レイリー・プレセット方程式
− 1 ρ L ∂ P ∂ r = ∂ u ∂ t + u ∂ u ∂ r − ν L [ 1 r 2 ∂ ∂ r ( r 2 ∂ u ∂ r ) − 2 u r 2 ] {\displaystyle -{\frac {1}{\rho _{L}}}{\frac {\partial P}{\partial r}}={\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial r}}-\nu _{L}\left[{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial u}{\partial r}}\right)-{\frac {2u}{r^{2}}}\right]}
ここで質量保存から求めた u ( r , t ) {\displaystyle u(r,t)} を代入し、
− 1 ρ L ∂ P ∂ r = 2 R r 2 ( d R d t ) 2 + R 2 r 2 d 2 R d t 2 − 2 R 4 r 5 ( d R d t ) 2 = 1 r 2 ( 2 R ( d R d t ) 2 + R 2 d 2 R d t 2 ) − 2 R 4 r 5 ( d R d t ) 2 {\displaystyle -{\frac {1}{\rho _{L}}}{\frac {\partial P}{\partial r}}={\frac {2R}{r^{2}}}\left({\frac {\mathrm {d} R}{\mathrm {d} t}}\right)^{2}+{\frac {R^{2}}{r^{2}}}{\frac {\mathrm {d} ^{2}R}{\mathrm {d} t^{2}}}-{\frac {2R^{4}}{r^{5}}}\left({\frac {\mathrm {d} R}{\mathrm {d} t}}\right)^{2}={\frac {1}{r^{2}}}\left(2R\left({\frac {\mathrm {d} R}{\mathrm {d} t}}\right)^{2}+R^{2}{\frac {\mathrm {d} ^{2}R}{\mathrm {d} t^{2}}}\right)-{\frac {2R^{4}}{r^{5}}}\left({\frac {\mathrm {d} R}{\mathrm {d} t}}\right)^{2}}