Rotating Hele-Shaw Cell Research Articles

Pinch-off singularities in rotating Hele-Shaw flows at high viscosity contrast

We study the evolution of a family of dumbbell-shaped liquid patches surrounded by air inside a rotating Hele-Shaw cell with lubrication methods and numerical simulations. Depending on initial conditions, the dumbbell either stretches to infinity, pinches off at the neck to form a droplet, or collects into a circular drop at the center of rotation. Whether or not pinch-off occurs results from a subtle interplay between centrifugal and capillary forces. In particular, rotation may delay or even prevent pinch-off from occurring owing to stretching and smoothing of the fluid neck. However, frequently rotation may have the opposite effect leading to pinch-off where the relaxation toward a circular drop would be observed in an ordinary Hele-Shaw cell.

Coriolis effects on fingering patterns under rotation

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.

Geometric approach to stationary shapes in rotating Hele–Shaw flows

We consider the geometry of a special family of curves associated with the motion of a two-fluid interface in a rotating Hele–Shaw cell. This family of stationary exact solutions with surface tension consists of interface shapes which balance exactly the competing capillary and centrifugal forces. The result is the formation of patterned structures presenting fingers that eventually assume a teardrop-like shape, and tend to be detached from the main body of the rotating fluid (occurrence of a pinch-off event). By using the vortex–sheet formalism, we approach the problem analytically, and show that the curvature of these particular curves can be very simply expressed as a function of the radial distance to the rotation axis. Motivated by this fact, and through a simple geometric interpretation, we show that the exact solutions for the rotating Hele–Shaw problem satisfy a first-order ordinary differential equation which is readily solved, so that the shape solutions are determined up to quadratures. The ability to probe a number of key morphological features of such solutions analytically is demonstrated, and a gallery of plots is provided to highlight these findings. The possibility of accessing a criterion to predict the occurrence of pinch-off is also discussed. Finally, we prove that the general solutions are dense, and use this fact to guarantee the very existence of the neat, highly symmetric physical patterns.

Dynamics of viscous fingers in rotating Hele-Shaw cells with Coriolis effects

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.

Numerical study of pattern formation in miscible rotating Hele-Shaw flows

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.

Viscosity contrast effects on fingering formation in rotating Hele-Shaw flows

The different finger morphologies that arise at the interface separating two immiscible fluids in a rotating Hele-Shaw cell are studied numerically. The whole range of viscosity contrast is analyzed and a variety of fingering patterns systematically introduced, including the case in which the inner fluid is less viscous than the outer one. Our numerical results demonstrate that both the magnitude and the sign of the viscosity contrast strongly affect the shape of the emerging fingers, and also their length distribution. We have also found that the occurrence and location of pinch-off singularities are remarkably modified when the inner fluid is less viscous: instead of generating an isolated detaching drop, a full finger is disconnected from the interface. Finally, we have verified that the finger competition phenomena revealed by our simulations are correctly predicted by a weakly nonlinear analysis of the pattern development, showing that such important finger competition dynamics is already set at relatively early stages of interfacial evolution.

Numerical simulations of miscible fluids on a rotating Hele–Shaw cell with effects of Coriolis forces

AbstractThe effects of Coriolis forces to interfacial stabilities on miscible rotating Hele–Shaw flows are investigated by means of highly accurate numerical schemes. Two major influences on the interfaces, i.e. the stabilization of circumferential fingerings and the unstable body distortion of droplets, are observed. These two phenomena can be represented by two characteristic measurements, i.e. interfacial length and radius of gyration, respectively. Without an additional injection, the miscible interface is found to be more stable at higher Coriolis factors, based on both the judgments of growths in interfacial length and the radius of gyration. However, with an additional injection, a visible body distortion with suppressed interfacial fingerings at higher Coriolis factors leads to inconsistent developments in these two characteristic measurements. The stabilized effect of interfacial fingering delays and weakens the growth in interfacial length at higher Coriolis factors, while faster and greater growth in the radius of gyration is observed because of the significant body distortion. As a result, the effects of the Coriolis forces on the miscible interfacial instabilities should be addressed with great caution. Copyright © 2005 John Wiley & Sons, Ltd.

