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Apply the Unsteady Bernoulli equation at $r=a(t)$ and $r=b(t)$ and subtract, thereby eliminating its time-dependent constant. For $A=a\dot{a}$ and $\varphi =A\log r+C(t)$,where each partial derivative is taken at fixed $r$. Since $u_r=A/r$ and $p(b,t)=p_{\infty }$,Equivalently, $\dot{A}={\dot{a}}^2+a\ddot{a}$. This is the pressure in radially symmetric annular potential flow.