11.3 Bubble dynamics

Bubble dynamics is a fascinating aspect of fluid mechanics, exploring how gas pockets form, grow, and move within liquids. From bubble formation to coalescence, this topic delves into the physics behind everyday phenomena like boiling water and carbonated drinks.

Understanding bubble behavior is crucial for many engineering applications, including heat transfer, cavitation, and chemical reactors. By examining factors like surface tension, pressure, and fluid properties, we can predict and control bubble dynamics in various systems.

Bubble formation and growth

• Bubble formation and growth are fundamental processes in fluid dynamics, involving the creation and expansion of gas or vapor bubbles within a liquid medium
• Understanding bubble formation and growth is crucial for various applications, such as boiling heat transfer, cavitation, and chemical reactors
• The process of bubble formation and growth is governed by complex interplay of surface tension, pressure, and heat transfer

Nucleation sites

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• Nucleation sites are preferential locations where bubbles tend to form and grow
• Nucleation can occur homogeneously within the bulk liquid or heterogeneously at solid surfaces, impurities, or pre-existing gas pockets
• Surface roughness, crevices, and hydrophobic surfaces act as favorable nucleation sites by reducing the energy barrier for bubble formation
• The presence and distribution of nucleation sites significantly influence the bubble formation process and the resulting bubble size distribution

Bubble growth rate

• Once a bubble nucleates, its growth rate depends on various factors such as the surrounding liquid properties, temperature, and pressure
• Bubble growth can be driven by heat transfer, where the bubble expands due to the evaporation of the liquid at the bubble interface
• The growth rate is also influenced by the diffusion of dissolved gases into the bubble, which can contribute to its expansion
• In the early stages of growth, the bubble radius typically follows a square root dependence on time, known as the $R \propto t^{1/2}$ growth law

Factors affecting bubble growth

• Bubble growth is affected by several physical properties of the liquid, including viscosity, surface tension, and thermal conductivity
• Higher liquid viscosity tends to slow down bubble growth by exerting drag forces on the expanding bubble
• Surface tension acts to minimize the bubble surface area and opposes bubble growth, especially for small bubbles
• Thermal conductivity of the liquid determines the rate of heat transfer to the bubble interface, influencing the evaporation and growth processes
• External factors such as pressure fluctuations, flow velocity, and the presence of dissolved gases also impact bubble growth dynamics

Bubble shapes and oscillations

• Bubbles exhibit various shapes and undergo oscillations due to the interplay of surface tension, viscosity, and external forces
• The study of bubble shapes and oscillations is important for understanding bubble behavior, stability, and interactions in fluid systems
• Bubble shape and oscillation characteristics have implications for mass and heat transfer, acoustic properties, and bubble breakup

Spherical vs non-spherical bubbles

• In many cases, bubbles assume a spherical shape due to the minimization of surface energy by surface tension forces
• However, bubbles can deviate from a perfect sphere and exhibit non-spherical shapes under certain conditions
• Non-spherical bubbles can arise due to high bubble velocity, fluid flow, bubble-bubble interactions, or the presence of surfactants
• Ellipsoidal, cap-shaped, and skirted bubbles are examples of non-spherical bubble shapes observed in different flow regimes

Natural frequency of oscillations

• Bubbles have a natural frequency of oscillation that depends on their size, the properties of the surrounding liquid, and the ambient pressure
• The natural frequency of a spherical bubble is given by the Minnaert frequency: $f_0 = \frac{1}{2\pi R_0} \sqrt{\frac{3\gamma P_0}{\rho}}$, where $R_0$ is the equilibrium bubble radius, $\gamma$ is the specific heat ratio of the gas, $P_0$ is the ambient pressure, and $\rho$ is the liquid density
• Bubbles can be excited at their natural frequency by acoustic or pressure perturbations, leading to resonance and enhanced oscillations
• The natural frequency of oscillation plays a role in bubble acoustics, cavitation, and bubble-induced vibrations

Damping effects on oscillations

• Bubble oscillations are subject to damping mechanisms that dissipate energy and attenuate the oscillation amplitude over time
• Viscous damping arises from the shear stresses at the bubble-liquid interface and is more significant for smaller bubbles and higher liquid viscosities
• Thermal damping occurs due to heat transfer between the bubble and the surrounding liquid during compression and expansion cycles
• Acoustic radiation damping is caused by the emission of sound waves from the oscillating bubble, which carries energy away from the bubble
• The presence of surfactants or impurities on the bubble surface can also introduce additional damping effects by modifying the surface properties and rheology

