A general theory for the dynamics of thin viscous sheets

Abstract

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 T0 and principal curvatures k1 and k2, loaded by an harmonic normal stress with wavenumbers q1 and q2 in the directions of principal curvature. Two distinct types of deformation can occur: an ‘inextensional’ (bending) mode when [mid ]L3(k1q22 + k2q21)[mid ] [Lt ] ε, and a ‘membrane’ (stretching) mode when [mid ]L3(k1q22 + k2q21)[mid ] [Gt ] ε, where L ≡ (q21 + q22)−1/2 and ε = T0/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 ∼ (εL)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.