[Appendix 2.5] Bubble collapse time, tm

(dynamically controlled region)

From Rayleigh Equation : [eqn: rm ρ_l over g_c ~[``R ``ddot R ~+~ 3 over 2 ``dot R ^2 ``] ~=~ - P'_l]

∴[eqn: rm d over dt ``(`` R^3 ``dot R ^2 ``) ~ =~ - {2 ``g_c ``P'_l } over ρ_l ``R^2 ``dot R] …………………………………	(1)

Using I.C [eqn: rm CASES{ LPILE {dot R `(`0`) ~=~ 0} # {R`(`0`) ~=~ R} }] , integrate equation (1)

:[eqn: rm R^3 dot R^2 - 0 = - {2 g_c } over {ρ_l} P' _l int _o ^t R^2 dR over dt dt], where, [eqn: rm R^2 dR over dt = 1 over 3 dR^3 over dt]

∴[eqn: rm R^3 dot R^2 ~-~ 0 ~=~ {2`` g_c } over {3`` ρ_l} ``P' _l ~[R_o ^3 ~-~ R^3 ]]……………………………	(2)

Let [eqn: rm R^+ = R(t) over R_o]

:[eqn: rm (R^+ ) ^{3over2} R_o ```LEFT ( dR^* over dt RIGHT ) ~=~ LEFT ( {2``g_c ``P'_l } over 3ρ_l RIGHT ) ^{1over2} `` root { 1 - (R^+ ) ^3 }]

Integrating Equation (2),

∴ [eqn: rm t over R_o = root { {3 ρ_l} over { 2 g_c P'_l } } INT _{ 1}^{R^+ } {(r^* ) ^{3over2} } over root { 1 - (r^+ ) ^3 } dr^+]

When [eqn: rm t = t_m], [eqn: rm R^+ (t_m ) = 0]

:[eqn: rm t_m over R_o ~ =~root { ρ_l over {6``g_c ``P'_l} } ```` {Γ ( 1over2 ) `Γ ( 5 over 6 ) } over {Γ ( 4 over 3 ) }]

[eqn: rm t _m over R_o ~≃~ 0.915 ```root { ρ_l over { 6 ``g_c ``P'_l } }]