Bubble rise and motion

• Bubbles in a liquid medium tend to rise due to the buoyancy force resulting from the density difference between the gas and the liquid
• The rise and motion of bubbles are influenced by various factors, including bubble size, shape, and the properties of the surrounding fluid
• Understanding bubble rise and motion is important for applications such as bubble columns, gas-liquid separators, and multiphase flow systems

Terminal velocity of rising bubbles

• As a bubble rises, it experiences a drag force that opposes its motion, and the bubble eventually reaches a terminal velocity when the drag force balances the buoyancy force
• The terminal velocity of a rising bubble depends on its size, shape, and the properties of the liquid (density and viscosity)
• For small bubbles in low Reynolds number flows, the terminal velocity is given by Stokes' law: $v_t = \frac{2}{9} \frac{(\rho_l - \rho_g) g R^2}{\mu}$, where $\rho_l$ and $\rho_g$ are the densities of the liquid and gas, $g$ is the gravitational acceleration, $R$ is the bubble radius, and $\mu$ is the liquid viscosity
• For larger bubbles and higher Reynolds numbers, the drag coefficient and bubble shape play a significant role in determining the terminal velocity

Drag coefficient and bubble shape

• The drag coefficient quantifies the resistance experienced by a rising bubble due to the surrounding fluid
• The drag coefficient depends on the bubble shape, which can vary depending on the bubble size and the flow conditions
• For spherical bubbles, the drag coefficient is a function of the Reynolds number and can be estimated using empirical correlations (Schiller-Naumann, Clift-Grace-Weber)
• Non-spherical bubbles, such as ellipsoidal or cap-shaped bubbles, have different drag coefficients compared to spherical bubbles
• The relationship between drag coefficient and bubble shape is important for accurate modeling and prediction of bubble rise velocities

Bubble trajectory and path instability

• As bubbles rise, they may exhibit different trajectories and path instabilities depending on the flow conditions and bubble characteristics
• In stagnant or low-velocity flows, bubbles tend to follow a straight vertical path due to the balance between buoyancy and drag forces
• At higher velocities or in the presence of shear flows, bubbles can undergo lateral motion and exhibit zigzag or helical trajectories
• Path instability can arise from the interaction between the bubble wake and the surrounding fluid, leading to vortex shedding and oscillatory motion
• Bubble trajectory and path instability have implications for bubble dispersion, mixing, and mass transfer in multiphase flow systems

Bubble coalescence and breakup

• Bubble coalescence and breakup are important phenomena in multiphase flows, affecting the bubble size distribution and overall system behavior
• Coalescence occurs when two or more bubbles collide and merge to form a larger bubble, while breakup involves the fragmentation of a bubble into smaller ones
• Understanding the mechanisms and factors influencing bubble coalescence and breakup is crucial for controlling and optimizing multiphase processes

Bubble collision and coalescence

• Bubble coalescence is initiated by the collision of two or more bubbles, which can occur due to various mechanisms such as turbulence, wake interactions, or buoyancy-driven motion
• Upon collision, the thin liquid film between the bubbles must drain and rupture for coalescence to occur
• The coalescence efficiency depends on factors such as bubble size, relative velocity, liquid properties (viscosity, surface tension), and the presence of surface-active agents
• Coalescence can be enhanced by high bubble concentration, low liquid viscosity, and the absence of surface-active impurities that stabilize the liquid film

Mechanisms of bubble breakup

• Bubble breakup occurs when the disruptive forces acting on a bubble overcome the cohesive forces holding it together
• The main mechanisms of bubble breakup include turbulent fluctuations, shear stresses, and interfacial instabilities
• In turbulent flows, bubble breakup is caused by the interaction of bubbles with turbulent eddies, leading to deformation and fragmentation
• Shear-induced breakup occurs when the shear stresses exerted by the surrounding fluid exceed the surface tension forces, causing the bubble to elongate and eventually break
• Interfacial instabilities, such as Rayleigh-Taylor or Kelvin-Helmholtz instabilities, can also lead to bubble breakup when the interface becomes sufficiently distorted

