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Register a new userControlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects
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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 ${dot b}(t)$ 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 ${dot b}(t)$] in comparison to the non-wetting 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.
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Droplet migration in a Hele--Shaw cell is a fundamental multiphase flow problem which is crucial for many microfluidics applications. We focus on the regime at low capillary number and three-dimensional direct numerical simulations are performed to investigate the problem. In order to reduce the computational cost, an adaptive mesh is employed and high mesh resolution is only used near the interface. Parametric studies are performed on the droplet horizontal radius and the capillary number. For droplets with an horizontal radius larger than half the channel height the droplet overfills the channel and exhibits a pancake shape. A lubrication film is formed between the droplet and the wall and particular attention is paid to the effect of the lubrication film on the droplet velocity. The computed velocity of the pancake droplet is shown to be lower than the average inflow velocity, which is in agreement with experimental measurements. The numerical results show that both the strong shear induced by the lubrication film and the three-dimensional flow structure contribute to the low mobility of the droplet. In this low-migration-velocity scenario the interfacial flow in the droplet reference frame moves toward the rear on the top and reverses direction moving to the front from the two side edges. The velocity of the pancake droplet and the thickness of the lubrication film are observed to decrease with capillary number. The droplet velocity and its dependence on capillary number cannot be captured by the classic Hele--Shaw equations, since the depth-averaged approximation neglects the effect of the lubrication film.
We report experimental results for the Kelvin-Helmholtz instability between two immiscible fluids in parallel flow in a Hele-Shaw cell. We focus our interest on the influence of the gap size between the walls on the instability characteristics. Experimental results show that the instability threshold, the critical wavelength, the phase velocity and the spatial growth rate depend on this gap size. These results are compared to both the previous two-dimensional analysis of Gondret and Rabaud (1997) and the three-dimensional analysis recently derived by Plourabou=E9 and Hinch (2001), showing that the agreement is still not complete especially when gap size increases.
We study here experimentally, numerically and using a lubrication approach; the shape, velocity and lubrication film thickness distribution of a droplet rising in a vertical Hele-Shaw cell. The droplet is surrounded by a stationary immiscible fluid and moves purely due to buoyancy. A low density difference between the two mediums helps to operate in a regime with capillary number $Ca$ lying between $0.03-0.35$, where $Ca=mu_o U_d /gamma$ is built with the surrounding oil viscosity $mu_o$, the droplet velocity $U_d$ and surface tension $gamma$. The experimental data shows that in this regime the droplet velocity is not influenced by the thickness of the thin lubricating film and the dynamic meniscus. For iso-viscous cases, experimental and three-dimensional numerical results of the film thickness distribution agree well with each other. The mean film thickness is well captured by the Aussillous & Quere (2000) model with fitting parameters. The droplet also exhibits the catamaran shape that has been identified experimentally for a pressure-driven counterpart (Huerre $textit{et al}$. 2015). This pattern has been rationalized using a two-dimensional lubrication equation. In particular, we show that this peculiar film thickness distribution is intrinsically related to the anisotropy of the fluxes induced by the droplets motion.
We adopt a boundary integral method to study the dynamics of a translating droplet confined in a Hele-Shaw cell in the Stokes regime. The droplet is driven by the motion of the ambient fluid with the same viscosity. We characterize the three-dimensional (3D) nature of the droplet interface and of the flow field. The interface develops an arc-shaped ridge near the rear-half rim with a protrusion in the rear and a laterally symmetric pair of higher peaks; this pair of protrusions has been identified by recent experiments (Huerre et al., Phys. Rev. Lett., vol. 115 (6), 2015, 064501) and predicted asymptotically (Burgess and Foster, Phys. Fluids A, vol. 2 (7), 1990, pp. 1105-1117). The mean film thickness is well predicted by the extended Bretherton model (Klaseboer et al., Phys. Fluids, vol. 26 (3), 2014, 032107) with fitting parameters. The flow in the streamwise wall-normal middle plane is featured with recirculating zones, which are partitioned by stagnation points closely resembling those of a two-dimensional droplet in a channel. Recirculation is absent in the wall-parallel, unconfined planes, in sharp contrast to the interior flow inside a moving droplet in free space. The preferred orientation of the recirculation results from the anisotropic confinement of the Hele-Shaw cell. On these planes, we identify a dipolar disturbance flow field induced by the travelling droplet and its $1/r^2$ spatial decay is confirmed numerically. We pinpoint counter-rotating streamwise vortex structures near the lateral interface of the droplet, further highlighting the complex 3D flow pattern.
We study microfluidic self digitization in Hele-Shaw cells using pancake droplets anchored to surface tension traps. We show that above a critical flow rate, large anchored droplets break up to form two daughter droplets, one of which remains in the anchor. Below the critical flow velocity for breakup the shape of the anchored drop is given by an elastica equation that depends on the capillary number of the outer fluid. As the velocity crosses the critical value, the equation stops admitting a solution that satisfies the boundary conditions; the drop breaks up in spite of the neck still having finite width. A similar breaking event also takes place between the holes of an array of anchors, which we use to produce a 2D array of stationary drops in situ.
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