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 \L-3(k(1)q(2)(2) + k(2)q(1)(2))\ much less than epsilon, and a 'membrane' (stretching) mode when \L-3(k(1)q(2)(2) + k(2)q(1)(2))\ much greater than epsilon, where L equivalent to (q(1)(2) + q(2)(2))(-1/2) and epsilon = T-0/L much less than 1. The scales revealed by the shallow-sheet solution together with asympotic expansions in powers of epsilon 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 similar to (epsilonL)(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.