The flow in a Hele-Shaw cell with a time-increasing gap poses a unique shrinking interface problem. When the upper plate of the cell is lifted perpendicularly at a prescribed speed, the exterior less viscous fluid penetrates the interior more viscous fluid, which generates complex, time-dependent interfacial patterns through the Saffman-Taylor instability. The pattern formation process sensitively depends on the lifting speed and is still not fully understood. For some lifting speeds, such as linear or exponential speed, the instability is transient and the interface eventually shrinks as a circle. However, linear stability analysis suggests there exist shape invariant shrinking patterns if the gap $b(t)$ is increased more rapidly: $\displaystyle b(t)=\left(1-\frac{7}{2}τ\mathcal{C} t\right)^{-{2}/{7}}$, where $τ$ is the surface tension and $\mathcal{C}$ is a function of the interface perturbation mode $k$. Here, we use a spectrally accurate boundary integral method together with an efficient time adaptive rescaling scheme, which for the first time makes it possible to explore the nonlinear limiting dynamical behavior of a vanishing interface. When the gap is increased at a constant rate, our numerical results quantitatively agree with experimental observations (Nase et al., Phys. Fluids, vol. 23, 2011, pp. 123101). When we use the shape invariant gap $b(t)$, our nonlinear results reveal the existence of $k$-fold dominant, one-dimensional, web-like networks, where the fractal dimension is reduced to almost one at late times. We conclude by constructing a morphology diagram for pattern selection that relates the dominant mode $k$ of the vanishing interface and the control parameter $\mathcal{C}$.
We study the Hele-Shaw immiscible displacements when all surfaces tensions on the interfaces are zero. The Saffman-Taylor instability occurs when a less viscous fluid is displacing a more viscous one, in a rectangular Hele-Shaw cell. We prove that an intermediate liquid with a variable viscosity can almost suppress this instability. On the contrary, a large number of constant viscosity liquid-layers inserted between the initial fluids gives us boundless growth rates with respect to the wave numbers of perturbations. The same amount of intermediate liquid is used in both cases.
A mathematical model describing motion of an inhomogeneous incompressible fluid in a Hele-Shaw cell is considered. Linear stability analysis of shear flow class is provided. The role of inertia, linear friction and impermeable boundaries in Kelvin--Helmholtz instability development is studied. Hierarchy of simplified one-dimensional models of viscosity- and density-stratified flows is obtained in long-wave approximation. Interpretation of Saffman--Taylor instability development is given in the framework of these models.
We present direct numerical simulations (DNS) study of confined buoyancy-driven bubbly flows in a Hele-Shaw setup. We investigate the spectral properties of the flow and make comparisons with experiments. The energy spectrum obtained from the gap-averaged velocity field shows $E(k) \sim k$ for $k < k_d$, $E(k) \sim k^{-5}$ for $k > k_d$, and an intermediate scaling range with $E(k) \sim k^{-3}$ around $k \sim k_d$. We perform an energy budget analysis to understand the dominant balances and explain the observed scaling behavior. We also show that the Navier-Stokes equation with a linear drag c…
We calculate the shape and the velocity of a bubble rising in an infinitely large and closed Hele-Shaw cell using Park and Homsy's boundary condition which accounts for the change of the three dimensional structure in the perimeter zone. We first formulate the problem in the form of a variational problem, and discuss the shape change assuming that the bubble takes elliptic shape. We calculate the shape and the velocity of the bubble as a function of the bubble size, gap distance and the inclination angle of the cell. We show that the bubble is flattened as it rises. This result is in agreement…
Exact solutions are reported for a stream of asymmetric bubbles steadily moving in a Hele-Shaw channel. From the periodicity along the streamwise direction, the flow region is reduced to a rectangular unit cell containing one bubble, which is conformally mapped to an annulus in an auxiliary complex plane. Analytic expressions for the bubble shape as well as for the velocity field are obtained in terms of the generalized Schwarz-Christoffel formula for doubly connected domains.