Numerical simulations of interfacial instabilities on a rotating miscible magnetic droplet with effects of Korteweg stresses

Interfacial instability of a miscible magnetic droplet in a rotating Hele-Shaw cell is simulated numerically. The influence of magnetic strengths, the Korteweg stresses, and their coupled effects are first discussed qualitatively by fingering patterns and streamlines. Quantitative measurements are evaluated by interfacial length L, number of fingers n, and diameter of gyration Dg. The results confirm with coupling rotational effects more vigorous fingering instability occurs on stronger magnetic strengths and less effective surface tensions (Korteweg stresses). Without the effects of Korteweg stresses, significant nonlinear fingering merges occur which lead to reduction in fingering number, early decay of interfacial length and reversed plane trajectories. Before the occurrence of fingering merges, monotonic growths of interfacial lengths, constant fingering numbers, and nearly linear pattern trajectories are observed. If the significant Korteweg stresses are taken into account, the nonlinear merge is prevented and the features of fingering patterns resemble the immiscible situations remarkably. The fingering behavior can be approximated by a master line of dL∕dDg≈0.386n+0.13 within the linear fingering region.

Coriolis effects in a rotating Hele-Shaw cell

Following Schwartz [Phys. Fluids A 1, 167 (1989)], we consider flow in a rotating Hele-Shaw cell where the Coriolis effect is significant. In that paper the Coriolis term is included in an ad hoc manner. We derive the model with the exact form of the Coriolis term, and show where it differs from that of Schwartz. The linear stability analysis carried out by Schwartz is repeated and the results for the two models are compared.

Finger competition dynamics in rotating Hele-Shaw cells

We report analytical results for the development of interfacial instabilities in rotating Hele-Shaw cells. We execute a mode-coupling approach to the problem and examine the morphological features of the fluid-fluid interface at the onset of nonlinear effects. The impact of normal stresses is accounted for through a modified pressure jump boundary condition. A differential equation describing the early nonlinear evolution of the interface is derived, being conveniently written in terms of three relevant dimensionless parameters: viscosity contrast A , surface tension B , and gap spacing b . We focus our study on the influence of these parameters on finger competition dynamics. It is deduced that the link between finger competition and A , B , and b can be revealed by a mechanism based on the enhanced growth of subharmonic perturbations. Our results show good agreement with existing experimental and numerical investigations of the problem both in low and high A<0 limits. In particular, it is found that the condition of vanishing A suppresses the dynamic competition between fingers, regardless of the value of B and b . Moreover, our study enables one to extract analytical information about the problem by exploring the whole range of allowed values for A , B , and b . Specifically, it is verified that pattern morphology is significantly modified when the viscosity contrast -1< or =A< or =1 varies: increasingly larger values of A>0 (A<0) lead to enhanced competition of outward (inward) fingers. Within this context the role of B and b in determining different finger competition behaviors is also discussed.

Fingering instabilities of a miscible magnetic droplet on a rotating Hele–Shaw cell

Highly accurate numerical simulations of a magnetic droplet on a rotating Hele–Shaw cell are presented. Under a perpendicular magnetic field, the centrifugally induced fingerings are enhanced, while the azimuthal field tends to damp the instabilities. The influences of these two magnetic fields are also demonstrated quantitatively, which confirm the opposite effects to the interfacial fingerings.

Controlling fingering instabilities in rotating ferrofluids.

We perform a detailed analytic and numerical study of the evolution of a ferrofluid drop confined to a rotating Hele-Shaw cell in the presence of an azimuthal magnetic field. Our results demonstrate that the centrifugally driven interfacial instabilities can be simply controlled with the use of a current-carrying wire. We compare an analytic linear analysis to our computational results and show that a number of observed features cannot be explained by linear theory alone, including a "diamond ring" instability that results when a droplet is nearly stabilized.

Interfacial instabilities of miscible fluids in a rotating Hele–Shaw cell

Numerical simulations of interfacial stability for miscible droplet in a rotating Hele–Shaw cell by means of high-accurate numerical schemes are presented. The interfacial instability is dominated by four control parameters, such as rotating speed, viscosity contrast, diffusion between the species and injecting strength. More vigorous interfacial fingerings are found at higher rotating speed and lower viscosity contrast. The effect of stronger diffusion is not significant to the fingering patterns, but leads only to milder instability. Injection in general reinforces viscous stabilization. For the case of annulus, different mechanisms of instability are observed on the inner and outer fronts. The injecting strength tends to stabilize the outer interface, and perturbs the inner.

