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 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 a semi-infinite bubble into a tube of elliptical or rectangular cross-section

2002 ◽  
Vol 470 ◽  
pp. 91-114 ◽  
Author(s):  
ANDREW L. HAZEL ◽  
MATTHIAS HEIL
Keyword(s):  
Aspect Ratio ◽  
Cross Section ◽  
Capillary Number ◽  
Stokes Equations ◽  
Rectangular Cross Section ◽  
Scaling Law ◽  
Propagation Speed ◽  
Cross Sectional ◽  
Steady Propagation ◽  
Rectangular Tubes

This paper investigates the propagation of an air finger into a fluid-filled, axially uniform tube of elliptical or rectangular cross-section with transverse length scale a and aspect ratio α. Gravity is assumed to act parallel to the tube's axis. The problem is studied numerically by a finite-element-based direct solution of the free-surface Stokes equations.In rectangular tubes, our results for the pressure drop across the bubble tip, Δp, are in good agreement with the asymptotic predictions of Wong et al. (1995b) at low values of the capillary number, Ca (ratio of viscous to surface-tension forces). At larger Ca, Wong et al.'s (1995b) predictions are found to underestimate Δp. In both elliptical and rectangular tubes, the ratio Δp(α)/Δp(α = 1) is approximately independent of Ca and thus equal to the ratio of the static meniscus curvatures.In non-axisymmetric tubes, the air-liquid interface develops a noticeable asymmetry near the bubble tip at all values of the capillary number. The tip asymmetry decays with increasing distance from the bubble tip, but the decay rate becomes very small as Ca increases. For example, in a rectangular tube with α = 1.5, when Ca = 10, the maximum and minimum finger radii still differ by more than 10% at a distance 100a behind the finger tip. At large Ca the air finger ultimately becomes axisymmetric with radius r∞. In this regime, we find that r∞ in elliptical and rectangular tubes is related to r∞ in circular and square tubes, respectively, by a simple, empirical scaling law. The scaling has the physical interpretation that for rectangular and elliptical tubes of a given cross-sectional area, the propagation speed of an air finger, which is driven by the injection of air at a constant volumetric rate, is independent of the tube's aspect ratio.For smaller Ca (Ca < Ca), the air finger is always non-axisymmetric and the persisting draining flows in the thin film regions far behind the bubble tip ultimately lead to dry regions on the tube wall. Ca increases with increasing α and for α > αˆ dry spots will develop on the tube walls at all values of Ca.

scholarly journals Flow onset for a single bubble in a yield-stress fluid

2021 ◽  
Vol 933 ◽  
Author(s):  
Ali Pourzahedi ◽  
Emad Chaparian ◽  
Ali Roustaei ◽  
Ian A. Frigaard
Keyword(s):  
Surface Tension ◽  
Yield Stress ◽  
Computational Methods ◽  
Capillary Number ◽  
Single Bubble ◽  
Two Dimensional ◽  
Bubble Shape ◽  
Static Limit ◽  
Yield Stress Fluid

We use computational methods to determine the minimal yield stress required in order to hold static a buoyant bubble in a yield-stress liquid. The static limit is governed by the bubble shape, the dimensionless surface tension ( $\gamma$ ) and the ratio of the yield stress to the buoyancy stress ( $Y$ ). For a given geometry, bubbles are static for $Y > Y_c$ , which we determine for a range of shapes. Given that surface tension is negligible, long prolate bubbles require larger yield stress to hold static compared with oblate bubbles. Non-zero $\gamma$ increases $Y_c$ and for large $\gamma$ the yield-capillary number ( $Y/\gamma$ ) determines the static boundary. In this limit, although bubble shape is important, bubble orientation is not. Two-dimensional planar and axisymmetric bubbles are studied.

Instability of two-layer creeping flow in a channel with parallel-sided walls

1997 ◽  
Vol 351 ◽  
pp. 139-165 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Surface Tension ◽  
Reynolds Number ◽  
Stokes Flow ◽  
Capillary Number ◽  
Travelling Waves ◽  
Boundary Integral ◽  
Creeping Flow ◽  
Two Dimensional ◽  
Two Fluids ◽  
Axial Pressure Gradient

