[Appendix 2.0] Derivations of Governing Equations for Bubble

Dynamics

(1) Continuity Equation

[eqn: dot{O~} -~ dot{I~} +~ dot{S~} =~0]

[eqn: ∴~~ 1 over r^2 partial over partialr (r^2 rho_l ~v_r )~ +~ partialrho_l over partialt ~=~0]

[eqn: 4pir^2 rho_l~ v_r~ &+~ partial over partialr (4pir^2 rho_l~ v_r )dr~ ##-~ 4pir^2 rho_l~~ v_r~ &+~ partial over partialt (4pir^2 rho_l~ dr)~ =~ 0]

(1)

(2) Momentum Equation

[eqn: -~~partialP/partialr ~=~ 1overg sub c~ rho subl~ Dv subr over Dt]

(Proof) [eqn: dot O~ +~ dot I~ +~ dot S~ =~ 0]

[eqn: 1 over g_c [(4pir^2 rho_l v_r) + partial over partialr (4pir^2 rho_l v_r )dr~ ] - 1 over g_c [(4pir^2 rho_l ) v_r~ ] + 1 over g_c partial over partialt (4pir^2 rho_l dr)]

[eqn: = 4pir^2 P ~-~[4pir^2 P + partial over partialr (4pir^2 P)dr] ~+~(P + 1 over 2 partialP over partialr dr) partial over partialr (4pir^2 ) dr]

[eqn: = -4pir^2 P partialP over partialr dr -8pir Pdr~+~8pirPdr ~+~ 8pi over 2 r(dr)^2 partialP over partialr]

Dividing with 4πr2dr

[eqn: 1 over g_c partial over partialt (rho_l v_r )+ 1 over g_c 1 over r^2 partial over partialr (rho_l r^2 v_r ^2 ) = partialP over partialr]

Expanding and apply the mass continuity

[eqn: 1 over g_c v_r [ partialrho_l over partialt + 1 over r^2 partial over partialr ( rho_l r^2 v_r )] + 1 over g_c rho_l ( partialv_r over partialt + v_r partialv_r over partialt )]

[eqn: =~ - partialP over partialr]

[eqn: ∴ - partialP over partialr ~=~ 1 over g_c rho_l Dv_r over Dt]

(2)

For incompressible and inviscid liquid, from Eq.(1) for steady state

[eqn: INT from R to r partial over partialr (r^2 u_r )dr ~=~0]

[eqn: r^2 u(r) - R^2 u(R) ~=~0]

[eqn: ∴~u(r) ~ =~ u(R) ( R over r ) ^ 2]