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.

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.

A New Model for Predicting the Maximum Gas Entrapment Concentration in a Yield Stress Fluid

SPE Journal ◽  
2022 ◽  
pp. 1-15
Author(s):  
Shaowei Pan ◽  
Zhiyuan Wang ◽  
Baojiang Sun
Keyword(s):  
Yield Stress ◽  
Maximum Error ◽  
Critical Distance ◽  
Single Bubble ◽  
Average Error ◽  
Two Phase ◽  
Yield Stress Fluid ◽  
Multiple Bubbles ◽  
Gas Entrapment ◽  
First Time

Summary Gas entrapment is a typical phenomenon in gas-yield stress fluid two-phase flow, and most of the related research focuses on the entrapped condition of the single bubble. However, the amount of entrapped gas, which is more meaningful for engineering, is rarely involved. In this paper, a theoretical model for calculating the maximum gas entrapment concentration (MGEC) is established for the first time. The critical distance between horizontal and vertical entrapped bubbles was determined by the yielded region caused by the buoyancy and the coupled stress field around the multiple bubbles. The MGEC is the ratio of a single bubble volume to its domain volume, which is calculated from the distance between the vertical and the horizontal bubbles. By comparing with the experimental results, the average error of MGEC calculated by this model is 4.42%, and the maximum error is 7.32%. According to the prediction results of the model, an empirical equation that can be conveniently used for predicting MGEC is proposed.

Deformation of a bubble or drop in a uniform flow

1980 ◽  
Vol 101 (4) ◽  
pp. 673-686 ◽  
Author(s):  
Jean-Marc Vanden-Broeck ◽  
Joseph B. Keller
Keyword(s):  
Surface Tension ◽  
Liquid Drop ◽  
Pressure Ratio ◽  
Bubble Surface ◽  
Stagnation Pressure ◽  
Two Dimensional ◽  
Bubble Shape ◽  
Fluid Density ◽  
Circular Arc ◽  
Steady Potential

Steady potential flow around a two-dimensional bubble with surface tension, either free or attached to a wall, is considered. The results also apply to a liquid drop. The flow and the bubble shape are determined as functions of the contact angle β and the dimensionless pressure ratio γ = (pb − ps)/½ρU2. Here pb is the pressure in the bubble, ps = p∞ + ½ρU2 is the stagnation pressure, p∞ is the pressure at infinity, ρ is the fluid density and U is the velocity at infinity. The surface tension σ determines the dimensions of the bubble, which are proportional to 2σ/ρU2. As γ tends to ∞, the bubble surface tends to a circle or circular arc, and as γ decreases the bubble elongates in the direction normal to the flow. When γ reaches a certain value γ0(β), opposite sides of the bubble touch each other. The problem is formulated as an integrodifferential equation for the bubble surface. This equation is discretized and solved numerically by Newton's method. Bubble profiles, the bubble area, the surface energy and the kinetic energy are presented for various values of β and γ. In addition a perturbation solution is given for γ large when the bubble is nearly a circular arc, and a slender-body approximation is presented for β ∼ ½π and γ ∼ γ0(β), when the bubble is slender.

Interacting two-dimensional bubbles and droplets in a yield-stress fluid

2008 ◽  
Vol 20 (4) ◽  
pp. 040901 ◽  
Author(s):  
John P. Singh ◽  
Morton M. Denn
Keyword(s):  
Yield Stress ◽  
Two Dimensional ◽  
Yield Stress Fluid

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.

scholarly journals Yield-stress fluid deposition in circular channels

2017 ◽  
Vol 818 ◽  
pp. 838-851 ◽  
Author(s):  
Benoît Laborie ◽  
Florence Rouyer ◽  
Dan E. Angelescu ◽  
Elise Lorenceau
Keyword(s):  
Yield Stress ◽  
Newtonian Fluid ◽  
Capillary Number ◽  
Pure Shear ◽  
Scaling Law ◽  
Classical Problem ◽  
Nonlinear Effects ◽  
Shear Thinning ◽  
Quantitative Agreement ◽  
Yield Stress Fluid

Since the pioneering works of Taylor and Bretherton, the thickness $h$ of the film deposited behind a long bubble invading a Newtonian fluid is known to increase with the capillary number power $2/3$ ($h\sim RCa^{2/3}$), where $R$ is the radius of the circular tube and $Ca$ is the capillary number, comparing the viscous and capillary effects. This law, known as Bretherton’s law, is valid only in the limit of $Ca<0.01$ and negligible inertia and gravity. We revisit this classical problem when the fluid is a yield-stress fluid (YSF) exhibiting both a yield stress and a shear-thinning behaviour. First, we provide quantitative measurement of the thickness of the deposited layer for Carbopol, a Herschel–Bulkley fluid, in the limit where the yield stress is of a similar order of magnitude to the capillary pressure and for $0.1<Ca<1$. To understand our observations, we use scaling arguments to extend the analytical expression of Bretherton’s law to YSFs in circular tubes. In the limit of $Ca<0.1$, our scaling law, in which the adjustable parameters are set using previous results concerning non-Newtonian fluids, successfully retrieves several features of the literature. First, it shows that (i) the thickness deposited behind a Bingham YSF (exhibiting a yield stress only) is larger than for a Newtonian fluid and (ii) the deposited layer increases with the amplitude of the yield stress. This is in quantitative agreement with previous numerical results concerning Bingham fluids. It also agrees with results concerning pure shear-thinning fluids in the absence of yield stress: the shear-thinning behaviour of the fluid reduces the deposited thickness as previously observed. Last, in the limit of vanishing velocity, our scaling law predicts that the thickness of the deposited YSF converges towards a finite value, which presumably depends on the microstructure of the YSF, in agreement with previous research on the topic performed in different geometries. For $0.1<Ca<1$, the scaling law fails to describe the data. In this limit, nonlinear effects must be taken into account.

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.

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