= Finite-stroke coating profile
{title2=$H_\infty(z)=C_0(z-z_0)^{4/3}\quad(0<z-z_0<1)$}
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 $z_0<\xi\leq z_0+1$; 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.
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