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 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.

Fractal Characteristics of Viscous Fingering

Fractals ◽  
1997 ◽  
Vol 05 (02) ◽  
pp. 221-227 ◽  
Author(s):  
Songyue Tang ◽  
Zhonglei Wei
Keyword(s):  
Surface Tension ◽  
Monte Carlo ◽  
Stochastic Simulation ◽  
Viscosity Ratio ◽  
Fractal Dimensions ◽  
Viscous Fingering ◽  
Experimental Results ◽  
Fractal Characteristics ◽  
Hele Shaw Cell ◽  
Good Agreement

Viscous fingering is investigated by experiment in a 2-dimensional radial Hele-Shaw cell and Monte Carlo stochastic simulation. Experimental results show that viscosity ratio between the driving and driven fluids determines whether or not viscous fingerings occur and that surface tension makes the viscous fingering patterns "fatter". Simulation patterns are in good agreement with experimental ones. The fractal dimensions of the viscous fingering patterns by both experiments and simulations are about Df=1.2-1.6.

scholarly journals Viscous-fingering mechanisms under a peeling elastic sheet

2019 ◽  
Vol 864 ◽  
pp. 1177-1207
Author(s):  
Gunnar G. Peng ◽  
John R. Lister
Keyword(s):  
Surface Tension ◽  
Stability Analysis ◽  
Cell Walls ◽  
Viscous Fingering ◽  
Base Flow ◽  
Fingering Instability ◽  
Linear Perturbations ◽  
The Stability ◽  
Rigid Walls ◽  
Hele Shaw Cell

We study the mechanisms affecting the viscous-fingering instability in an elastic-walled Hele-Shaw cell by considering the stability of steady states of unidirectional peeling-by-pulling and peeling-by-bending. We demonstrate that the elasticity of the wall influences the steady base state but has a negligible direct effect on the behaviour of linear perturbations, which thus behave like in the ‘printer’s instability’ with rigid walls. Moreover, the geometry of the cell can be very well approximated as a triangular wedge in the stability analysis. We identify four distinct mechanisms – surface tension acting on the horizontal and the vertical interfacial curvatures, kinematic compression in the longitudinal base flow, and the films deposited on the cell walls – that each contribute to stabilizing the system. The vertical curvature is the dominant stabilizing mechanism for small capillary numbers, but all four mechanisms have a significant effect in a large region of parameter space.

Global instability of viscous fingering in a Hele–Shaw cell: formation of oscillatory fingers

1991 ◽  
Vol 2 (2) ◽  
pp. 105-132 ◽  
Author(s):  
Jian-Jun Xu
Keyword(s):  
Surface Tension ◽  
Travelling Waves ◽  
Cell Formation ◽  
Stable State ◽  
Viscous Fingering ◽  
Instability Mechanism ◽  
Global Instability ◽  
Trapped Wave ◽  
Global Modes ◽  
Hele Shaw Cell

This work deals with the global instability mechanism of viscous fingering in a Hele–Shaw cell with the inclusion of surface tension. We investigate the interaction and propagation of travelling waves in the system, and obtain two discrete sets of global wave modes: symmetrical modes and anti-symmetrical modes. We call the instability mechanism determined by these global modes the global trapped wave (GTW) instability. A unique global, neutrally stable state of the system is found; it explains the formation of the narrow, oscillatory fingers discovered by Couder el al. (1986) and by Kopf-Sill & Homsy (1987).

Characterization of viscous fingering in a radial Hele-Shaw cell by image analysis

1999 ◽  
Vol 26 (1-2) ◽  
pp. 153-160 ◽  
Author(s):  
M.-N. Pons ◽  
E. M. Weisser ◽  
H. Vivier ◽  
D. V. Boger
Keyword(s):  
Image Analysis ◽  
Viscous Fingering ◽  
Hele Shaw Cell

scholarly journals Steady rimming flows with surface tension

2008 ◽  
Vol 597 ◽  
pp. 91-118 ◽  
Author(s):  
E. S. BENILOV ◽  
M. S. BENILOV ◽  
N. KOPTEVA
Keyword(s):  
Thin Film ◽  
Surface Tension ◽  
Viscous Fluid ◽  
Small Parameter ◽  
Outer Region ◽  
Lubrication Approximation ◽  
Steady Flows ◽  
Matched Asymptotics ◽  
Weak Surface ◽  
Rimming Flows

We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the ‘rising’ side of the cylinder and, for large ones, pools at the cylinder's bottom. The paper examines rimming flows with a pool affected by weak surface tension. Using the lubrication approximation and the method of matched asymptotics, we find a solution describing the pool, the ‘outer’ region, and two transitional regions, one of which includes a variable (depending on the small parameter) number of asymptotic zones.

Effect of gas generation by chemical reaction on viscous fingering in a Hele–Shaw cell

2021 ◽  
Vol 33 (9) ◽  
pp. 093104
Author(s):  
Weicen Wang ◽  
Chunwei Zhang ◽  
Anindityo Patmonoaji ◽  
Yingxue Hu ◽  
Shintaro Matsushita ◽  
...  
Keyword(s):  
Chemical Reaction ◽  
Viscous Fingering ◽  
Gas Generation ◽  
Hele Shaw Cell
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