34 research outputs found
Viscous fingering and dendritic growth under an elastic membrane
We investigate the viscous fingering instability that arises when air is
injected from the end of an oil-filled, compliant channel. We show that induced
axial and transverse depth gradients foster novel pattern formation. Moreover,
the steady propagation of the interface allows us to elucidate the nonlinear
saturation of a fingering pattern first observed in a time-evolving system
(Pihler-Puzovic et al. PRL 108, 074502, 2012): the wavelength is set by the
viscous fingering mechanism, but the amplitude is inversely proportional to the
tangent of the compliant wall's inclination angle
The propagation of air fingers into an elastic branching network
We study experimentally the propagation of an air finger through the
Y-bifurcation of an elastic, liquid-filled Hele-Shaw channel, as a benchtop
model of airway reopening. With channel compliance provided by an elastic upper
boundary, we can impose collapsed channel configurations into which we inject
air with constant volume-flux. We typically observe steady finger propagation
in the main channel, which is lost ahead of the Y-bifurcation but subsequently
recovered in the daughter channels. At low levels of initial collapse, steady
finger shapes and bubble pressure in the daughter channels map onto those in
the main channel, despite small differences in initial collapse in different
parts of the Y-channel. However, at higher levels of initial collapse where the
elastic sheet almost touches the bottom boundary of the channel, experimentally
indistinguishable fingers in the main channel can lead to multiple states of
reopening of the daughter channels. The downstream distance at which steady
propagation is recovered in the daughter channels also varies considerably with
injection flow rate and initial collapse because of a transition in the
mechanics regulating finger propagation. We find that the characteristic time
and length-scales of this recovery are largest in the regime where viscous and
surface tension forces dominate at low flow rate and/or low initial collapse,
and that they decrease towards a constant plateau reached in the limit where
elastic and surface tension forces balance at high flow rate and/or high
initial collapse. Our findings suggest that practical networks are unlikely to
comprise long enough channels for steady state propagation to remain
established.Comment: 36 pages, 13 finger
Viscous fingering in a radial elastic-walled Hele-Shaw cell
We study the viscous fingering instability in a radial Hele-Shaw cell in which the top boundary has been replaced by a thin elastic sheet. The introduction of wall elasticity delays the onset of the fingering instability to much larger values of the injection flow rate. Furthermore, when the instability develops, the fingers that form on the expanding air-liquid interface are short and stubby, in contrast with the highly-branched patterns observed in rigid-walled cells (Pihler-Puzovi c et al. 2012).
We report the outcome of a comprehensive experimental study of this problem and compare the experimental observations to the predictions from a theoretical model that
is based on the solution of the Reynolds lubrication equations, coupled to the F oppl-von-K arm an equations which describe the deformation of the elastic sheet. We perform
a linear stability analysis to study the evolution of small-amplitude non-axisymmetric perturbations to the time-evolving base
flow. We then derive a simpli ed model by exploiting the observations (i) that the non-axisymmetric perturbations to the sheet are very small and (ii) that perturbations to the flow occur predominantly in a small wedge-shaped
region ahead of the air-liquid interface. This allows us to identify the various
physical mechanisms by which viscous fi ngering is weakened (or even suppressed) by the
presence of wall elasticity.
We show that the theoretical predictions for the growth rate of small amplitude perturbations are in good agreement with experimental observations for injection flow rates that are slightly larger than the critical
flow rate required for the onset of the
instability. We also characterize the large-amplitude fingering patterns that develop at larger injection
flow rates. We show that the wavenumber of these patterns is still well
predicted by the linear stability analysis, and that the length of the fingers is set by the local geometry of the compliant cell.EPSR
Self-similar and disordered front propagation in a radial Hele-Shaw channel with time-varying cell depth
The displacement of a viscous fluid by an air bubble in the narrow gap
between two parallel plates can readily drive complex interfacial pattern
formation known as viscous fingering. We focus on a modified system suggested
recently by [1], in which the onset of the fingering instability is delayed by
introducing a time-dependent (power-law) plate separation. We perform a
complete linear stability analysis of a depth-averaged theoretical model to
show that the plate separation delays the onset of non-axisymmetric
instabilities, in qualitative agreement with the predictions obtained from a
simplified analysis by [1]. We then employ direct numerical simulations to show
that in the parameter regime where the axisymmetrically expanding air bubble is
unstable to nonaxisymmetric perturbations, the interface can evolve in a
self-similar fashion such that the interface shape at a given time is simply a
rescaled version of the shape at an earlier time. These novel, self-similar
solutions are linearly stable but they only develop if the initially circular
interface is subjected to unimodal perturbations. Conversely, the application
of non-unimodal perturbations (e.g. via the superposition of multiple linearly
unstable modes) leads to the development of complex, constantly evolving finger
patterns similar to those that are typically observed in constant-width
Hele-Shaw cells.
[1] Z. Zheng, H. Kim, and H. A. Stone, Controlling viscous fingering using
time-dependent strategies, Phys. Rev. Lett. 115, 174501 (2015).Comment: 12 pages, 8 figure
Displacement flows under elastic membranes. Part 1: Experiments and direct numerical simulations
Peeling fingers in an elastic Hele-Shaw channel
Using experiments and a depth-averaged numerical model, we study
instabilities of two-phase flows in a Hele-Shaw channel with an elastic upper
boundary and a non-uniform cross-section prescribed by initial collapse.
Experimentally, we find increasingly complex and unsteady modes of air-finger
propagation as the dimensionless bubble speed, Ca, and level of collapse are
increased, including pointed fingers, indented fingers and the feathered modes
first identified by Cuttle et al.(J. Fluid Mech., vol. 886, 2020, A20).
By introducing a measure of the viscous contribution to finger propagation,
we identify a Ca threshold beyond which viscous forces are superseded by
elastic effects. Quantitative prediction of this transition between 'viscous'
and 'elastic' reopening regimes across levels of collapse establishes the
fidelity of the numerical model. In the viscous regime, we recover the
non-monotonic dependence on Ca of the finger pressure, which is characteristic
of benchtop models of airway reopening. To explore the elastic regime
numerically, we extend the depth-averaged model introduced by Fontana et al.
(J. Fluid Mech., vol. 916, 2021, A27) to include an artificial disjoining
pressure which prevents the unphysical self-intersection of the interface.
Using time simulations, we capture for the first time the majority of
experimental finger dynamics, including feathered modes. We show that these
disordered states continually evolve, with no evidence of convergence to steady
or periodic states. We find that the steady bifurcation structure
satisfactorily predicts the bubble pressure as a function of Ca, but that it
does not provide sufficient information to predict the transition to unsteady
dynamics which appears strongly nonlinear.Comment: 28 pages, 15 figure
A review of current techniques for the evaluation of powder mixing
Blending a mixture of powders to a homogeneous system is a crucial step in many manufacturing processes. To achieve a high quality of the end product, powder mixtures should be made with high content uniformity. For instance, producing uniform tablets depends on the homogeneous dispersion of active pharmaceutical ingredient (API), often in low level quantities, into excipients. To control the uniformity of a powder mixture, the first required step is to estimate the powder content information during blending. There are several powder homogeneity evaluation techniques which differ in accuracy, fundamental basis, cost and operating conditions. In this article, emerging techniques for the analysis of powder content and powder blend uniformity, are explained and compared. The advantages and drawbacks of all the techniques are reviewed to help the readers to select the appropriate equipment for the powder mixing evaluation. In addition, the paper highlights the recent innovative on-line measurement techniques used for the non-invasive evaluation of the mixing performance