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 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.

Surprises in viscous fingering

2000 ◽  
Vol 409 ◽  
pp. 273-308 ◽  
Author(s):  
S. TANVEER
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Flow Field ◽  
Viscous Fingering ◽  
Structural Instability ◽  
Local Inhomogeneity ◽  
Explicit Model ◽  
Leakage Term ◽  
Finger Tip ◽  
Hele Shaw Cell

In this paper, we review some aspects of viscous fingering in a Hele-Shaw cell that at first sight appear to defy intuition. These include singular effects of surface tension relative to the corresponding zero-surface-tension problem both for the steady and unsteady problem. They also include a disproportionately large influence of small effects like local inhomogeneity of the flow field near the finger tip, or of the leakage term in boundary conditions that incorporate realistic thin-film effects. Through simple explicit model problems, we demonstrate how such properties are not unexpected for a system approaching structural instability or ill-posedness.

scholarly journals The effect of surface tension on the shape of fingers in a Hele Shaw cell

1981 ◽  
Vol 102 ◽  
pp. 455-469 ◽  
Author(s):  
J. W. McLean ◽  
P. G. Saffman
Keyword(s):  
Surface Tension ◽  
Experimental Results ◽  
Two Dimensional ◽  
Singular Perturbation Analysis ◽  
Linearized Stability ◽  
Finger Width ◽  
The Stability ◽  
Hele Shaw Cell ◽  
Effect Of Surface ◽  
Small Disturbances

The experimental results of Saffman & Taylor (1958) and Pitts (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the interface. Using free streamline techniques, the shape of the free surface is expressed as the solution of a nonlinear integro-differential equation. The equation is solved numerically and the solutions are compared with experimental results. The shapes of the profiles are very well predicted, but the dependence of finger width on surface tension is not quantitatively accurate, although the qualitative behaviour is correct. A conflict between the numerics and a formal singular perturbation analysis is noted but not resolved. The stability of the steady finger to small disturbances is also examined. Linearized stability analysis indicates that the two-dimensional fingers are not stabilized by the surface-tension effect, which disagrees with the experimental observations. A possible reason for the discrepancy between theory and experiment is suggested.

scholarly journals The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

2017 ◽  
Vol 473 (2201) ◽  
pp. 20170050 ◽  
Author(s):  
Christopher C. Green ◽  
Christopher J. Lustri ◽  
Scott W. McCue
Keyword(s):  
Surface Tension ◽  
Numerical Solutions ◽  
Bubble Velocity ◽  
Number Of Solutions ◽  
Branch Number ◽  
Single Solution ◽  
Countably Infinite ◽  
Prime Function ◽  
Hele Shaw Cell ◽  
Effect Of Surface

New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.

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