Multiple states of finger propagation in partially occluded tubes

Hazel, Andrew L.; Pailha, Mickaël; Cox, Simon; Juel, Anne

doi:10.1063/1.4811176

Cited by 11 publications

(27 citation statements)

References 16 publications

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“…B. Thompson, A. L. Hazel and A. Juel of the channel height (de Lózar et al 2009;Pailha et al 2012;Hazel et al 2013). Thompson et al (2014) found that all finger propagation modes in partially occluded Hele-Shaw channels were qualitatively reproduced in an adapted version of the depthaveraged model used by McLean & Saffman (1981) to show that finite surface tension selects the half-width finger in unoccluded Hele-Shaw channels.…”

Section: Discussionmentioning

confidence: 99%

“…The bifurcation moves towards lower capillary numbers and eventually becomes subcritical as the occlusion height increases further. For yet higher occlusions, the asymmetric solution smoothly evolves into a localised asymmetric solution upon decrease of Ca, which can persist even at Ca = 0, the capillary static limit (Hazel et al 2013).…”

Section: Experimental Methodsmentioning

confidence: 97%

“…Localised fingers, which arise for higher occlusions or very low Ca, were not investigated in detail. Investigations of the dynamics of localised states can be found in Hazel et al (2013) and Thompson et al (2014). In each figure, the different modes of propagation encountered as α h is increased are illustrated with inset experimental images.…”

Section: Comparison Between Experimental and Numerical Modes Of Fingementioning

confidence: 99%

“…More recently, motivated by the physiological problem of airway reopening, de Lózar et al (2009), Pailha et al (2012) and Hazel et al (2013) investigated two-phase flow through rectangular tubes with aspect ratio α 13 that were partially occluded by axially uniform rectangular blocks occupying 50 % and 33.3 % of the tube height. Multiple stable modes of propagation were found in these geometries including asymmetric and oscillatory modes as well as the expected symmetric mode.…”

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confidence: 99%

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Sensitivity of Saffman–Taylor fingers to channel-depth perturbations

et al. 2016

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We examine the sensitivity of Saffman-Taylor fingers to controlled variations in channel depth by investigating the effects of centred, rectangular occlusions in Hele-Shaw channels. For large occlusions, the geometry is known to support symmetric, asymmetric and oscillatory propagation states when air displaces a more viscous fluid from within the channel. A previously developed depth-averaged model is found to be in quantitative agreement with laboratory experiments once the aspect ratio (width/height) of the tube's cross-section reaches a value of 40. We find that the multiplicity of solutions at finite occlusion heights arises through interactions of the single stable and multiple unstable solutions already present in the absence of the occlusion: the classic Saffman-Taylor viscous fingering problem. The sequence of interactions that occurs with increasing occlusion height is the same for all aspect ratios investigated, but the occlusion height required for each interaction decreases with increasing aspect ratio. Thus, the system becomes more sensitive as the aspect ratio increases in the sense that multiple solutions are provoked for smaller relative depth changes. We estimate that the required depth changes become of the same order as the typical roughnesses of the experimental system (1 µm) for aspect ratios beyond 155, which we conjecture underlies the extreme sensitivity of experiments conducted in such Hele-Shaw channels.

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Section: Discussionmentioning

confidence: 99%

Section: Experimental Methodsmentioning

confidence: 97%

Section: Comparison Between Experimental and Numerical Modes Of Fingementioning

confidence: 99%

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Sensitivity of Saffman–Taylor fingers to channel-depth perturbations

et al. 2016

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“…We solve for the perturbation function f (ζ ) numerically by writing it as a truncated Taylor series 42) and computing the N real coefficients {a j } N−1 j=0 , and the two real parameters a and U. We chose N = 200 for all our calculations after checking that this value of N gives the same solution for N = 400 (up to at least eight decimal places).…”

Section: (D) Numerical Schemementioning

confidence: 99%

The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

2017

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New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.

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