Influence of fluid properties

• The fluid properties, such as viscosity, surface tension, and density, play a significant role in bubble coalescence and breakup processes
• Higher liquid viscosity tends to promote coalescence by slowing down the drainage of the liquid film between colliding bubbles
• Surface tension acts as a cohesive force, resisting bubble deformation and breakup
• The viscosity ratio between the dispersed phase (bubble) and the continuous phase (liquid) influences the breakup behavior and the resulting bubble size distribution
• The presence of surface-active agents (surfactants) can modify the interfacial properties, affecting coalescence and breakup rates

Heat and mass transfer

• Bubbles play a crucial role in heat and mass transfer processes in various engineering applications, such as boiling heat transfer, gas-liquid reactors, and bubble columns
• The presence of bubbles enhances heat and mass transfer rates by increasing the interfacial area and promoting mixing and turbulence
• Understanding the mechanisms of heat and mass transfer across bubble interfaces is essential for optimizing and controlling these processes

Heat transfer across bubble interface

• Heat transfer occurs between the bubble and the surrounding liquid due to the temperature difference across the bubble interface
• The main mechanisms of heat transfer include conduction, convection, and latent heat transfer (evaporation or condensation)
• Conductive heat transfer occurs through the thin thermal boundary layer surrounding the bubble, driven by the temperature gradient
• Convective heat transfer is enhanced by the motion of the bubble and the associated liquid circulation, which promotes mixing and reduces the thermal boundary layer thickness
• Latent heat transfer involves the evaporation of liquid at the bubble interface during bubble growth and the condensation of vapor during bubble collapse, leading to significant heat exchange

Mass transfer and gas diffusion

• Mass transfer occurs between the bubble and the surrounding liquid due to the concentration difference of dissolved gases or other species
• Gas diffusion plays a crucial role in bubble dynamics, as it governs the transfer of gas molecules across the bubble interface
• The rate of mass transfer is influenced by the concentration gradient, the interfacial area, and the diffusion coefficient of the gas in the liquid
• Bubble growth can be driven by the diffusion of dissolved gases into the bubble, while bubble dissolution occurs when the gas concentration in the liquid is below the equilibrium value
• Mass transfer across bubble interfaces is important in applications such as gas-liquid reactions, bubble columns, and wastewater treatment

Bubble collapse and micromixing

• Bubble collapse is a rapid and violent process that occurs when the pressure surrounding the bubble suddenly increases or the bubble encounters a high-pressure region
• During collapse, the bubble can generate high local temperatures and pressures, leading to intense heat and mass transfer
• Bubble collapse can also induce strong liquid jets and shock waves, which enhance mixing and turbulence in the surrounding fluid
• The collapse of bubbles near solid surfaces can lead to cavitation erosion and damage due to the high-velocity liquid jets and pressure waves
• Bubble collapse-induced micromixing is exploited in applications such as sonochemistry, where the extreme conditions generated during collapse are used to initiate chemical reactions or enhance mixing at the microscale

Cavitation and bubble dynamics

• Cavitation is the formation and subsequent collapse of vapor bubbles in a liquid due to rapid changes in pressure
• Cavitation occurs when the local pressure in a liquid falls below its vapor pressure, leading to the nucleation and growth of vapor bubbles
• The study of cavitation and bubble dynamics is crucial for understanding and predicting the behavior of bubbles in various engineering applications, such as hydraulic systems, propellers, and turbomachinery

Cavitation inception and bubble formation

• Cavitation inception refers to the initial formation of vapor bubbles in a liquid when the local pressure drops below the vapor pressure
• Cavitation can be initiated by various mechanisms, such as hydrodynamic cavitation (due to pressure variations in flowing liquids), acoustic cavitation (induced by high-intensity sound waves), or optic cavitation (triggered by high-energy laser pulses)
• The inception of cavitation depends on factors such as the liquid properties, the presence of nucleation sites (e.g., impurities, dissolved gases), and the magnitude and duration of the pressure drop
• Once cavitation bubbles are formed, they can grow rapidly due to the evaporation of the surrounding liquid, leading to the development of cavitation clouds or cavitation bubble clusters

