Pressure-driven flow of a thin viscous sheet

Abstract

Systematic asymptotic expansions are used to find the leading-order equations for the pressure-driven flow of a thin sheet of viscous fluid. Assuming the fluid geometry to be slender with non-negligible curvatures, the Navier–Stokes equations with appropriate free-surface conditions are simplified to give a ‘shell-theory’ model. The fluid geometry is not known in advance and a time-dependent coordinate frame has to be employed. The effects of surface tension, gravity and inertia can also be incorporated in the model.

Models for thin viscous sheets

“…The shape of this surface is determined by the forces acting on the sheet (contrast, for example, a two-dimensional sheet placed under tension by pulling its ends apart which will straighten, with a twodimensional sheet acting under a pressure drop which will form a circular arc -see van de Fliert et al (1995) for details). Once it has adopted this shape, stretching will commence and the equations of section 4.3 will then apply.…”

“…Gravity, surface tension and non-constant viscosity can easily be incorporated into our models; this has been carried out by Howell (1994) and some details of the surface tension calculations are given in the appendix. The inclusion of inertia in the "viscous shell" equations of section 4.3 is also considered by Howell (1994) and van de Fliert et al (1995). In order to be applicable to diverse industrial processes, the theory could be extended to allow for coupled heat transfer, as well as more complicated rheology than incompressible Newtonian fluid; in principle our methods should still allow simplified leading-order equations to be derived.…”

“…Then our asymptotic analysis is based on the assumptions (equivalent to those of section 2.3) that the thickness of the sheet is of order L, while the curvature is of order L −1 , where L is a typical lengthscale. The leading-order equations are derived in detail in Howell (1994), and applied to the motion of a viscous sheet under an applied pressure drop in van de Fliert et al (1995). Denoting by u 1 and u 2 the velocity components in the e 1 -and e 2 -directions relative to the velocity of the centre-surface, the leading-order stresses in the sheet are given by…”

“…One aspect of this is that the form of the coordinate system is not known in advance. However, some analytical progress can be made when some symmetries of the geometry are preserved as the sheet evolves; examples of this may be found in Howell (1994) and van de Fliert et al (1995).…”

“…Models for axisymmetric viscous sheets have been derived by and Yarin, Gospodinov, & Roussinov (1994). A fully nonlinear model for the evolution of a three-dimensional sheet of arbitrary geometry has been derived by Howell (1994) and applied to the blowing of glass sheets by van de Fliert, Howell & Ockendon (1995).…”