The toroidal bubble

1968 ◽  
Vol 32 (1) ◽  
pp. 97-112 ◽  
Author(s):  
T. J. Pedley
Keyword(s):  
Surface Tension ◽  
Viscous Fluid ◽  
Energy Equation ◽  
Bubble Surface ◽  
Inviscid Fluid ◽  
Initial Instant ◽  
Bubble Volume ◽  
Toroidal Bubble ◽  
History Of ◽  
Impulse Equation

It has been observed by Walters & Davidson (1963) that release of a mass of gas in water sometimes produces a rising toroidal bubble. This paper is concerned with the history of such a bubble, given that at the initial instant the motion is irrotational everywhere in the water. The variation of its overall radius a with time may be predicted from the vertical impulse equation, and it should be possible to make the same prediction by equating the rate of loss of combined kinetic and potential energy to the rate of viscous dissipation. This is indeed seen to be the case, but not before it is recognized that in a viscous fluid vorticity will continually diffuse out from the bubble surface, destroying the irrotationality of the motion, and necessitating an examination of the distribution of vorticity. The impulse equation takes the same form as in an inviscid fluid, but the energy equation is severely modified. Other results include an evaluation of the effect of a hydrostatic variation in bubble volume, and a prediction of the time which will have elapsed before the bubble becomes unstable under the action of surface tension.

Numerical simulations of sink flow in the Hele-Shaw cell with small surface tension

1997 ◽  
Vol 8 (6) ◽  
pp. 533-550 ◽  
Author(s):  
E. D. KELLY ◽  
E. J. HINCH
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Viscous Fluid ◽  
Numerical Simulations ◽  
Boundary Integral Equation ◽  
Numerical Algorithm ◽  
Boundary Integral ◽  
Inviscid Fluid ◽  
Small Surface ◽  
Hele Shaw Cell

The motion of an initially circular drop of viscous fluid surrounded by inviscid fluid in a Hele-Shaw cell withdrawn from an eccentric point sink is considered. Using a numerical algorithm based on a boundary integral equation, the solution for small, finite surface tension is observed. It is found that the zero-surface-tension formation of a cusp is avoided, and instead a narrow finger of inviscid fluid forms, which then rapidly propagates towards the sink. The scaling of the finger in the sink vicinity is determined.

On driving a viscous fluid out of a tube

1962 ◽  
Vol 14 (1) ◽  
pp. 81-96 ◽  
Author(s):  
B. G. Cox
Keyword(s):  
Surface Tension ◽  
Viscous Fluid ◽  
Equations Of Motion ◽  
Horizontal Tube ◽  
Tube Diameter ◽  
Experimental Results ◽  
Inviscid Fluid ◽  
Asymptotic Value ◽  
Two Fluids ◽  
Interfacial Surface

Two problems are considered. First, it is shown experimentally that the amount of viscous fluid left on the walls of a horizontal tube, when it is expelled by an inviscid fluid, reaches an asymptotic value of 0.60 of the amount required to fill the tube, when the parameter μU/T is increased, μ and T being the coefficients of viscosity and interfacial surface tension respectively, and U the velocity of the interface between the two fluids. Secondly, by neglecting the inertia terms in the equations of motion and the effect of gravity, a theory for the passage of this type of bubble is presented, together with experimental results in support of the theory. It is shown that such a solution is only valid under certain other restrictions, and then only to within half a tube diameter of the nose of the bubble.

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.

scholarly journals Application of Depletion Attraction in Mineral Flotation: II. Effects of Depletant Concentration

Minerals ◽  
2018 ◽  
Vol 8 (10) ◽  
pp. 450 ◽  
Author(s):  
Gahee Kim ◽  
Junhyun Choi ◽  
Sowon Choi ◽  
KyuHan Kim ◽  
Yosep Han ◽  
...  
Keyword(s):  
Surface Tension ◽  
Polymer Brush ◽  
Bubble Surface ◽  
Mass Ratio ◽  
Attraction Force ◽  
Surface Acting ◽  
Pure Silica ◽  
Mineral Flotation ◽  
Air Water Interface ◽  
High Concentrations

Along with the accompanying theory article, we experimentally investigate the effect of the depletion attraction force on the flotation of malachite. While varying the concentration of the depletion agent (polyethylene glycol), three different systems are studied: pure malachite, pure silica and a 1:1 mass ratio of malachite and silica binary system. We find that the recovery increases significantly as the concentration of the depletion reagents increases for all three systems. However, the recovery suddenly decreases in a certain concentration range, which corresponds to the onset of the decreased surface tension when high concentrations of the depletion agent are used. The decreased surface tension of the air/water interface suggests that the recovery rate is lowered due to the adsorption of the depletion agent to the bubble surface, acting as a polymer brush. We also perform experiments in the presence of a small amount of a collector, sodium oleate. An extremely small amount of the collector (10−10–10−5 M) leads to the increase in the overall recovery, which eventually reaches nearly 100 percent. Nevertheless, the grade worsens as the depletant provides the force to silica particles as well as target malachite particles.

scholarly journals Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

2019 ◽  
Vol 877 ◽  
pp. 1063-1097 ◽  
Author(s):  
Liam C. Morrow ◽  
Timothy J. Moroney ◽  
Scott W. McCue
Keyword(s):  
Viscous Fluid ◽  
Linear Prediction ◽  
Applied Mathematics ◽  
Viscous Fingering ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Inviscid Fluid ◽  
Interfacial Instabilities ◽  
Hele Shaw Cell ◽  
Injection Rates

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.

