Quasisteady sliding collar on a coated cylinder

A large collar translating slowly at speed $c$ over a film $h_{+}$ and leaving a thinner film $h_{-}$ must grow: its leading excess volume increases at rate $c(h_{+}-h_{-})$. An exact travelling wave connecting unequal stationary films is impossible, since integration of the thin-film equation would give a common constant $q-ch=-c{h}_{+}=-c{h}_{-}$. In a slowly evolving collar core the leading pressure gradient vanishes. Therefore $h^{\prime }+h^{\prime \prime \prime }=0$, and a centred, reflection-symmetric collar has $h=A(1+\cos x)+B$ on $-\pi <x<\pi$. Its leading volume is $2\pi A$, so for constant small $B$, $2\pi \dot{A}\simeq c(h_{+}-h_{-})$. The speed is supplied by collar-to-film capillary matching.

Near a collar edge adjoining a film of thickness $h_f$, local mass conservation gives $h^3{h}^{\prime \prime \prime }=c(h-h_f)$. The scaling $h=h_{f}H$, $x-x_f=h_{f}X/c^{1/3}$ yields the displayed equation. Its linearization at $H=1$ has exponents $1$ and $-1/2\pm i\sqrt{3}/2$. The positive growing rear branch approaching one as $X\to -\infty$ has one amplitude, removable by translation; the front's decaying oscillatory branches have two amplitudes, leaving one parameter after translation. This dimension count describes local branches; global positivity and selection require matching. If their core asymptotes, after removing linear terms by translation, are $H_{-}\sim a_{-}{X}^2+b_{-}$ and $H_{+}\sim a_{+}(\lambda )X^2+b_{+}(\lambda )$, matching to the collar gives $A/2=a_{\pm }{c}^{2/3}/h_{\pm }$ and $B=b_{\pm }{h}_{\pm }$. Consequently $a_{+}{b}_{+}=a_{-}{b}_{-}$; a selected root with $\alpha =a_{+}({\lambda }_0)>a_{-}$ yields $c=(A{h}_{+}/(2\alpha ){)}^{3/2}$, $B=h_{+}{b}_{+}({\lambda }_0)$ and $h_{-}/h_{+}=a_{-}/\alpha$. Local linearization alone does not establish uniqueness of this global matching root.