Bubble Hydrodynamics

Bubble Hydrodynamics

1. Introduction to Bubble Hydrodynamics 2. Temperature and Bubbles 3. Surfactants and Bubbles		4. Oscillations and Bubbles 5. Measurement Techniques
1. Introduction After formation, a bubble rapidly accelerates to its terminal velocity, V_B. The value of V_B is determined by the balance between the buoyant rise force, and the drag force. While it is easy to calculate the buoyancy force for a bubble, the drag force varies with bubble size. For small, spherical bubbles, the drag force can be calculated, and when combined with the buoyancy force, yielding Stoke's Law: Stoke's Law is only applicable to small bubbles with an immobile surface. An extension of Stoke's Law for a bubble with a mobile surface was derived by Hadamard-Rybczynski and is where the dirty case reduces, of course, to Stoke's Law. Additionally, Hadamard-Rybcszynski is only applicable to spherical bubbles (Reynolds number < 1). As the bubble rises, new interface is created at the upstream hemisphere and flows down towards the bottom stream hemisphere where it disappears. It is important for the proper conceptual model to correctly describe the bubble interface. Specifically, the interface is a curved region with water molecules on one side, air molecules on the other, and, if present in the water, surfactant molecules. The water molecules at the interface are continually exchanging with water molecules in the bulk fluid, and also with water vapor molecules in the gas (according to the equilibrium vapor pressure). Thus to create fresh interface, water molecules at the upstream hemisphere from the bulk fluid arrive at the interface.
This process is shown schematically to the right. The schematic is not to any scale, nor does it show the correct orientation of water molecules on the interface ( a result of the dipole moment ). Water vapor molecules can be seen in the bubble, and dissolved Nitrogen gas molecules in the water.		H - Black O - White N - Gray
		Schematic of Bubble Interface.
The no slip condition means that the motion of molecules on both sides of the interface must match. Water molecules will exchange with the water molecules in the bulk fluid, Gas molecules with gas in the bubble, and water and gas molecules are continually creating interface at the upstream pole, which is also known as the stagnation point, since the flow velocity at the pole itself must be zero. An animation of the flow around a clean bubble is compared with the flow around a contaminated bubble in the Surfactant section. Larger bubbles deform from sphericity, into ellipsoidal shapes. For bubbles larger than a critical radius, bubbles oscillate both in shape and trajectory. Numerical techniques have been used to solve for V_B for some non-oscillating bubbles; however, only empirical parameterizations are available for larger laminar flow bubbles and oscillating bubbles. Fluid motions As the bubble rises, if there are motions in the fluid, it's trajectory, behavior, and evolution will be different from the stagnant fluid assumption inherent in the parameterization of V_B. For example, a stream of bubbles causes an upwelling flow which causes bubbles in the bubble stream to rise faster than if they were alone. If the fluid is sheared, the bubbles will experience a "lift" force similar to that of an airplane wing and move perpendicular to the velocity shear. And turbulence motions in the fluid will affect the bubble trajectory, with the affect being strongest for bubbles whose rise is less than the turbulence velocity scale. Another factor affecting bubble hydrodynamics is the fluid properties. The fluid viscosity, and density, both of which very with temperature, affect the drag force. 2. Temperature and Bubbles Temperature affects bubble hydrodynamics in several ways. By altering the fluids viscosity and density, the friction force on the bubble is changed and as a result, bubbles rise faster with increasing temperature. However, bubbles larger than approximately 750 microns in radius oscillate as they rise. Because of the oscillations, the bubbles oscillate faster, and move faster but rise slower with increasing temperature. This is in direct contradiction with the prediction of the wave analogy of bubble motion.
A plot of the variation of rise speed with radius for a clean bubble at 20C is shown at the right. Bubbles larger than the peak in the figure to the right oscillate, while bubbles smaller than the peak do not. The oscillations are also accompanied by shape oscillations. Initially, the shape oscillations are simple harmonics, but become multimodal and more complex with increasing size. For bubbles larger than approximately 2000 microns radius, the trajectory oscillations are relatively unimportant while shape oscillations are very complex. As a result, bubble rise velocity becomes independant of temperature. This figure is an adaptation of data from Clift et al., 1978, Leifer et al., 1999 and others. Interestingly, this implies
3. Oscillations and Bubbles A movie of a 2500-µm bubble rising at 250 fps (504k). Movie For clean bubbles larger than about 700 microns radius at 20C, oscillations are very important to understanding the hydrodynamics of bubbles. Oscillations are thought to begin from instabilities in the bubble wake, and are observable in both the trajectory and shape. Because of oscillations, the bubble rise speed is affected, as is mass transfer. Additionally, the turbulence created by the rising bubbles will affect other bubbles, and mass transfer in the fluid. Additionally, bubble oscillations are sensitive to the presence of surfactants.