The Saffman-Taylor instability occurs when a less viscous fluid is displacing a more viscous one in a rectangular Hele-Shaw cell. A surface tension on the interface between the two fluids is improving the stability. The multi-layer Hele - Shaw model, consisting of $N$ intermediate fluids with constant viscosities, was studied in some previous papers and very low growth constants were obtained for large $N$. We prove that this model leads us to a significant instability, even if $N$ is very large. The maximum value of growth constants can not decrease under a certain value, not depending on the…
The Saffman-Taylor instability occurs when { a} Stokes fluid is displaced by a less viscous one { in} a Hele-Shaw cell. { This model is useful to study the secondary oil recovery from a pororus medum.} Since 1960, { polymer solutions} were used as displacing fluids; moreover, the oil in a porous reservoir can often be considered a non-Newtonian fluid. Motivated by this fact, in this paper we study the linear stability of the displacement of two Oldroyd-B fluids in a rectilinear Hele-Shaw cell, even if the direct relevance for the flow through porous media is not so evident. We get an { approxi…
A hierarchy of mathematical models describing viscosity-stratified flow in a Hele-Shaw cell is constructed. Numerical modelling of jet flow and development of viscous fingers with the influence of inertia and friction is carried out. One-dimensional multi-layer flows are studied. In the framework of three-layer flow the interpretation of the Saffman--Taylor instability is given. A kinematic-wave model of viscous fingering taking into account friction between the fluid layers is proposed. Comparison with calculations on the basis of two-dimensional equations shows that this model allows to dete…
Exact solutions are presented for a doubly-periodic array of steadily moving bubbles in a Hele-Shaw cell when surface tension is neglected. It is assumed that the bubbles either are symmetrical with respect to the channel centreline or have fore-and-aft symmetry, or both, so that the relevant flow domain can be reduced to a simply connected region. By using conformal mapping techniques, a general solution with any number of bubbles per unit cell is obtained in integral form. Several examples are given, including solutions for multi-file arrays of bubbles in the channel geometry and doubly-peri…
This is a response to the comment cited as PoF2023,35:029101. We show that the depth will have an impact on the wave height and the scaling law we obtained before is more feasible to use as a prior to give an initial guess of the wave height without any experimental information. This response strongly supports our previous work and the capability of the scaling law.
In this fluid dynamics video, we study a two-phase flow in an elastic Hele-Shaw cell that involves two distinct fluid- and solid-based instabilities: viscous fingering and sheet buckling. We show that the relative importance of the two instabilities is controlled by a single non-dimensional parameter, which provides a measure of the elasticity of the flexible wall. We employ numerical simulations to show that for relatively stiff [soft] walls, the system's behaviour is dominated by viscous fingering [sheet buckling]. Strong interactions between the two instabilities arise in an intermediate re…
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 well-studied selection problems involving Saffman-Taylor fingers or Taylor-Saffman bubbles in a Hele-Shaw channel are prototype examples of pattern selection. Exact solutions to the corresponding zero-surface-tension problems exist for an arbitrary finger or bubble speed, but the addition of surface tension leads to a discrete set of solution branches, all of which approach a single solution in the limit the surface tension vanishes. In this sense, the surface tension selects a single physically meaningful solution from the continuum of zero-surface-tension solutions. Recently, we provided…
We consider experimentally the instability and mass transport of a porous-medium flow in a Hele-Shaw geometry. In an initially stable configuration, a lighter fluid (water) is located over a heavier fluid (propylene glycol). The fluids mix via diffusion with some regions of the resulting mixture being heavier than either pure fluid. Density-driven convection occurs with downward penetrating dense fingers that transport mass much more effectively than diffusion alone. We investigate the initial instability and the quasi steady state. The convective time and velocity scales, finger width, wave n…
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…
We experimentally and theoretically investigate the behavior of Leidenfrost droplets inserted in a Hele-Shaw cell. As a result of the confinement from the two surfaces, the droplet has the shape of a flattened disc and is thermally isolated from the surface by the two evaporating vapor layers. An analysis of the evaporation rate using simple scaling arguments is in agreement with the experimental results. Using the lubrication approximation we numerically determine the shape of the droplets as a function of its radius. We furthermore find that the droplet width tends to zero at its center when…
We present an analytical study, validated by numerical simulations, of electroosmotic flow in a Hele-Shaw cell with non-uniform surface charge patterning. Applying the lubrication approximation and assuming thin electric double layer, we obtain a pair of uncoupled Poisson equations which relate the pressure and the stream function, respectively, to gradients in the zeta potential distribution parallel and perpendicular to the applied electric field. We solve the governing equations for the fundamental case of a disk with uniform zeta potential and show that the flow-field in the outer region t…
We analyze flow of non-Newtonian fluids in a Hele-Shaw cell, subjected to spatially non-uniform electroosmotic slip. Motivated by their potential use for increasing the characteristic pressure fields, we specifically focus on power-law fluids with wall depletion properties. We derive a p-Poisson equation governing the pressure field, as well as a set of linearized equations representing its asymptotic approximation for weakly non-Newtonian behavior. To investigate the effect of non-Newtonian properties on the resulting fluidic pressure and velocity, we consider several configurations in one- a…
Time-dependent injection strategies are commonly employed to control the number of viscous fingers emerging at the interface separating two fluids during radial displacement in Hele-Shaw flows. Here we demonstrate theoretically that such a usual controlling method is significantly improved by taking advantage of an electro-osmotic flow generated by applying an external electric field. More specifically, under the coupled action of time-varying electric currents and injection rates, we design a strategy capable of controlling not only the number of fingers emerging at the interface but also whe…
