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Write the Cauchy stress tensor in the local fibre frame as $\sigma =se\otimes e+h(e\otimes n+n\otimes e)+tn\otimes n$. The compatible geometry has $\nabla \theta =\kappa e$, $\nabla \cdot e=0$ and $\nabla \cdot n=-\kappa$. With zero body force, force balance becomes $e\cdot \nabla s=2\kappa h-n\cdot \nabla h$ and $n\cdot \nabla t-\kappa t=-e\cdot \nabla h-\kappa s$. For specified shear stress $h$, the first equation determines $s$ along fibres, then the second determines $t$ along transverse curves. Pressure and fibre tension supply these two normal constraint reactions. This is a local stress construction; boundary tractions must be chosen consistently with its integration data.

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 a nonadhesive obstacle, $g$ is the signed separation and $r$ its outward normal contact force. Nonpenetration gives $g\ge 0$, absence of adhesion gives $r\ge 0$, and reaction is possible only at contact, giving $gr=0$. In a small-slope taut-string model, $g=w-f$ and force balance gives $r=p-T{w}^{\prime \prime }$ for downward applied load $p$. The three conditions then become $(w-f)(T{w}^{\prime \prime }-p)=0$, $w\ge f$, $T{w}^{\prime \prime }-p\le 0$. Multiplication by admissible displacement differences and integration gives the associated variational inequality, connecting a local contact law to constrained energy minimization.

Viscous-sheet drawing stretches a thin ribbon between rollers. Mass conservation, axial force balance, lateral surface tension, and edge kinematics jointly determine its half-thickness, half-width, and axial speed.

An axial force balance on fluid between two remote cross-sections of a cylindrical tube giveswhen every immersed body between the sections is force-free.