Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells

Abstract

Recent experiments by Pihler-Puzovic et al. (Phys. Rev. Lett., vol. 108, 2012, article 074502) have shown that the onset of viscous fingering in circular Hele-Shaw cells in which an air bubble displaces a viscous fluid is delayed considerably when the top boundary of the cell is replaced by an elastic membrane. Non-axisymmetric instabilities are only observed at much larger flow rates, and the large-amplitude fingers that develop are fundamentally different from the highly branched fingers in rigid-walled cells. We explain the mechanism for the suppression of the instability using a combination of linear stability analysis and direct numerical simulations, based on a theoretical model that couples a depth-averaged lubrication equation for the fluid flow to the Föppl-von Kármán equations, which describe the deformation of the elastic membrane. We show that fluid-structure interaction affects the instability primarily via two changes to the axisymmetric base flow: the axisymmetric inflation of the membrane prior to the onset of any instabilities slows down the expansion of the air bubble and forces the air-liquid interface to propagate into a converging fluid-filled gap. Both of these changes reduce the destabilizing viscous effects that drive the fingering instability in a rigid-walled cell. In contrast, capillary effects only play a very minor role in the suppression of the instability.

Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations

“…ForV > 300 ml.min −1 the system was found to be unstable to non-axisymmetric fingering instabilities studied earlier by Pihler-Puzović et al (2012; forV < 50 ml.min −1 , the air-liquid interface remained circular but drifted slowly across the cell in a rigid-body mode (with unit azimuthal wavenumber). This latter instability is due to the finite extent of the HeleShaw cell, and is always present for rigid-walled cells, even when the flow rate is so small that all other non-axisymmetric modes are suppressed (Pihler-Puzović et al 2013). At larger flow rates, the geometry of the elastic-walled cell is drastically different, and this mode is no longer observed to be unstable.…”

“…The Föppl-von Kármán equations were discretised by a mixed finite-element method in which we approximated the vertical displacements, ζ, the Airy stress functions φ, and their respective Laplacians by piecewise quadratic basis functions, defined on threenoded, one-dimensional elements. Again, we omit details but refer to Pihler-Puzović et al (2013) for a more detailed discussion of the implementation. The fully-coupled, implicit discretisation of the fluid and solid equations produces a large system of nonlinear algebraic equations at each timestep.…”

“…The lubrication-theory model was discretised by a second-order accurate finitedifference scheme and implemented in MatLab; see Pihler-Puzović et al (2013) for details. The inflation of the membrane under constant pressure, discussed in §4.1 below, was also solved using the same finite-difference discretisation in MatLab.…”

“…A curve fit to the experimental data shows that the power-law exponent α varies slightly between the two sets of experiments (α = 0.368±0.005 and 0.387±0.004 for the latex and polypropylene sheets, respectively, where the error corresponds to the standard deviation over six and four least-square power-law fits to the experimental data in figures 10 (a) and 11, respectively; the difference between these two values is significant when compared with the experimental and fitting errors). Motivated by this observation we re-examined the computational data in figure 3 (c) of Pihler-Puzović et al (2013) that shows the evolution of the bubble radius as a function of time for various values of the elastic modulus. When re-plotted on log-log axes this data also displays a power-law behaviour, with the power-law exponent again depending on the properties of the elastic membrane.…”

Reconstruction by fluorescence imaging of the spatio-temporal evolution of the viscosity field in Hele-Shaw flows

“…As a result, interest in the control of the properties of this instability by mechanical, [1][2][3][4][5][6][7] thermal, [8][9][10][11][12][13] or chemical [14][15][16][17][18][19][20][21][22][23][24] ways for instance are of constant interest in the literature. Numerous experimental and theoretical works have already shown that the properties of the fingers (such as their mixing length or their area) can be strongly affected by in situ changes in the viscosity field distribution induced by gradients of temperature or composition in the solutions.…”