Hele–Shaw flow with a point sink: generic solution breakdown

2004 ◽  
Vol 15 (1) ◽  
pp. 1-37 ◽  
Author(s):  
L. J. CUMMINGS ◽  
J. R. KING
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Complex Plane ◽  
Arbitrary Number ◽  
Blow Up ◽  
Arbitrary Angle ◽  
Small Surface ◽  
Singularity Structure ◽  
Numerical Evidence ◽  
Special Complex

Recent numerical evidence [8, 28, 33] suggests that in the Hele–Shaw suction problem with vanishingly small surface tension $\gamma$, the free boundary generically approaches the sink in a wedge-like configuration, blow-up occurring when the wedge apex reaches the sink. Sometimes two or more such wedges approach the sink simultaneously [33]. We construct a family of solutions to the zero-surface tension (ZST) problem in which fluid is injected at the (coincident) apices of an arbitrary number $N$ of identical infinite wedges, of arbitrary angle. The time reversed suction problem then models what is observed numerically with non-zero surface tension. We conjecture that (for a given value of $N$) a particular member of this family of ZST solutions, with special complex plane singularity structure, is selected in the limit $\gamma\,{\to}\,0$.

scholarly journals On the deflection of a liquid jet by an air-cushioning layer

2018 ◽  
Vol 846 ◽  
pp. 711-751 ◽  
Author(s):  
M. R. Moore ◽  
J. P. Whiteley ◽  
J. M. Oliver
Keyword(s):  
Surface Tension ◽  
Equations Of Motion ◽  
Constant Thickness ◽  
Liquid Jet ◽  
Pressure Jump ◽  
Length Scales ◽  
Small Surface ◽  
Numerical Studies ◽  
Impact Problems ◽  
Systematic Derivation

A hierarchy of models is formulated for the deflection of a thin two-dimensional liquid jet as it passes over a thin air-cushioning layer above a rigid flat impermeable substrate. We perform a systematic derivation of the leading-order equations of motion for the jet in the distinguished limit in which the air pressure jump, surface tension and gravity affect the displacement of the centreline of the jet, but not its thickness or velocity. We identify thereby the axial length scales for centreline deflection in regimes in which the air layer is dominated by viscous or inertial effects. The derived length scales and reduced equations aim to expand the suite of tools available for future analyses of the evolution of lamellae and ejecta in impact problems. Assuming that the jet is sufficiently long that tip and entry effects can be neglected, we demonstrate that the centreline of a constant-thickness jet moving with constant axial speed is destabilised by the air layer for sufficiently small surface tension. Expressions for the fastest-growing modes are obtained in both the viscous-dominated air and inertia-dominated air regimes. For a finite-length jet emanating from a nozzle, we show that, in one particular asymptotic limit, the evolution of the jet centreline is akin to the flapping of an unfurling flag above a thin air layer. We discuss the distinguished limit in which tip retraction can be neglected and perform numerical investigations into the resulting model. We show that the cushioning layer causes the jet centreline to bend, leading to rupture of the air layer. We discuss how our toolbox of models can be adapted and utilised in the context of recent experimental and numerical studies of splash dynamics.

scholarly journals Solution selection of axisymmetric Taylor bubbles

2018 ◽  
Vol 843 ◽  
pp. 518-535 ◽  
Author(s):  
A. Doak ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Constant Velocity ◽  
Numerical Scheme ◽  
Solution Space ◽  
Selection Mechanism ◽  
Small Surface ◽  
Taylor Bubble ◽  
Solution Selection ◽  
Selection Of ◽  
Taylor Bubbles

A finite difference scheme is proposed to solve the problem of axisymmetric Taylor bubbles rising at a constant velocity in a tube. A method to remove singularities from the numerical scheme is presented, allowing accurate computation of the bubbles with the inclusion of both gravity and surface tension. This paper confirms the long-held belief that the solution space of the axisymmetric Taylor bubble for small surface tension is qualitatively similar to that of the plane Taylor bubble. Furthermore, evidence suggesting that the solution selection mechanism associated with plane bubbles also occurs in the axisymmetric case is presented.

scholarly journals Droplet spreading and permeating on the hybrid-wettability porous substrates: a lattice Boltzmann method study

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 483-491 ◽  
Author(s):  
Wen-Kai Ge ◽  
Gui Lu ◽  
Xin Xu ◽  
Xiao-Dong Wang
Keyword(s):  
Surface Tension ◽  
Contact Angle ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Droplet Spreading ◽  
Small Surface ◽  
Pore Sizes ◽  
Fluid Properties ◽  
Boltzmann Method ◽  
Porous Substrates

AbstractThe spreading and permeation of droplets on porous substrates is a fundamental process in a variety of applications, such as coating, dyeing, and printing. The spreading and permeating usually occur synchronously but play different roles in the practical applications. The mechanisms of the competition between spreading and permeation is significant but still unclear. A lattice Boltzmann method is used to study the spreading and permeation of droplets on hybrid-wettability porous substrates, with different wettability on the surface and the inside pores. The competition between the spreading and the permeation processes is studied in this work from the effects of the substrate and the fluid properties, including the substrate wettability, the porous parameters, as well as the fluid surface tension and viscosity. The results show that increasing the surfacewettability and the porosity contact angle both inhibit the spreading and the permeation processes. When the inside porosity contact angle is larger than 90° (hydrophobic), the permeation process does not occur. The droplets suspend on substrates with Cassie state. The droplets are more easily to permeate into substrates with a small inside porosity contact angle (hydrophilic), as well as large pore sizes. Otherwise, the droplets are more easily to spread on substrate surfaces with small surface contact angle (hydrophilic) and smaller pore sizes. The competition between droplet spreading and permeation is also related to the fluid properties. The permeation process is enhanced by increasing of surface tension, leading to a smaller droplet lifetime. The goals of this study are to provide methods to manipulate the spreading and permeation separately, which are of practical interest in many industrial applications.

Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions

2009 ◽  
Vol 465 (2109) ◽  
pp. 2725-2749 ◽  
Author(s):  
M. J. Ablowitz ◽  
T. S. Haut
Keyword(s):  
Surface Tension ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Asymptotic Series ◽  
High Order ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Small Surface ◽  
Petviashvili Equation ◽  
Dimensional Series

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs–Thomson relation

2015 ◽  
Vol 26 (4) ◽  
pp. 401-425 ◽  
Author(s):  
FUJUN ZHOU ◽  
JUNDE WU
Keyword(s):  
Surface Tension ◽  
Free Boundary ◽  
Boundary Problem ◽  
Stationary Solution ◽  
Free Boundary Problem ◽  
Bifurcation Analysis ◽  
Surface Tension Coefficient ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis ◽  
A Free Boundary Problem

Of concern is the stability and bifurcation analysis of a free boundary problem modelling the growth of multi-layer tumours. A remarkable feature of this problem lies in that the free boundary is imposed with nonlinear boundary conditions, where a Gibbs–Thomson relation is taken into account. By employing a functional approach, analytic semigroup theory and bifurcation theory, we prove that there exists a positive threshold value γ* of surface tension coefficient γ such that if γ > γ* then the unique flat stationary solution is asymptotically stable under non-flat perturbations, while for γ < γ* this unique flat stationary solution is unstable and there exists a series of non-flat stationary solutions bifurcating from it. The result indicates a significant phenomenon that a smaller value of surface tension coefficient γ may make tumours more aggressive.

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