The effect of surfactant on the motion of long bubbles in horizontal capillary tubes (original) (raw)
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The effect of surfactant on long bubbles rising in vertical capillary tubes
Journal of Statistical Mechanics: Theory and Experiment, 2011
Abstract� In this letter we investigate the effect of interfacial surfactant on the motion of an air bubble rising in a vertical capillary tube filled with a viscous fluid and sealed at one end. A thin layer of liquid, with almost constant thickness b, exists between the bubble interface and the tube wall. The fluid displaced by the front meniscus flows down through this layer because the tube is sealed far up at the top. The steady rising velocity U of the bubble is related to the thickness b. An upper bound for U is obtained in terms of b and other physical data of the problem, which is in good agreement with previous experimental results. It is proved here analytically that the presence of surfactant on the bubble interface causes a thinning and a delay effect: the thickness of the liquid layer behind the bubble and the rise velocity of the bubble are smaller than those for the 'clean' case. Exactly the opposite effect of surfactant in the horizontal case has been derived analytically by Daripa and Pasa (2010 J� Stat� Mech� L02002) and numerically by Ratulowski and Chang (1990 J� Fluid Mech� 21� 303). These effects of interfacial surfactant are consistent with previous experimental and numerical results.
Effect of surfactants on the stability of films between two colliding small bubbles
Colloids and Surfaces A: Physicochemical and Engineering Aspects, 2000
The stability of partially mobile drainage thin liquid film formed between two slightly deformed approaching bubbles or drops is studied. The intervening film is assumed to be thermodynamically unstable. The material properties of the interfaces (surface viscosity, Gibbs elasticity, surface and bulk diffusion) are taken into account. To examine the stability of the thin film we consider the coupling between the drainage and the disturbance flows. The velocity and pressure distributions due to the drainage flow are obtained by using the lubrication approximation. The disturbance flow is examined by imposing small perturbations on the film interfaces and liquid flow. The long wave approximation is applied. We solved the linear problem for the evolution of the fluctuations in the local film thickness, interfacial velocity and pressure. The linear stability analysis of the gap region allows us to calculate the critical thickness, at which the system becomes unstable. Quantitative explanation of the following effects is proposed, (i) the increase of critical thickness with the increase of the interfacial mobility; (ii) the role of surface viscosity, compared with that of the Gibbs elasticity; (iii) the significant destabilization of the gap region with the decreasing droplet radius in the case of buoyancy driven motion. The analytical expressions for critical thickness in the case of negligible surface viscosity and tangentially immobile interfaces are presented. : S 0 9 2 7 -7 7 5 7 ( 0 0 ) 0 0 6 2 1 -X
Surfactants role on the deformation of colliding small bubbles
Colloids and Surfaces A: Physicochemical and Engineering Aspects, 1999
The mutual approach of two bubbles and the rate of thinning and the deformation of the partially mobile thin liquid film intervening between them is studied. The material properties of the interfaces (surface viscosity, Gibbs elasticity, surface and/or bulk diffusivity) are taken into account. In the normal stress balance at the fluid interfaces we include the contribution of the intermolecular forces. To obtain the liquid velocity and pressure distribution the lubrication approximation is used. From the normal stress boundary condition the first order (with respect to the capillary number) shape function is derived. It provides information on the inversion thickness, at which the curvature in the gap between the drops changes from convex to concave, and the pimple thickness, at which the curvature of the interfaces spontaneously increases due to the action of the attractive intermolecular forces. The analytical and numerical investigations reveal significant influence of the disjoining pressure and the surfactant on both thicknesses. Explanation of the following effects is proposed: (i) increase of the pimple thickness and decrease of the inversion thickness with the increase of the interfacial mobility; (ii) role of the surface viscosity; (iii) role of the van der Waals interaction.
The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow
Journal of Fluid Mechanics, 1995
Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure-velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The 'piston' is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments.