Rayleigh-Plesset equation

• The Rayleigh-Plesset equation is a fundamental equation that describes the dynamics of a spherical bubble in an infinite liquid medium
• The equation relates the bubble radius to the pressure difference across the bubble interface, taking into account the effects of surface tension, viscosity, and the surrounding pressure field
• The Rayleigh-Plesset equation is given by: $\rho (R \ddot{R} + \frac{3}{2} \dot{R}^2) = P_b - P_\infty - \frac{2\sigma}{R} - \frac{4\mu \dot{R}}{R}$, where $R$ is the bubble radius, $\rho$ is the liquid density, $P_b$ is the pressure inside the bubble, $P_\infty$ is the far-field pressure, $\sigma$ is the surface tension, and $\mu$ is the liquid viscosity
• The equation captures the growth and collapse phases of a cavitation bubble, including the effects of inertia, pressure, surface tension, and viscous damping
• The Rayleigh-Plesset equation forms the basis for modeling and simulating cavitation bubble dynamics in various applications

Cavitation damage and erosion

• Cavitation can cause significant damage and erosion to solid surfaces exposed to cavitating flows
• Cavitation damage occurs when collapsing bubbles near a solid surface generate high-velocity liquid jets and intense pressure waves that impact the surface
• The repeated impact of these liquid jets and pressure waves can lead to material removal, pitting, and surface erosion over time
• Cavitation erosion is a major concern in hydraulic machinery, such as pumps, valves, and turbines, where it can reduce efficiency, cause component failure, and shorten equipment lifespan
• The extent of cavitation damage depends on factors such as the bubble size, collapse intensity, material properties, and the frequency and duration of cavitation events
• Strategies to mitigate cavitation damage include designing components to minimize cavitation inception, using cavitation-resistant materials, and implementing flow control techniques to suppress cavitation

Acoustic and ultrasonic cavitation

• Acoustic cavitation refers to the formation and dynamics of bubbles in a liquid medium under the influence of high-intensity sound waves (ultrasound)
• Ultrasonic cavitation is widely used in various applications, such as cleaning, sonochemistry, and materials processing, due to its ability to generate intense localized effects and enhance physical and chemical processes
• The study of acoustic and ultrasonic cavitation involves understanding the interaction between sound waves and bubbles, as well as the resulting cavitation phenomena and their effects

Acoustic cavitation threshold

• The acoustic cavitation threshold is the minimum acoustic pressure amplitude required to initiate cavitation in a liquid
• The threshold depends on factors such as the liquid properties (e.g., vapor pressure, surface tension), the frequency of the sound waves, and the presence of pre-existing nucleation sites
• Liquids with higher vapor pressures and lower surface tensions generally have lower cavitation thresholds
• The cavitation threshold is also influenced by the frequency of the sound waves, with higher frequencies typically requiring higher pressure amplitudes to induce cavitation
• Knowledge of the acoustic cavitation threshold is important for designing and optimizing ultrasonic systems and processes

Ultrasonic bubble dynamics

• Ultrasonic bubble dynamics refers to the behavior and characteristics of bubbles generated by high-intensity ultrasound
• Under the influence of ultrasonic waves, bubbles undergo rapid expansion and compression cycles, leading to highly nonlinear and transient cavitation phenomena
• The dynamics of ultrasonic bubbles are governed by the acoustic pressure field, the bubble size, and the properties of the liquid medium
• Ultrasonic bubbles can exhibit various behaviors, such as stable cavitation (sustained oscillations), transient cavitation (rapid growth and violent collapse), and acoustic streaming (fluid motion induced by the oscillating bubbles)
• The collapse of ultrasonic bubbles can generate high local temperatures and pressures, leading to the formation of hot spots and the emission of shock waves and liquid jets

Applications in cleaning and processing

• Ultrasonic cavitation has found widespread applications in cleaning and processing industries due to its ability to generate intense physical and chemical effects
• In ultrasonic cleaning, the collapse of cavitation bubbles near solid surfaces generates high-velocity liquid jets and shock waves that dislodge and remove contaminants and debris
• Ultrasonic cavitation enhances the cleaning efficiency by providing a mechanical scrubbing action and increasing the penetration of cleaning agents into crevices and small spaces
• In sonochemistry, the extreme conditions generated during bubble collapse (high temperatures and pressures) are used to initiate and accelerate chemical reactions, leading to the formation of unique products or the enhancement of reaction rates
• Ultrasonic processing is employe