On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

Khalid, AH; McDonald, N. Robb; Vanden‐Broeck, J.‐M.

doi:10.1063/1.4905582

Cited by 14 publications

(15 citation statements)

References 42 publications

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“…We have applied techniques in exponential asymptotics to describe these solution branches in the limit B → 0. In particular, our analysis leads to the solvability condition (32), which is a relationship that provides an estimate for U in terms of B and m. By comparing with numerical solutions, we have demonstrated that this estimate works very well for small to moderate values of surface tension. The techniques in exponential asymptotics that we employ were developed by Chapman, King & Adams [11] and subsequently used in a variety of problems in applied mathematics.…”

Section: Discussionmentioning

confidence: 82%

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

Lustri¹,

Green²,

McCue³

2020

SIAM J. Appl. Math.

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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 numerical evidence to suggest the selection problem for a bubble propagating in an unbounded Hele-Shaw cell behaves in an analogous way to other finger and bubble problems in a Hele-Shaw channel; however, the selection of the ratio of bubble speeds to background velocity appears to follow a very different surface tension scaling to the channel cases. Here we apply techniques in exponential asymptotics to solve the selection problem analytically, confirming the numerical results, including the predicted surface tension scaling laws. Further, our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. These results have ramifications for exotic-shaped Saffman-Taylor fingers as well as for the time-dependent evolution of bubbles propagating in Hele-Shaw cells.

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Section: Discussionmentioning

confidence: 82%

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

Lustri¹,

Green²,

McCue³

2020

SIAM J. Appl. Math.

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“…It was generally accepted that surface tension was necessary for selecting the pattern [6,26,50,52]. However, recent studies [30,39,60] have demonstrated, by using time-dependent exact solutions without surface tension, that the steady solution with U = 2 is the only attractor for the non-singular unsteady solutions, thus showing that velocity selection in a Hele-Shaw cell is inherently determined by the zero surface tension dynamics. It has been conjectured [60] that a similar scenario holds in domains of arbitrary connectivity.…”

Section: Discussionmentioning

confidence: 99%

Multiple bubbles and fingers in a Hele-Shaw channel: complete set of steady solutions

Vasconcelos¹

2015

J. Fluid Mech.

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Analytical solutions for both a finite assembly and a periodic array of bubbles steadily moving in a Hele-Shaw channel are presented. The particular case of multiple fingers penetrating into the channel and moving jointly with an assembly of bubbles is also analysed. The solutions are given by a conformal mapping from a multiply connected circular domain in an auxiliary complex plane to the fluid region exterior to the bubbles. In all cases the desired mapping is written explicitly in terms of certain special transcendental functions, known as the secondary Schottky-Klein prime functions. Taken together, the solutions reported here represent the complete set of solutions for steady bubbles and fingers in a horizontal Hele-Shaw channel when surface tension is neglected. All previous solutions under these assumptions are particular cases of the general solutions reported here. Other possible applications of the formalism described here are also discussed. * Electronic address: giovani.vasconcelos@ufpe.br

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“…The introduction of surface tension selects both a main, symmetric branch of linearly stable steady solutions that persists for all fluxes, along with a countably infinite sequence of unstable 'exotic' bubble shapes; these exist in channels bounded by parallel side walls [19] as well as in unbounded channels [21,22]. Numerical simulations of the time-dependent problem have also been reported [23,24]. Figure 2a shows the bubble shape and speed for the first three solution branches in channels with a rectangular cross section.…”

Section: Introductionmentioning

confidence: 99%

The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel

et al. 2019

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We hypothesize that dynamical systems concepts used to study the transition to turbulence in shear flows are applicable to other transition phenomena in fluid mechanics. In this paper, we consider a finite air bubble that propagates within a Hele-Shaw channel containing a depth-perturbation. Recent experiments revealed that the bubble shape becomes more complex, quantified by an increasing number of transient bubble tips, with increasing flow rate. Eventually, the bubble changes topology, breaking into multiple distinct entities with non-trivial dynamics. We demonstrate that qualitatively similar behaviour to the experiments is exhibited by a previously established, depth-averaged mathematical model and arises from the model’s intricate solution structure. For the bubble volumes studied, a stable asymmetric bubble exists for all flow rates of interest, while a second stable solution branch develops above a critical flow rate and transitions between symmetric and asymmetric shapes. The region of bistability is bounded by two Hopf bifurcations on the second branch. By developing a method for a numerical weakly nonlinear stability analysis we show that unstable periodic orbits (UPOs) emanate from the first Hopf bifurcation. Moreover, as has been found in shear flows, the UPOs are edge states that influence the transient behaviour of the system.

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On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

Cited by 14 publications

References 42 publications

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

Multiple bubbles and fingers in a Hele-Shaw channel: complete set of steady solutions

The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel

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