	The importance of oscillations is shown in the figure to the left which demonstrates that very small changes in the density and viscosity of water are able to change the trajectory of a 677 micron bubble from linear (T>27C) to oscillatory. With the onset of oscillations, the rise velocity begins to decrease. The experiment was conducted in millipore distilled water. With increasing size, the bubble surface oscillations change from a simple sinusoidal oscillation to higher order modes, the trajectory changes from a simple helix to more complex trajectories. With these changes, the rise velocity becomes less dependant upon temperature.
4. Surfactants and Bubbles Surfactants are surface active substances whose molecules have a hydrophobic and hydrophilic part. The hydrophobic part tends to align itself in the air, while the hydrophilic part aligns itself in the water, thus surfactants tend to be found at interfaces. Dish washing soap is a familiar surfactant, but many substances, including salt and fatty acids are surfactants. An important effect of surfactants is that they alter the surface tension, which alters the hydrodynamcis of the interface. Another effect of surfactants is to increase the longevity of bubbles on the surface, and alter the size distribution of bubbles produced in a breaking wave or jet of water. Animations of both surfactant free (Left) and surfactant contaminated (Right) rising bubbles in the bubble frame of reference are shown below. Flow velocities are representative, THIS IS NOT OUTPUT FROM A FLUID DYNAMIC MODEL. `If Images are still, Reload page and scroll to the animation.`
Surfactant Free Bubble Movie	Surfactant Contaminated Bubble Movie
The clean bubble rises faster because its entire surface is mobile. However, for a bubble in surfactant contaminated water, the situation is very different. Surfactant molecules (black circles) diffuse to the bubble where they accumulate at the interface and are pushed downstream (convection) by the flow. Repulsion between surfactant molecules primarily in the downstream hemisphere (high surfactant concentration) counters the squeezing by the flow. Additionally, the flow is not uniform with angle on the bubble, increasing from zero at the upstream pole stagnation point to a maximum near the equator, and decreasing again to zero at the downstream pole. As a result, convection (pushing) decreases towards both poles. Thus in steady state, there is a balance between diffusion to the upstream hemisphere where the surfactant concentration on the surface is lower than equilibrium with the bulk fluid, and diffusion from the bubble in the downstream hemisphere where convection (pushing) by the flow has increased the concentration to levels above equilibrium. Similarly, the concentration of surfactant on the surface is determined by the diffusional process just described, the effect of convection by the flow towards the downstream hemisphere, and the repulsion between surfactant molecules, particularly where they are pushed into higher concentration. As a result, the surfactant concentration on the downstream hemisphere is elevated, and because of convection, most of the gradient occurs in the downstream hemisphere. The surfactant concentration gradient causes a gradient in the surface tension. And this gradient of surface tension has the important effect of decreasing surface mobility. As a result, the internal circulation to the rear of the bubble is supressed, and the flow at the interface is deceased (or stopped) and the bubble experiences a higher drag. The angle from the downstream pole to which the interface is immobilized is called the stagnant cap angle from the stagnant cap model. Surfactant Summary Interface creation causes the upstream hemisphere to be clean. Surfactants diffuse to the upstream hemisphere. Surfactants diffuse from the downstream hemisphere. Surfactant gradients cause a surface tension gradient. Surface tension gradients decreas surface mobility. Bubbles in contaminated water rise slower. 5. Measurement Techniques There are many aspects to bubble hydrodynamics measurements. Bubble hydrodynamics measurements generally consist of making single bubbles through a capillary tube, and video imaging (high speed or normal video camera) followed by computer digitization and analysis. Perhaps the most important is to control for surfactants, either by using distilled water, by using a known dilute surfactant solution (e.g., sodium dodecyl sulfate) or choosing a specific water (such as lake, sea, etc.). Also important is to avoid bubble-bubble interactions due to producing the bubbles too fast. It may take 2 to 5 seconds for secondary fluid motions from even a single bubble to disappear (depending upon size). Another issue of potential concern (particularly for higher T) is gas presaturation, otherwise the bubbles will rapidly grow due to water vapor, and the size of the bubbles at release and several centimeters above may be different by several percent. The previous paragraph only briefly discussed some of the many issues surrounding these measurements. Many details are presented in a Leifer et al. (2003) - bubble measurement system paper, while others are presented in a paper on the temperature dependency of the Leifer et al. (2000) - bubble rise velocity as well as on the bubbleology website including figures and images not included in the aforemention papers.