A general theory for the dynamics of thin viscous sheets

A model for the deformation of thin viscous sheets of arbitrary shape subject to arbitrary loading is presented. The starting point is a scaling analysis based on an analytical solution of the Stokes equations for the flow in a shallow (nearly planar) sheet with constant thickness T 0 and principal curvatures k 1 and k 2 , loaded by an harmonic normal stress with wavenumbers q 1 and q 2 in the directions of principal curvature. Two distinct types of deformation can occur: an ‘inextensional’ (bending) mode when [mid ] L 3 ( k 1 q 2 2 + k 2 q 2 1 )[mid ] [Lt ] ε, and a ‘membrane’ (stretching) mode when [mid ] L 3 ( k 1 q 2 2 + k 2 q 2 1 )[mid ] [Gt ] ε, where L ≡ ( q 2 1 + q 2 2 ) −1/2 and ε = T 0 / L [Lt ] 1. The scales revealed by the shallow-sheet solution together with asympotic expansions in powers of ε are used to reduce the three-dimensional equations for the flow in the sheet to a set of equivalent two-dimensional equations, valid in both the inextensional and membrane limits, for the velocity U of the sheet midsurface. Finally, kinematic evolution equations for the sheet shape (metric and curvature tensors) and thickness are derived. Illustrative numerical solutions of the equations are presented for a variety of buoyancy-driven deformations that exhibit buckling instabilities. A collapsing hemispherical dome with radius L deforms initially in a compressional membrane mode, except in bending boundary layers of width ∼ (ε L ) 1/2 near a clamped equatorial edge, and is unstable to a buckling mode which propagates into the dome from that edge. Buckling instabilities are suppressed by the extensional flow in a sagging inverted dome (pendant drop), which consequently evolves entirely in the membrane mode. A two-dimensional viscous jet falling onto a rigid plate exhibits steady periodic folding, the frequency of which varies with the jet height and extrusion rate in a way similar to that observed experimentally.