Introduction Gas bubbles can grow or collapse in flow Introducing many important phenomena General assumptions in this chapter Flow far away from bubble is at rest -Spherical symmetric bubble Single bubble
Introduction • Gas bubbles can grow or collapse in flow • Introducing many important phenomena • General assumptions in this chapter – Flow far away from bubble is at rest – Spherical symmetric bubble – Single bubble
Rayleigh-Plesset equation Looking for a function for bubble radius R(t) ·Assumptions: Liquid temperature Too constant Liquid pressure Po()is a known input Liquid density pL constant Liquid viscosity constant and uniform Bubble is homogeneous,Ta(t)and pE()are uniform
Rayleigh-Plesset equation • Looking for a function for bubble radius • Assumptions: – Liquid temperature constant – Liquid pressure is a known input – Liquid density constant – Liquid viscosity constant and uniform – Bubble is homogeneous , and are uniform
Rayleigh-Plesset equation 。Conservation of mass -4(,)=F F()related to boundary condition of bubble -On the bubble surface,u(R,t)=dR/dt -good for vapor density is much smaller u(r,t) LIQUID p(r,t) FAR FROM BUBBLE T(r,t) Poo(t],Too ∠-R(t)+ VAPOR GAS Pe(t).Te(t] -BUBBLE SURFACE
Rayleigh-Plesset equation • Conservation of mass – related to boundary condition of bubble – On the bubble surface, – , good for vapor density is much smaller
Rayleigh-Plesset equation N-S equation in r direction 1∂p0u,0u pLar=t+“ar {品 ·Introducing u=F(t)/r2 1 Op 1 dF 2F2 PL Orr2 dt r5 ·Integrate to get p-Poo 1dF 1F2 PL r dt 2r4 Net force on the surface radially outward 2S (Orr)r=R+PB-R Orr =-p+2uLou/Or. AuL dR PB -(p)r-R-R dt 2S replacing (p),=R R pB(t)-p∞() d2R 3 dR 2 AvL dR 2S PL -2 dt R dt PLR
Rayleigh-Plesset equation • N-S equation in r direction • Introducing • Integrate to get • Net force on the surface radially outward • replacing
Bubble content effect Assume bubble contains non-condensable gas Partial pressure poo at reference size and temp. pa)=ma+o(是)(货) Te need to be determined,rewrite the R-P equation: pv(T)-Poe(t)pv(TB)-pv(To) PL PL ()() d2R 3 dR AvL dR 2S =R d2+2( +R dt PLR First term is driving term Thermal term,which is considered later
Bubble content effect • Assume bubble contains non-condensable gas – Partial pressure at reference size and temp. • TB need to be determined, rewrite the R-P equation: – First term is driving term – Thermal term, which is considered later
Bubble content Taylor expansion on the first term Pv(TB)-PV(Too)=A(TB-Too) PL -A can be derived from Clausius-Clapeyron relation A=1= Pv(To)L(Too) PL dT PLToo 。Determine(Ts-To) -First step:heat diffusion equation,relates (Ta-T(ar/an +()买-是() Second step:energy balance,relates (aT/ar),-R R(t). dR kL dt PyC r-R (TB-Too) R(t)
Bubble content • Taylor expansion on the first term – A can be derived from Clausius-Clapeyron relation • Determine – First step: heat diffusion equation, relates – Second step: energy balance, relates
Bubble content However,the heat diffusion equation is hard to solve analytically Approximated solution for RT。-IB(器)),R mo-()/r盟 du -Using results from former 人) [R(x)2 dx [fR4(g)d划 It can introduce to R-P equation
Bubble content • However, the heat diffusion equation is hard to solve analytically • Approximated solution – for – Using results from former – It can introduce to R-P equation
Bubble content Focusing on bubble size change,then R=Rt" R*and n are constants. 。Former equation T。-TBd=CovR--iCm) PLCPLD ca=n)产f Thermal term (Ta-T)t--(Ts )C(n) PLToo C2py ∑(T)= PiePLTooDE
Bubble content • Focusing on bubble size change, then • Former equation Thermal term
Bubble growth without thermal effect Inertially controlled behavior Gas behavior,simplified from former equation 100 ()() ·NoWf 80 Pv(1 2S 60 PLR Can etc. 40 -Give 20- 0 200 400 600 800 1000 DIMENSIONLESS TIME,U/R
Bubble growth without thermal effect • Inertially controlled behavior • Gas behavior, simplified from former equation • Now R-P equation – Can be solved numerically – Given constants and initial conditions