Viscous sheet

A viscous sheet is a thin fluid layer whose motion is controlled by viscous fluid flow and whose two broad faces can move freely. Unlike a no-slip wall film, an extensional sheet with shear-free faces has leading tangential velocity nearly uniform across its thickness. The in-plane stresses depend on extension in both horizontal directions.

A symmetric thin viscous sheet with shear-free faces has leading longitudinal velocity independent of thickness coordinate $z$, and incompressibility gives $w=-z{u}_x$. Young–Laplace equation and the Newtonian fluid stress tensor give ${\sigma }_{zz}=-p_a+\gamma h_{xx}/2$ and ${\sigma }_{xx}=-p_a+\gamma h_{xx}/2+4\mu u_x$. The effective longitudinal tension per unit transverse width is $\mathcal{T}=4\mu h{u}_x+\gamma h{h}_{xx}/2-\gamma h_x^2/4+2\gamma$. The last two terms include the horizontal projections of the two surface tension pulls. Force balance gives ${\mathcal{T}}_x=0$, yielding the displayed longitudinal equation; the face kinematic conditions give the displayed conservation of mass equation.

For a thin planar viscous sheet with zero effective gravity and prescribed tensile force $F(t)$ per unit transverse width, force balance gives $4\mu h{w}_z=F(t)$. Conservation of mass gives $D_{t}h=-h{w}_z=-F/(4\mu )$, proving uniform subtraction from the initial material thickness. The material map satisfies $dz/d{z}_0=h_0/[h_0-\Delta ]$. Thus the same material thinning creates greater extension wherever the initial sheet is thinner.

For initial thickness $H(1+k^2{z}_0^2)$ and constant positive force, the material-map integral is dominated by a shrinking region around the minimum thickness as $\Delta \uparrow H$. With $k\ne 0$, the integral of $1/[H-\Delta +H{k}^2{z}_0^2]$ gives the displayed inverse-square-root divergence, whose coefficient is independent of the initial half-length. If $k=0$, every material point thins equally and $L=H{L}_0/(H-\Delta )$, with the stronger inverse-time divergence $L\sim 4\mu H{L}_0/[F(t^{\ast }-t)]$. These are ideal slender-sheet predictions up to their singular limit.

Two cylindrical bubbles separated by a uniform flat liquid film of half-length $L$ drain through short curvature-transition regions at the ends. The elevated gas pressure implies a film liquid pressure approximately $\gamma /a$ above the outer liquid pressure. The transition length is $\delta \sim \sqrt{a{h}_0}$, and its volume flux is nearly constant because $\delta /L\ll 1$. Write $uh=U{h}_0$ and $\xi =(x-L)/\delta$. The planar viscous-sheet stretching equations reduce to $h{h}^{\prime \prime \prime }=C(\log h{)}^{\prime \prime }$, with $C=8\mu U{h}_0\delta /\gamma$. Two integrations, matched to a flat film at the inner end, give $h^{\prime }/\sqrt{h}=(16\mu U\sqrt{a}/(3\gamma ))[1-(h_0/h{)}^{3/2}]$. Matching the curvature to $1/a$ gives the displayed speed. Uniform-film conservation of mass then gives ${\dot{h}}_0=-U{h}_0/L$, hence $h_0=[h_i^{-1/2}+3\gamma t/(16\mu L\sqrt{a}){]}^{-2}$. There is no finite-time contact in this ideal continuum model.

For thickness $h(r,t)$, radial velocity $u(r,t)$ and external pressure $p_{\mathrm{ext}}$, the leading Newtonian fluid stress tensor and incompressibility giveRadial force balance on an annular sector includes the inward projection of hoop traction and external pressure on the sloping broad faces. Together with conservation of mass it yields

For external pressure $\Delta \rho gh$ and edge condition $h{\sigma }_{rr}=-\Delta \rho g{h}^2/2$, the fixed-volume similarity solution is a uniformly thick disc:Conservation of mass fixes the similarity velocity. The radial equation then forces a differentiable thickness profile to be constant, and the edge stress fixes that constant. The singular point-release solution describes the spreading scaling; a finite initial uniform disc corresponds to shifting time.

For a uniform annular sheet attached to a fixed outer rim $R_0$, two surface tension pulls give $h{\sigma }_{rr}=-2\gamma$ at the hole radius $R$. The zero-pressure radial equation gives $u=Ar+B/r$, with $B=-A{R}_0^2$ and $A=-\gamma R^2/[\mu h(R_0^2+3R^2)]$. The thickness remains uniform by conservation of mass. With negligible initial hole volume, $h=h_0/(1-x^2)$ andA positive seed hole grows initially exponentially. An exactly zero initial radius remains zero within this continuum evolution law.