When two fluid drops touch, they coalesce due to surface tension. At early times, there is only a relatively small fluid bridge joining the drops. An asymptotic solution is presented for an inertial ...regime of early-time coalescence, in which inertial forces balance surface tension at leading order. It is demonstrated that viscosity nevertheless has a leading-order effect. Radial momentum is created at the tightly curved edge of the fluid bridge by the net force $2\gamma$ (per unit length) due to surface tension. This momentum is left behind the radially expanding bridge edge in a thin viscous wake. The divergent volume flux in the wake entrains fluid from above and below the bridge, and drives an inviscid irrotational flow in the drops on the scale of the bridge radius. This flow widens the gap between the drops ahead of the bridge, and the larger gap width results in a lower rate of coalescence. Including viscosity in this way improves the agreement between theory and the available experimental and numerical data.
When two small fluid drops are sufficiently close, the van der Waals force overcomes surface tension and deforms the surfaces into contact, initiating coalescence. The dynamics of surface deformation ...across an inviscid gap falls into two distinct regimes (Stokes and inertial–viscous) characterized by the forces that balance the van der Waals attraction at leading order (viscosity, and both inertia and viscosity). The previously studied Stokes regime holds for very viscous drops but fails for less viscous drops as inertia becomes significant before contact is reached. We show that the subsequent inertial–viscous dynamics is self-similar as contact is approached, with the gap width decreasing as $t{'^{3/8}}$ and the radial scale of the deformed region decreasing as $t{'^{1/2}}$ as $t{'}\to 0$, for time until contact $t'$. The self-similar behaviour is universal and is the generic asymptotic behaviour observed in time-dependent simulations. The unique self-similar gap profile of the inertial–viscous regime suggests new initial conditions for the coalescence of the drops after contact.
Plethora of transitions during breakup of liquid filaments Castrejón-Pita, José Rafael; Castrejón-Pita, Alfonso Arturo; Thete, Sumeet Suresh ...
Proceedings of the National Academy of Sciences - PNAS,
04/2015, Letnik:
112, Številka:
15
Journal Article
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Significance Fluid flows, governed by nonlinear equations, permit formation of singularities. Often, singularities are artifacts of neglecting physical effects. However, free-surface flows exhibit ...observable singularities including filament pinch-off. As filaments thin, slightly (highly) viscous filaments are expected from theory to transition from an inertial (viscous) regime where viscosity (density) is negligible to an inertial–viscous regime where viscous and inertial effects are important. Previous works show this transition either does not occur or occurs for filament radii well below theoretical predictions. We demonstrate that thinning filaments unexpectedly pass through a number of intermediate transient regimes, thereby delaying onset of the final regime. The findings raise the question if similar dynamical transitions arise in problems that are not necessarily hydrodynamic in nature.
Thinning and breakup of liquid filaments are central to dripping of leaky faucets, inkjet drop formation, and raindrop fragmentation. As the filament radius decreases, curvature and capillary pressure, both inversely proportional to radius, increase and fluid is expelled with increasing velocity from the neck. As the neck radius vanishes, the governing equations become singular and the filament breaks. In slightly viscous liquids, thinning initially occurs in an inertial regime where inertial and capillary forces balance. By contrast, in highly viscous liquids, initial thinning occurs in a viscous regime where viscous and capillary forces balance. As the filament thins, viscous forces in the former case and inertial forces in the latter become important, and theory shows that the filament approaches breakup in the final inertial–viscous regime where all three forces balance. However, previous simulations and experiments reveal that transition from an initial to the final regime either occurs at a value of filament radius well below that predicted by theory or is not observed. Here, we perform new simulations and experiments, and show that a thinning filament unexpectedly passes through a number of intermediate transient regimes, thereby delaying onset of the inertial–viscous regime. The new findings have practical implications regarding formation of undesirable satellite droplets and also raise the question as to whether similar dynamical transitions arise in other free-surface flows such as coalescence that also exhibit singularities.
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-Puzović et al., Phys. Rev. Lett., vol. 108, 2012, 074502). 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 Föppl–von-Kármán 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 simplified 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 fingering 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.
We examine the motion in a shear flow at zero Reynolds number of particles with two planes of symmetry. We show that in most cases the rotational motion is qualitatively similar to that of a ...non-axisymmetric ellipsoid, and characterised by a combination of chaotic and quasiperiodic orbits. We use Kolmogorov–Arnold–Moser (KAM) theory and related ideas in dynamical systems to elucidate the underlying mathematical structure of the motion and thence to explain why such a large class of particles all rotate in essentially the same manner. Numerical simulations are presented for curved spheroids of varying centreline curvature, which are found to drift persistently across the streamlines of the flow for certain initial orientations. We explain the origin of this migration as the result of a lack of symmetries of the particle’s orientation orbit.