Various thermodynamical phenomena have occurred with change of pressure and temperature, volume. We can choose these parameters but not these constraints, in order to need the thermodynamics with physical properties in the fields of various situations. The Hele-Shaw flow is one of the classical fluid dynamics problems in chemical engineering. Indeed, the Stokes flow is represented as the Stokes equations which is simplified from the famous Navier-Stokes equations. However we have often to deal with the Hele-Shaw flow with the Darcy's law. This law is derived by the Stokes equations. The basic …
We study the transient dynamics of a viscous liquid contained in a narrow gap between a rigid surface and a parallel elastic plate. The elastic plate is deformed due to an externally applied time-varying pressure-field. We model the flow-field via the lubrication approximation, and the plate deformation by the Kirchhoff-Love plate theory. We obtain a self-similarity solution for the case of an external point force acting on the elastic plate. The pressure and deformation field during and after the application of the external force are derived and presented by closed form expressions. We examin…
A plane turbulent mixing in a shear flow of an ideal homogeneous fluid confined between two relatively close rigid walls is considered. The character of the flow is determined by interaction of vortices arising at the nonlinear stage of the Kelvin--Helmholtz instability development and by turbulent friction. In the framework of the shallow water theory and a three-layer representation of the flow, one-dimensional models of a mixing layer are proposed. The obtained equations allow one to determine averaged boundaries of the region of intense fluid mixing. Stationary solutions of the governing e…
School of Mathematical Sciences, Queensland University of Technology（昆士兰科技大学理学院）；School of Mathematics & Statistics, University of Melbourne（墨尔本大学数学与统计学院）
The rise of a single bubble confined between two vertical plates is investigated over a wide range of Reynolds numbers. In particular, we focus on the evolution of the bubble speed, aspect ratio and drag coefficient during the transition from the viscous to the inertial regime. For sufficiently large bubbles, a simple model based on power balance captures the transition for the bubble velocity and matches all the experimental data despite strong time variations of bubble aspect ratio at large Reynolds numbers. Surprisingly, bubbles in the viscous regime systematically exhibit an ellipse elonga…
Various microfluidic systems, such as chemical and biological lab-on-a-chip devices, involve motion of multiple droplets within an immersing fluid in shallow micro-channels. Modeling the dynamics of such systems requires calculation of the forces of interaction between the moving droplets. These forces are commonly approximated by superposition of dipoles solutions, which requires an assumption of sufficiently large distance between the droplets. In this work we obtain exact solutions (in the Hele-Shaw limit) for two moving droplets, and a droplet within a droplet, located within a moving imme…
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 fingeri…
We study the effects of some injection policies used in oil recovery process. The Saffman-Taylor instability occurs when a less viscous fluid is displacing a more viscous one, in a rectangular Hele-Shaw cell. The injection of $N$ successive intermediate phases with constant viscosities (the multi-layer Hele-Shaw model) was studied in some recent papers, where a minimization of the Saffman-Taylor instability was obtained for large enough $N$. However, in this paper we get a particular eigenfunction of the linear stability system which leads to eigenvalues which become infinite for large wave nu…
We study the stability of multi-layer radial flows in porous media within the Hele-Shaw model. We perform a linear stability analysis for radial flows consisting of an arbitrary number of fluid layers with interfaces separating fluids of constant viscosity and with positive viscosity jump at each interface in the direction of flow. Several different time-dependent injection strategies are analyzed including the maximal injection rate that maintains a stable flow. We find numerically that flows with more fluid layers can be stable with faster time-dependent injection rates than comparable flows…
This research has found a novel computational efficient method of modelling flow at low Reynolds number through fracture networks. The numerical analysis was performed by connecting Hele-Shaw cells to investigate the effect of the intersection to the pressure field and hydraulic resistance for given inlet and outlet pressure values. In this analysis, the impact of intersecting length, intersecting angle and fracture aperture on the fluid flow was studied. For this purpose, two models with different topologies were established. The Hele-Shaw simulation results for hydraulic resistance, pressure…
Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn-Hilliard-Navier-Stokes model introduced by Abels, Garcke and Grün (Math. Models Methods Appl. Sci. 2012), which uses a volume averaged velocity, we derive a diffuse interface model in a Hele-Shaw geometry, which in the case of non-matched densities, simplifies an earlier model of Lee, Lowengrub and Goodman (Phys. Fluids 2002). We recover the classical Hele-Shaw model as a sharp…
The three-layer Saffman-Taylor problem introduces two coupled moving interfaces separating the three fluids. A very recent weakly nonlinear analysis of this problem in a radial Hele-Shaw cell setup has shown that the morphologies of the emerging fingering patterns strongly depend on the initial thickness of the intermediate layer connecting the two interfaces. Here we go beyond the weakly nonlinear regime and explore full nonlinear interfacial dynamics using a spectrally accurate boundary integral method. We quantify the nonlinear instability of both interfaces as the relevant physical paramet…
Let the interface between two immiscible fluids in a Hele-Shaw cell have, at t=0, a wedge shape. As a wedge is scale-free, the fluid relaxation dynamics are self-similar. We find the dynamic exponent of this self-similar flow and show that the interface shape is given by the solution of an unusual inverse problem of potential theory. We solve this inverse problem analytically for an almost flat wedge, and numerically otherwise. The wedge solution is useful for analysis of pinch-off singularities.
