scholarly journals Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: An Exact Result

1998 ◽  
Vol 80 (10) ◽  
pp. 2113-2116 ◽  
Author(s):  
Mark Mineev-Weinstein
Keyword(s):  
Surface Tension ◽  
Exact Result ◽  
Finger Width ◽  
Selection Of

scholarly journals Rigorous results in existence and selection of Saffman–Taylor fingers by kinetic undercooling

2018 ◽  
Vol 30 (1) ◽  
pp. 63-116
Author(s):  
XUMING XIE
Keyword(s):  
Surface Tension ◽  
Rigorous Results ◽  
Engineering Mathematics ◽  
Stokes Lines ◽  
Kinetic Undercooling ◽  
Finger Width ◽  
Linear Differential ◽  
Finger Tip ◽  
Selection Of ◽  
Hele Shaw Cell

The selection of Saffman–Taylor fingers by surface tension has been extensively investigated. In this paper, we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele–Shaw cell by small but non-zero kinetic undercooling (ε2). We rigorously conclude that for relative finger width λ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling ε2 → 0 if the Stokes multiplier for a relatively simple non-linear differential equation is zero. This Stokes multiplier S depends on the parameter $\alpha \equiv \frac{2 \lambda -1}{(1-\lambda)}\epsilon^{-\frac{4}{3}}$ and earlier calculations have shown this to be zero for a discrete set of values of α. While this result is similar to that obtained previously for Saffman–Taylor fingers by surface tension, the analysis for the problem with kinetic undercooling exhibits a number of subtleties as pointed out by Chapman and King (2003, The selection of Saffman–Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics, 46, 1–32). The main subtlety is the behaviour of the Stokes lines at the finger tip, where the analysis is complicated by non-analyticity of coefficients in the governing equation.

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 The steady propagation of an air finger into a rectangular tube

2008 ◽  
Vol 614 ◽  
pp. 173-195 ◽  
Author(s):  
ALBERTO DE LÓZAR ◽  
ANNE JUEL ◽  
ANDREW L. HAZEL
Keyword(s):  
Surface Tension ◽  
Capillary Number ◽  
Rectangular Cross Section ◽  
Two Dimensional ◽  
Governing Parameter ◽  
Pressure Drops ◽  
Finger Width ◽  
Steady Propagation ◽  
Finger Tip ◽  
Surface Tension Forces

The steady propagation of an air finger into a fluid-filled tube of uniform rectangular cross-section is investigated. This paper is primarily focused on the influence of the aspect ratio, α, on the flow properties, but the effects of a transverse gravitational field are also considered. The three-dimensional interfacial problem is solved numerically using the object-oriented multi-physics finite-element library oomph-lib and the results agree with our previous experimental results (de Lózar et al. Phys. Rev. Lett. vol. 99, 2007, article 234501) to within the ±1% experimental error.At a fixed capillary number Ca (ratio of viscous to surface-tension forces) the pressure drops across the finger tip and relative finger widths decrease with increasing α. The dependence of the wet fraction m (the relative quantity of liquid that remains on the tube walls after the propagation of the finger) is more complicated: m decreases with increasing α for low Ca but it increases with α at high Ca. Our results also indicate that the system is approximately quasi-two-dimensional for α ≥ 8, when we obtain quantitative agreement with McLean & Saffman's two-dimensional model for the relative finger width as a function of the governing parameter 1/B = 12α2Ca. The action of gravity causes an increase in the pressure drops, finger widths and wet fractions at fixed capillary number. In particular, when the Bond number (ratio of gravitational to surface-tension forces) is greater than one the finger lifts off the bottom wall of the tube leading to dramatic increases in the finger width and wet fraction at a given Ca.For α ≥ 3 a previously unobserved flow regime has been identified in which a small recirculation flow is situated in front of the finger tip, shielding it from any contaminants in the flow. In addition, for α ≳ 2 the capillary number, Cac, above which global recirculation flows disappear has been observed to follow the simple empirical law: Cac2/3α = 1.21.

scholarly journals Optimization of the endodontic retreatment

2020 ◽  
Vol 18 (1) ◽  
pp. 31-36
Author(s):  
Z. S. Khabadze ◽  
F. R. Ismailov
Keyword(s):  
Surface Tension ◽  
Essential Oil ◽  
Essential Oils ◽  
Sodium Hypochlorite ◽  
Root Canal ◽  
Root Canals ◽  
Gutta Percha ◽  
Electron Scanning Microscopy ◽  
Endodontic Retreatment ◽  
Selection Of

Aim. Optimization of an endodontic retreatment by performing the following tasks, including the selection of an optimal and safe solvent.Materials and methods. In order to choose the most suitable solvent for the problem, an experiment was conducted with clove, mint, eucalyptus, orange, and grapefruit oils. The surface tension coefficient was measured by the Rebinder apparatus. In order to justify the choice of an effective essential oil, as well as the dissolving ability and the absence of solvent and debris after irrigation with sodium hypochlorite in the root canal, a tube-shaped electron scanning microscopy was performed.Results. Grapefruit oil has the most dissolving activity (p < 0.05). In addition to the solubility of the filler and siller, grapefruit essential oil is able to dissolve the smear layer.Conclusion. Optimization of the endodontic retreatment with root canals previously filled with gutta-percha, is possible using essential oils, however, a grapefruit oil is the most effective.

scholarly journals Selection of the Taylor-Saffman bubble does not require surface tension

2014 ◽  
Vol 89 (6) ◽  
Author(s):  
Giovani L. Vasconcelos ◽  
Mark Mineev-Weinstein
Keyword(s):  
Surface Tension ◽  
Selection Of
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