Determining the number of fingers in the lifting Hele-Shaw problem

Stability Transition in Gap Expansion-Driven Interfacial Flow

We investigate interfacial instability in a lifting Hele-Shaw cell by experiments and theory. We characterize the unexplored transition from stable to unstable patterns under a wide range of controlling parameters. Surprisingly, we find that the perturbation growth rate-based criterion for the onset of instability from linear stability theory is too strict by over 3 orders of magnitude. To reconcile this striking discrepancy, we propose a new criterion based on perturbation amplitude, which is in excellent agreement with the experimental results. We further show that the fingering pattern evolves to produce a hierarchical fluid structure and derive a theoretical equation to predict the fingering evolution.

Controlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects

In this paper, the interfacial motion between two immiscible viscous fluids in the confined geometry of a Hele-Shaw cell is studied. We consider the influence of a thin wetting film trailing behind the displaced fluid, which dynamically affects the pressure drop at the fluid-fluid interface by introducing a nonlinear dependence on the interfacial velocity. In this framework, two cases of interest are analyzed: The injection-driven flow (expanding evolution), and the lifting plate flow (shrinking evolution). In particular, we investigate the possibility of controlling the development of fingering instabilities in these two different Hele-Shaw setups when wetting effects are taken into account. By employing linear stability theory, we find the proper time-dependent injection rate Q(t) and the time-dependent lifting speed b[over ̇](t) required to control the number of emerging fingers during the expanding and shrinking evolution, respectively. Our results indicate that the consideration of wetting leads to an increase in the magnitude of Q(t) [and b[over ̇](t)] in comparison to the nonwetting strategy. Moreover, a spectrally accurate boundary integral approach is utilized to examine the validity and effectiveness of the controlling protocols at the fully nonlinear regime of the dynamics and confirms that the proposed injection and lifting schemes are feasible strategies to prescribe the morphologies of the resulting patterns in the presence of the wetting film.

Elastic fingering patterns in confined lifting flows

The elastic fingering phenomenon occurs when two confined fluids are brought into contact, and due to a chemical reaction, the interface separating them becomes elastic. We study elastic fingering pattern formation in Newtonian fluids flowing in a lifting (time-dependent gap) Hele-Shaw cell. Using a mode-coupling approach, nonlinear effects induced by the interplay between viscous and elastic forces are investigated and the weakly nonlinear behavior of the fluid-fluid interfacial patterns is analyzed. Our results indicate that the existence of the elastic interface allows the development of unexpected morphological behaviors in such Newtonian fluid flow systems. More specifically, we show that depending on the values of the governing physical parameters, the observed elastic fingering structures are characterized by the occurrence of either finger tip splitting or side branching. The impact of the elastic interface on finger-competition events is also discussed.

Adhesion force in fluids: effects of fingering, wetting, and viscous normal stresses

Probe-tack measurements evaluate the adhesion strength of viscous fluids confined between parallel plates. This is done by recording the adhesion force that is required to lift the upper plate, while the lower plate is kept at rest. During the lifting process, it is known that the interface separating the confined fluids is deformed, causing the emergence of intricate interfacial fingering structures. Existing meticulous experiments and intensive numerical simulations indicate that fingering formation affects the lifting force, causing a decrease in intensity. In this work, we propose an analytical model that computes the lifting adhesion force by taking into account not only the effect of interfacial fingering, but also the action of wetting and viscous normal stresses. The role played by the system's spatial confinement is also considered. We show that the incorporation of all these physical ingredients is necessary to provide a better agreement between theoretical predictions and experiments.

Influence of wetting on fingering patterns in lifting Hele-Shaw flows

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.

Stretch flow of confined non-Newtonian fluids: nonlinear fingering dynamics

We employ a weakly nonlinear perturbative scheme to investigate the stretch flow of a non-Newtonian fluid confined in Hele-Shaw cell for which the upper plate is lifted. A generalized Darcy's law is utilized to model interfacial fingering formation in both the weak shear-thinning and weak shear-thickening limits. Within this context, we analyze how the interfacial finger shapes and the nonlinear competition dynamics among fingers are affected by the non-Newtonian nature of the stretched fluid.