Theory of exact solutions for the evolution of a fluid annulus in a rotating Hele-Shaw cell

Motivated by a series of recent experiments on the evolution of fluid annuli in rotating Hele-Shaw cells, this paper presents a new class of exact time-dependent solutions to a mathematical model of this nonlinear free boundary problem. For a certain class of initial conditions, the free boundary problem is reduced to the solution of a finite set of coupled nonlinear ordinary differential equations. These solutions can be explicitly studied and, despite the fact that the model problem is mathematically ill-posed, display the same qualitative features as the recent experiments. It is discussed how the present exact solutions might form an important basis for further study of the appropriately regularized model problem.

Magnetic fluid in a rotating Hele–Shaw cell

We investigate the instability of the interface separating two viscous fluids flowing in a rotating Hele–Shaw cell, when one of the fluids is a magnetic fluid and an external magnetic field is applied. The interplay between centrifugal and magnetic forces is examined. Linear stability analysis shows that a non-uniform field, applied tangential to the cell, tends to stabilize the interface, decreasing the number of interface ripples. The destabilizing role of a uniform, perpendicular field under rotating cell conditions is also considered.

Rotating hele-shaw cells with ferrofluids

We investigate the flow of two immiscible, viscous fluids in a rotating Hele-Shaw cell, when one of the fluids is a ferrofluid and an external magnetic field is applied. The interplay between centrifugal and magnetic forces in determining the instability of the fluid-fluid interface is analyzed. The linear stability analysis of the problem shows that a nonuniform, azimuthal magnetic field, applied tangential to the cell, tends to stabilize the interface. We verify that maximum growth rate selection of initial patterns is influenced by the applied field, which tends to decrease the number of interface ripples. We contrast these results with the situation in which a uniform magnetic field is applied normally to the plane defined by the rotating Hele-Shaw cell.

Interfacial instabilities of a fluid annulus in a rotating Hele–Shaw cell

We have studied the interfacial instabilities experienced by a liquid annulus as it moves radially in a circular Hele–Shaw cell rotating with angular velocity Ω. The instability of the leading interface (oil displacing air) is driven by the density difference in the presence of centrifugal forcing, while the instability of the trailing interface (air displacing oil) is driven by the large viscosity contrast. A linear stability analysis shows that the stability of the two interfaces is coupled through the pressure field already at a linear level. We have performed experiments in a dry cell and in a cell coated with a thin fluid layer on each plate, and found that the stability depends substantially on the wetting conditions at the leading interface. Our experimental results of the number of fingers resulting from the instability compare well with the predictions obtained through a numerical integration of the coupled equations derived from a linear stability analysis. Deep in the nonlinear regime we observe the emission of liquid droplets through the formation of thin filaments at the tip of outgrowing fingers.

Radial displacement of a fluid annulus in a rotating Hele–Shaw cell

The radial displacement of a fluid annulus in a rotating circular Hele–Shaw cell has been investigated experimentally. It has been found that the flow depends sensitively on the wetting conditions at the outer interface. Displacements in a prewet cell are well described by Darcy’s law in a wide range of experimental parameters, with little influence of capillary effects. In a dry cell, however, a more careful analysis of the interface motion is required; the interplay between a gradual loss of fluid at the inner interface, and the dependence of capillary forces at the outer interface on interfacial velocity and dynamic contact angle, result in a constant velocity for the interfaces. The experimental results in this case correlate in the form of an empirical scaling relation between the capillary number Ca and a dimensionless group, related to the ratio of centrifugal to capillary forces, which spans about three orders of magnitude in both quantities. Finally, the relative thickness of the coating film left by the inner interface, αi, is obtained as a function of Ca.

Experiments in a rotating Hele-Shaw cell.

Experiments in a rotating Hele-Shaw cell.

Instability and fingering in a rotating Hele–Shaw cell or porous medium

The simplified mathematical model for two-phase displacement in a Hele–Shaw cell, i.e., slow flow between closely spaced parallel plates, has been extended to include effects arising from a constant rate of rotation of the system. A linear stability analysis for an initially concentric circular drop reveals that it is unstable both to translation and, depending on the rotation rate, to a number of fingering modes. The nonlinear evolution with time of two of these modes is traced using a boundary-integral numerical scheme. Ultimately, the liquid mass will break up into a number of drops. Apart from small distortion due to the Coriolis force, the patterns formed are essentially independent of both the liquid viscosity and the cell plate spacing or porous-medium permeability. Potential application areas include certain aspects of coating flows, mold filling, and a class of liquid-filled spinning projectiles with permeable packing.