Finite-stroke coating profile ($H_{\infty }(z)=C_0(z-z_0{)}^{4/3}(0<z-z_0<1)$)

For the leading transport model $H_t+(1+t{)}^{-2}{H}_z=0$, initially dry coating above $z_0$ and matching boundary value $H(z_0,t)=C_0(1+t{)}^{-4/3}$, characteristics preserve $\xi =z+1/(1+t)$. Boundary entry at time $\tau$ gives $F(\xi )=C_0(\xi -z_0{)}^{4/3}$ for $z_0<\xi \le z_0+1$; the remaining initial-data branch is zero. The coated front is $z-z_0=t/(1+t)$. Its final profile is the displayed formula below unit stroke and zero above it. The final scaled film volume is $3C_0/7$, equal to the time-integrated boundary supply. This is a result within the leading transport approximation, not a uniform infinite-time assertion for every finite capillary parameter.