A Tangled Web: Young, Laplace, and the Surface Tension Law

Physiology ◽  
1993 ◽  
Vol 8 (4) ◽  
pp. 179-183
Author(s):  
S. Marsh Tenney
Keyword(s):  
Surface Tension ◽  
History Of

The nagging issue of priority complicated the important discovery by two great scientists. Their lives and the subsequent history of their work reveal amusing and sometimes unusual relationships.

Application of an Improved Ghost Fluid Method to the Collapse of Non-Spherical Bubbles in a Compressible Liquid

2011 ◽  
Author(s):  
Hiroyuki Takahira ◽  
Yoshinori Jinbo
Keyword(s):  
Surface Tension ◽  
Shock Waves ◽  
Thermal Diffusion ◽  
Level Set ◽  
Compressible Liquid ◽  
Maximum Temperature ◽  
Bubble Collapse ◽  
Ghost Fluid Method ◽  
Toroidal Bubble ◽  
The Impact

The ghost fluid method (GFM) is improved to investigate violent bubble collapse in a compressible liquid, in which the adaptive mesh refinement with multigrids, the surface tension, and the thermal diffusion through the bubble interface are taken into account. The improved multigrid GFM is applied to the interaction of an incident shock wave with a bubble. The multigrid GFM captures the fine interfacial and vortex structures of the toroidal bubble when the bubble collapses violently accompanied with the penetration of the liquid jet and the formation of the shock waves. The multigrid GFM is also applied to the bubble collapse near a tissue surface in which the tissue is modeled with gelatin in order to predict the tissue damage due to the bubble collapse; the motions of three phases for the gas inside the bubble, the liquid surrounding the bubble, and the gelatin boundary are solved directly by coupling the level set method with the improved GFM. Two kinds of level set functions are utilized for distinguishing the gas-liquid interface from the liquid-gelatin interface. It is shown that the impact of the shock waves generated from the collapsing bubble on the boundary leads to the formation of depression of the boundary; the toroidal bubble penetrates into the depression. Also, the surface tension effects are successfully included in the improved GFM. The thermal effects of internal gas on the bubble collapse are also discussed by considering the thermal diffusion across the interface in the GFM. The thermal boundary layers of the toroidal bubble are captured with the method. The result shows that the smaller the initial bubble radius becomes, the lower the maximum temperature inside the bubble becomes because of the thermal diffusion across the interface.

Cavitation of a viscous fluid in narrow passages

1963 ◽  
Vol 16 (4) ◽  
pp. 595-619 ◽  
Author(s):  
G. I. Taylor
Keyword(s):  
Surface Tension ◽  
Viscous Fluid ◽  
Hydrodynamic Lubrication ◽  
Surface Tension Force ◽  
Physical Description ◽  
Bubble Cavitation ◽  
Oil Flow ◽  
Hydrodynamic Separation ◽  
Flow Through ◽  
Air Column

The conditions which determine the existence and position of cavitation in the narrow passages of hydrodynamically lubricated bearings have been assumed to be the same as those which produce cavitation bubbles, namely a lowering of pressure below that at which gas separates out of fluid. This assumption enables certain predictions to be made which in some cases are verified, but it does not provide a physical description of the interface between oil and air. Theoretical analysis of the situation seems to be beyond our present capacity, and in none of the experiments so far published has it been possible to measure both the most important relevant data, namely the minimum clearance and the oil flow through it.A method is described here which enables this to be done. It turns out that two physically different kinds of cavitation can occur. One of these is well described by the existing theory and assumption. Surface tension plays no part in it, and in most text books on hydrodynamic lubrication is not even mentioned. The other kind, which is akin to hydrodynamic separation rather than bubble cavitation, depends essentially on surface tension. Both kinds appear clearly in published photographs taken through transparent bearings, but the experimenters do not seem to have distinguished between them.The reason why surface tension, which is only able to supply stresses that are exceedingly small compared with the pressure variation in the fluid itself, may have a large effect on the flow can be understood by considering the flow of a viscous fluid in a tube when blown out by air pressure applied at one end. For any given length of fluid the rate of outflow depends almost entirely on the pressure applied, the surface tension force being negligible; but the amount of fluid left in the tube after the air column has reached the end depends essentially on surface tension.

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