In this study, we investigate, using direct numerical simulation, the motion of a small bubble in a horizontal microchannel filled with a liquid containing surfactants. In particular, we study the combined effect of surfactants and bubble deformability on the bubble shape, bubble−liquid relative velocity, velocity field in the liquid, liquid velocity on the gas−liquid interface, and surfactant distribution on the interface. The level-set method is used to capture the gas−liquid interface. The surfactant transport equation on the gas−liquid interface is solved in an Eulerian framework and is coupled to an equation describing the transport of surfactants inside the liquid phase. The Marangoni stress, induced by surfactant concentration gradients, is computed using the continuum surface force model. The simulation results give insights into the complexity of the coupling of the different phenomena controlling the dynamics of the studied system. For instance, the results show that for values of the capillary number much smaller than unity, that is, for spherical bubbles, the bubble velocity decreases as the bubble diameter increases. Moreover, surfactants tend to decrease significantly the bubble velocity, when compared with a bubble with a clean surface. Indeed, they accumulate at a convergent stagnation point/circle on the bubble surface and deplete at a divergent stagnation point/circle. As a consequence, the velocity of the liquid adjacent to the bubble is reduced in between the convergent and divergent stagnation points/circles because of Marangoni stresses. It is shown that regarding the bubble−liquid relative velocity, the bubble behaves as a rigid sphere when the Langmuir number is larger than unity, at least for the range of parameters explored in this study. For values of the capillary number of the order of unity, the bubble can take a "bullet shape". In this case, the bubble velocity increases as the bubble diameter increases. This increase of the bubble−liquid relative velocity is linked to a drastic change in the liquid flow structure near the bubble. Surfactants are swept to the rear of the bubble and have less influence on the bubble dynamics than for spherical bubbles. Finally, it is shown that increasing the amount of surfactants adsorbing to the surface eventually leads to the bursting of the bubble.
The motion of long bubbles in polygonal capillaries. Part 1. Thin films
Journal of Fluid Mechanics, 1995
Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure-velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The 'piston' is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments. Part 1 of this work studies the fluid films deposited on capillary walls in the limit Ca-tO (Ca = ,uU/g, where , u is the fluid viscosity, U the bubble velocity, and the surface tension). Part 2 (Wong et al. 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x. For 1 < x < Ca-l, the film profile is frozen with a thickness of order Ca213 at the centre and order Ca at the sides. For x N Cap', surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca413. For x 9 Ca-', the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x N Ca-'I3, the height of the parabola is order Ca2/'. Finally, for x 9 Ca-''', the height decreases as Ca1/4x-114.
Marangoni effects on the motion of an expanding or contracting bubble pinned at a submerged tube tip
Journal of Fluid Mechanics, 1999
This work studies the motion of an expanding or contracting bubble pinned at a submerged tube tip and covered with an insoluble Volmer surfactant. The motion is driven by constant flow rate Q into or out of the tube tip. The purpose is to examine two central assumptions commonly made in the bubble and drop methods for measuring dynamic surface tension, those of uniform surfactant concentration and of purely radial flow. Asymptotic solutions are obtained in the limit of the capillary number Ca→0 with the Reynolds number Re=o(Ca−1, non-zero Gibbs elasticity (G), and arbitrary Bond number (Bo). (Ca=μQ/a2σc, where μ is the liquid viscosity, a is the tube radius, and σc is the clean surface tension.) This limit is relevant to dynamic-tension experiments, and gives M→∞, where M=G/Ca is the Marangoni number. We find that in this limit the deforming bubble at each instant in time takes the static shape. The surfactant distribution is uniform, but its value varies with time as the bubble are...
Effect of buoyancy on the motion of long bubbles in horizontal tubes
As a confined long bubble translates along a horizontal liquid-filled tube, a thin film of liquid is formed on the tube wall. For negligible inertial and buoyancy effects, respectively, small Reynolds (Re) and Bond (Bo) numbers, the thickness of the liquid film depends only on the flow capillary number (Ca). However, buoyancy effects are no longer negligible as the diameter of the tube reaches millimeter length scales, which corresponds to finite values of Bo. We perform experiments and theoretical analysis for a long bubble in a horizontal tube to investigate the effect of Bond number (0.05 < Bo < 0.5) on the thickness of the liquid film and the bubble orientation at different capillary numbers 10 −3 < Ca < 10 −1. We investigate several features of the lubricating film around the bubble. (i) Due to the gravitational effects, the film deposited on the upper wall of the channel is thinner than the film at the bottom wall. We extend the available theory for the film thickness at the front of the bubble in a two-dimensional geometry at low capillary numbers Ca < 10 −3 and finite Bo to account for the effect of larger Ca. The resulting model shows very good agreement with the present experimental measurements. (ii) Due to the asymmetry in the liquid film thickness and the consequent drainage of the liquid from the top to the bottom of the tube, the bubble is inclined relative to the channel centerline and our side-view visualizations allow direct quantification of the inclination angle, which increases with both Bo and Ca. While the inclination angle at the top is smaller than that at the bottom of the tube, the average of these two values follows the predictions of a mass balance analysis in the central region of the bubble. (iii) The inclination of the bubble causes the thickness of the thin film at the back of the bubble to depend on the length of the bubble, whereas the thickness at the front of the bubble does not depend on the bubble length.