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Cavitation and Bubble Dynamics Ch.2 Spherical Bubble Dynamics

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Cavitation and Bubble Dynamics Rayleigh-Plesset Equation Case with no thermal effects Stability/Perturbation of RPE Various fluid regime effects on growth –Mass diffusion –Thermal Effects –Nonequllibrium –Convection –Surface Roughness –Nonspherical Perturbation

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Rayleigh-Plesset Equation Generalized equation for bubble growth: Introduce a contaminant gas with Rayleigh-Plesset Eqn with included effects of T B & T I) Instantaneous tension, driving term at II) Thermal term, assumed small to eliminate heat tranfer from bulk fluid

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Rayleigh-Plesset Equation Assumptions of the generalized equation: –Zero mass transport across liquid-gas interface –Newtonian Fluid –Small temperature difference T B -T –All heat diffused to bubble is used to vaporize fluid (no heating/cooling) –Thermal boundary layer is small compared to bubble radius –No dissipation mechanisms like viscosity –Isothermal vapor inside bubble –Incompressible fluid flow during growth –Spherical symmetry Approximated by: R* and n constants

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No Thermal Effects Assume: –T B -T =0 –Vapor inside bubble is polytropic Asymptotic solution leads to: With P* = P (t>0)

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Stability and Perturbation Equilibrium is reached with a balance of forces at the interface: P GE partial pressure of the gas Small perturbation theory: substitute Results: –Unstable growth because of assumption 1) Bubbles only grow or shrink if time for mass diffusion is allowed –For fas gas pressure changes: Some critical radius R c where bubble is in euillibrium

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Stability and Perturbation Figure: –Isothermal –Pc goes to 4S/3R –2 equilibrium states if subcritical All unstable nuclei grow to a maximum size regardless of initial size Growth approximated by –Maximum size only slightly dependant on cavitation number

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Fluid regime effects Thermal effects: –Thermal term, even if intially zero, becomes significant at some critical growth time with thermodynamic effect: Ex: water @ 20C with tension of 10 Pa –t c1 =10s Bubble growth is inertially controlled –@ 100C: t c1 =10 s Bubble growth is thermally controlled Growth rate decreases after t c1 –R~t 1/2

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Fluid Regime Effects Nonequillibrium Effects: Result in using the assumption that the liquid at interface the temperature of saturated vapor inside the bubble Mass flux across interface and evaporation have little effect on thermally controlled bubble growth Convective Effects: Heat transfer across interface is affected by convection caused by relative motion of bubble and liquid –Effects are significant at small superheat/tension levels at low Temperature Peclet number = ratio of heat convection to heat diffusion –If Pe>1 Buoyancy motion is a primary cause

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Fluid Regime Effects Surface Roughening: Rapid growth induces instabilities on the surface of the bubble –inreased surface area increases heat transfer across the interface –further delay the onset of thermally controlled growth Nonspherical Perturbation Harmonic oscillations can cause the radius to exceed critical radius Rc and undergo unstable growth, which leads to collapse As the bubble grows the wavelength of the surface increases, lessening the amplitude, the reverse happens in collapse If inadequate time is available for perturbation to grow during the phase of acceleration of the radius, the bubble will remain unperturbed.

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