Hele–Shaw cell rotating at a constant angular velocity in the horizontal plane. Within an analytic framework, local existence and uniqueness of solutions is proved, and it is shown that the unique rotationally invariant equilibrium of the flow is unstable. There are, however, other time-independent solutions: using surface tension as a bifurcation parameter we establish the existence of global bifurcation branches consisting of stationary fingering patterns. The same results can be obtained by fixing the surface tension while varying the angular velocity. Finally, it is shown that the equilibria on a global bifurcation branch converge to a circle as the surface tension tends to infinity, provided they stay suitably bounded.