The evolution of the interface between two viscous fluid layers in a two-dimensional horizontal channel confined between two parallel walls is considered in the limit of Stokes flow. The motion is generated either by the translation of the walls, in a shear-driven or plane-Couette mode, or by an axial pressure gradient, in a plane-Poiseuille mode. Linear stability analysis for infinitesimal perturbations and fluids with matched densities shows that when the viscosities of the fluids are different and the Reynolds number is sufficiently high, the flow is unstable. At vanishing Reynolds number, the flow is stable when the surface tension has a non-zero value, and neutrally stable when the surface tension vanishes. We investigate the behaviour of the interface subject to finite-amplitude two-dimensional perturbations by solving the equations of Stokes flow using a boundary-integral method. Integral equations for the interfacial velocity are formulated for the three modular cases of shear-driven, pressure-driven, and gravity-driven flow, and numerical computations are performed for the first two modes. The results show that disturbances of sufficiently large amplitude may cause permanent interfacial deformation in which the interface folds, develops elongated fingers, or supports slowly evolving travelling waves. Smaller amplitude disturbances decay, sometimes after a transient period of interfacial folding. The ratio of the viscosities of the two fluids plays an important role in determining the morphology of the emerging interfacial patterns, but the parabolicity of the unperturbed velocity profile does not affect the character of the motion. Increasing the contrast in the viscosities of the two fluids, while keeping the channel capillary number fixed, destabilizes the interfaces; re-examining the flow in terms of an alternative capillary number that is defined with respect to the velocity drop across the more-viscous layer shows that this is a reasonable behaviour. Comparing the numerical results with the predictions of a lubrication-flow model shows that, in the absence of inertia, the simplified approach can only describe a limited range of motions, and that the physical relevance of the steadily travelling waves predicted by long-wave theories must be accepted with a certain degree of reservation.

Cusp formation for time-evolving bubbles in two-dimensional Stokes flow

2000 ◽  
Vol 412 ◽  
pp. 227-257 ◽  
Author(s):  
MICHAEL SIEGEL
Keyword(s):  
Surface Tension ◽  
Stokes Flow ◽  
Capillary Number ◽  
Broad Class ◽  
Finite Amplitude ◽  
Bubble Surface ◽  
Two Dimensional ◽  
Variable Surface ◽  
Cusp Formation

Analytical and numerical methods are applied to investigate the transient evolution of an inviscid bubble in two-dimensional Stokes flow. The evolution is driven by extensional incident flow with a rotational component, such as occurs for flow in a four-roller mill. Of particular interest is the possible spontaneous occurrence of a cusp singularity on the bubble surface. The role of constant as well as variable surface tension, induced by the presence of surfactant, is considered. A general theory of time- dependent evolution, which includes the existence of a broad class of exact solutions, is presented. For constant surface tension, a conjecture concerning the existence of a critical capillary number above which all symmetric steady bubble solutions are linearly unstable is found to be false. Steady bubbles for large capillary number Q are found to be susceptible to finite-amplitude instability, with the dynamics often leading to cusp or topological singularities. The evolution of an initially circular bubble at zero surface tension is found to culminate in unsteady cusp formation. In contrast to the clean flow problem, for variable surface tension there exists an upper bound Qc for which steady bubble solutions exist. Theoretical considerations as well as numerical calculations for Q > Qc verify that the bubble achieves an unsteady cusped formation in finite time. The role of a nonlinear equation of state and the influence of surface diffusion of surfactant are both considered. A possible connection between the observed behaviour and the phenomenon of tip streaming is discussed.

scholarly journals A numerical study of the Bénard cell

1971 ◽  
Vol 45 (4) ◽  
pp. 805-829 ◽  
Author(s):  
André Cabelli ◽  
G. de Vahl Davis
Keyword(s):  
Heat Transfer ◽  
Surface Tension ◽  
Numerical Method ◽  
Liquid Surface ◽  
Numerical Study ◽  
Three Dimensional ◽  
Critical Value ◽  
Two Dimensional ◽  
Surface Tension Forces ◽  
Biot Numbers

When a layer of liquid is heated from below at a rate which exceeds a certain critical value, a two- or three-dimensional motion is generated. This motion arises from the action of buoyancy and surface tension forces, the latter being due to variations in the temperature of the liquid surface.The two-dimensional form of the flow has been studied by a numerical method. It consists of a series of rolls, rotating alternately clockwise and anticlockwise, which are shown to be symmetrical about the dividing streamlines. As well as a detailed description of the motion and temperature of the liquid, and of the effects on these characteristics of variations in the Rayleigh, Marangoni, Prandtl and Biot numbers, a study has been made of the conditions under which the motion first starts, the wavelength of the rolls and the rate of heat transfer across the liquid layer.

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.

INFLUENCE OF SURFACE TENSION FORCES ON FILMWISE CONDENSATION INTENSITY

2016 ◽  
Vol 7 (1-2) ◽  
pp. 93-102
Author(s):  
Vasily Buz ◽  
Konstantin Goncharov ◽  
Henry F. Smirnov
Keyword(s):  
Surface Tension ◽  
Surface Tension Forces

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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