Nonlinear simulations of elastic fingering in a Hele-Shaw cell

Zhao, Meng; Belmonte, Andrew; Li, Shuwang; Li, Xiaofan; Lowengrub, John

doi:10.1016/j.cam.2015.11.016

Cited by 26 publications

(38 citation statements)

References 32 publications

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“…This might be due to the formation of more interfacial material at higher C, resulting in an increased thickness which would impede finger tip-splitting. An increased elasticity of the material with C could also lead to the suppression of tip-splitting, as shown numerically for fingers in a model where the interface is represented as a membrane with both elasticity and surface tension [36].…”

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confidence: 87%

Flow instabilities due to the interfacial formation of surfactant–fatty acid material in a Hele-Shaw cell

2017

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We present an experimental study of pattern formation during the penetration of an aqueous surfactant solution into a liquid fatty acid in a Hele-Shaw cell. When a solution of the cationic surfactant cetylpyridinium chloride is injected into oleic acid, a wide variety of fingering patterns are observed as a function of surfactant concentration and flow rate, which are strikingly different than the classic Saffman-Taylor (ST) instability. We observe evidence of interfacial material forming between the two liquids, causing these instabilities. Moreover, the number of fingers decreases with increasing flow rate Q, while the average finger width increases with Q, both trends opposite to the ST case. Bulk rheology on related mixtures indicates a gel-like state. Comparison of experiments using other oils indicates the importance of pH and the carboxylic head group in the formation of the surfactant-fatty acid material.

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confidence: 87%

Flow instabilities due to the interfacial formation of surfactant–fatty acid material in a Hele-Shaw cell

2017

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“…Mat.) Although this spreading process appears superficially similar to viscous fingering in a Hele-Shaw geometry [22,23], there are numerous distinguishing characteristics. First, the measured fractal dimension (D = 1.30 ± 0.05) is smaller than the value observed in viscous fingering (D = 1.7).…”

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confidence: 93%

Oxidation-Mediated Fingering in Liquid Metals

et al. 2017

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We identify and characterize a new class of fingering instabilities in liquid metals; these instabilities are unexpected due to the large interfacial tension of metals. Electrochemical oxidation lowers the effective interfacial tension of a gallium-based liquid metal alloy to values approaching zero, thereby inducing drastic shape changes, including the formation of fractals. The measured fractal dimension (D=1.3±0.05) places the instability in a different universality class than other fingering instabilities. By characterizing changes in morphology and dynamics as a function of droplet volume and applied electric potential, we identify the three main forces involved in this process: interfacial tension, gravity, and oxidative stress. Importantly, we find that electrochemical oxidation can generate compressive interfacial forces that oppose the tensile forces at a liquid interface. The surface oxide layer ultimately provides a physical and electrochemical barrier that halts the instabilities at larger positive potentials. Controlling the competition between interfacial tension and oxidative (compressive) stresses at the interface is important for the development of reconfigurable electronic, electromagnetic, and optical devices that take advantage of the metallic properties of liquid metals.

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“…To overcome these difficulties, we develop a spectrally accurate boundary integral method in which a new time and space rescaling is implemented. The rescaling idea [29,49,50] is to map the original time and space (x, t) into new coordinates (\= x, \= t) such that the interface can evolve at an arbitrary speed in the new rescaled frame. In particular, for the shrinking interface problem, we choose (1) the space scaling function R( \= t) so that the shrinking interface is always mapped back to its initial size, i.e., the interface does not shrink in the rescaled frame; (2) the time scaling function \rho ( \= t) to slow down the motion of the interface, especially at later times when the interface becomes very small and shrinks extremely rapidly.…”

Section: B1207mentioning

confidence: 99%

“…The time scaling function \rho (t) = \= \rho ( \= t) maps the original time t to the new time \= t and \rho (t) has to be positive and continuous. The evolution of the interface in the scaled frame can be accelerated [50,49] or decelerated by choosing a different \rho (t). A straightforward calculation shows the normal velocity in the new frame…”

Section: Rescaling Idea Introduce a New Framementioning

confidence: 99%

Computation of a Shrinking Interface in a Hele-Shaw Cell

Zhao

Ying

et al. 2018

SIAM J. Sci. Comput.

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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, where τ is the surface tension and 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 C.

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Nonlinear simulations of elastic fingering in a Hele-Shaw cell

Cited by 26 publications

References 32 publications

Flow instabilities due to the interfacial formation of surfactant–fatty acid material in a Hele-Shaw cell

Flow instabilities due to the interfacial formation of surfactant–fatty acid material in a Hele-Shaw cell

Oxidation-Mediated Fingering in Liquid Metals

Computation of a Shrinking Interface in a Hele-Shaw Cell

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