We study the spreading of viscous fluid injected under an elastic sheet, which is driven by gravity and by elastic bending and tension forces and resisted by viscous forces. The injected fluid forms ...a large blister and spreads outwards analogously to a viscous gravity current or a capillary droplet. The relative strengths of the three driving forces are determined by how the horizontal length scales of the system compare with three key transition length scales. Bending is dominant on small length scales, tension is dominant on intermediate length scales and gravity is dominant on large length scales. We show how to use the method of matched asymptotic expansions to predict the spreading rate and thickness profile of the blister of fluid in the seven possible asymptotic regimes, for both two-dimensional and axisymmetric geometries. Consideration of different physical effects at the fluid front increases the number of regimes yet further.
High-resolution numerical simulations of statistically steady convection in a three-dimensional porous medium are presented for Rayleigh numbers
$Ra \leqslant 2 \times 10^4$
. Measurements of the ...Nusselt number
$Nu$
in the range
$1750 \leqslant Ra \leqslant 2 \times 10^4$
are well fitted by a relationship of the form
$Nu = \alpha _3 Ra + \beta _3$
, for
$\alpha _3 = 9.6 \times 10^{-3}$
and
$\beta _3 = 4.6$
. This fit indicates that the classical linear scaling
$Nu \sim Ra$
is attained, and that
$Nu$
is asymptotically approximately
$40\, \%$
larger than in two dimensions. The dynamical flow structure in the range
$1750 \leqslant Ra \leqslant 2\times 10^4$
is analysed, and the interior of the flow is found to be increasingly well described as
$Ra \to \infty $
by a heat-exchanger model, which describes steady interleaving columnar flow with horizontal wavenumber
$k$
and a linear background temperature field. Measurements of the interior wavenumber are approximately fitted by
$k\sim Ra^{0.52 \pm 0.05}$
, which is distinguishably stronger than the two-dimensional scaling of
$k\sim Ra^{0.4}$
.
Propagation of a viscous fluid beneath an elastic sheet is controlled by local dynamics at the peeling front, in close analogy with the capillary-driven spreading of drops over a precursor film. Here ...we identify propagation laws for a generic elastic peeling problem in the distinct limits of peeling by bending and peeling by pulling, and apply our results to the radial spread of a fluid blister over a thin prewetting film. For the case of small deformations relative to the sheet thickness, peeling is driven by bending, leading to radial growth as t(7/22). Experimental results reproduce both the spreading behavior and the bending wave at the front. For large deformations relative to the sheet thickness, stretching of the blister cap and the consequent tension can drive peeling either by bending or by pulling at the front, both leading to radial growth as t(3/8). In this regime, detailed predictions give excellent agreement and explanation of previous experimental measurements of spread in the pulling regime in an elastic Hele-Shaw cell.
The stability of steady convective exchange flow with a rectangular planform in an unbounded three-dimensional porous medium is explored. The base flow comprises a balance between vertical advection ...with amplitude
$A$
in interleaving rectangular columns with aspect ratio
$\unicodeSTIX{x1D709}\leqslant 1$
and horizontal diffusion between the columns. Columnar flow with a square planform (
$\unicodeSTIX{x1D709}=1$
) is found to be weakly unstable to a large-scale perturbation of the background temperature gradient, irrespective of
$A$
, but to have no stronger instability on the scale of the columns. This result provides a stark contrast to two-dimensional columnar flow (Hewitt et al., J. Fluid Mech., vol. 737, 2013, pp. 205–231), which, as
$A$
is increased, is increasingly unstable to a perturbation on the scale of the columnar wavelength. For rectangular planforms with
$\unicodeSTIX{x1D709}<1$
, a critical aspect ratio is identified, below which a perturbation on the scale of the columns is the fastest growing mode, as in two dimensions. Scalings for the growth rate and the structure of this mode are identified, and are explained by means of an asymptotic expansion in the limit
$\unicodeSTIX{x1D709}\rightarrow 0$
. The difference between the stabilities of two-dimensional and three-dimensional exchange flow provides a potential explanation for the apparent difference in dominant horizontal scale observed in direct numerical simulations of two-dimensional and three-dimensional statistically steady ‘Rayleigh–Darcy’ convection at high Rayleigh numbers.
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