An useful approximation for the displacement of two immiscible fluids in a porous medium is the Hele-Shaw model. We consider several liquids with different constant viscosities, inserted between the displacing fluids. The linear stability analysis of this model leads us to an ill-posed problem. The growth rates (in time) of the perturbations exist iff some compatibility conditions on the interfaces are verified. We prove that these conditions cannot be fulfilled.
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 cha…
We investigate quasi-two-dimensional relaxation, by surface tension, of a long straight stripe of inviscid fluid trapped inside a viscous fluid in a Hele-Shaw cell. Combining analytical and numerical solutions, we describe the emergence of a self-similar dumbbell shape and find non-trivial dynamic exponents that characterize scaling behavior of the dumbbell dimensions.
An experimental study of methane, propane and dimethyl ether (DME) pre-mixed flames propagating in a quasi-two-dimensional Hele-Shaw cell placed horizontally is presented in this paper. The flames are ignited at the open end of the combustion chamber and propagate towards the closed end. Our experiments revealed two distinct propagation regimes depending on the equivalence ratio of the mixture as a consequence of the coupling between the heat-release rate and the acoustic waves. The primary acoustic instability induces a small-amplitude, of around 8 mm, oscillatory motion across the chamber th…
In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers [1, 2] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painleve I equation. We argue that the Painleve I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation sho…
A system of two-dimensional, two coupled Faraday interfacial waves is experimentally observed at the two interfaces of the three layers of fluids (air, pure ethanol and silicon oil) in a sealed Hele-Shaw cell with periodic vertical vibration. The upper and lower Faraday waves coexist: the upper vibrates vertically, but the crests of the lower one oscillate horizontally with unchanged wave height and a frequency equal to the half of the forcing one of the vertically vibrating basin, while the troughs of the lower one always stay in the same place (relative to the basin). Besides, they are stron…
We study steady-state oscillations of an elastic Hele-Shaw cell excited by traveling pressure waves over its upper surface. The fluid within the cell is bounded by two asymmetric elastic sheets which are connected to a rigid surface via distributed springs and modeled by the linearized plate theory. Modal analysis yields the frequency response of the configuration as a function of three parameters: the fluidic Womersley number and two ratios of elastic stress to viscous pressure for each of the sheets. These ratios, analogous to the Capillary number, combine the effects of fluid viscosity and …
The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman-Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical…
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 $Ca=μ_o U_d /γ$ is built with the surrounding oil viscosity $μ_o$, the droplet velocity $U_d$ and surface tension $γ$. The experimental data shows that in this regime the droplet …
We present a computational framework to address the flow of two immiscible viscous liquids which co-flow into a shallow rectangular container at one side, and flow out into a holding container at the opposite side. Assumptions based on the shallow depth of the domain are used to reduce the governing equations to one of Hele-Shaw type. The distinctive feature of the numerical method is the accurate modeling of the capillary effects. A continuum approach coupled with a volume-of-fluid formulation for computing the interface motion and for modeling the interfacial tension in Hele-Shaw flows is fo…
Granular materials constitute an intermediate state of matter between fluids and solids. Here we investigate the pattern formation when a grain is displaced by another type of grain in a radial Hele-Shaw cell. We show that several morphologies can occur, ranging from rounded to fingered patterns, interconnected by a continuous crossover. Fourier analyses shows that, in contrast to the rounded patterns, the fingered shapes present mode selection.
Exact solutions are reported for a periodic assembly of bubbles steadily co-travelling in a Hele-Shaw channel. The solutions are obtained as conformal mappings from a multiply connected circular domain in an auxiliary complex plane to the flow region in a period cell. The conformal mappings are constructed using the generalized Schwarz-Christoffel formula for multiply connected polygonal domains in terms of products of Schottky-Klein prime functions. It is shown that previous solutions for multiple steady bubbles in a Hele-Shaw cell are all particular cases of the solutions described herein. E…
We present a phase-field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. With asymptotic matching techniques we check the model to yield the right Hele-Shaw equations in the sharp-interface limit and compute the corrections to these equations to first order in the interface thickness. We also compute the effect of such corrections on the linear dispersion relation of the planar interface. We discuss in detail the conditions on the interface thickness to control the accuracy and convergence of the phase-fie…
