Wrinkling instability of an inhomogeneously stretched viscous sheet

Abstract

Motivated by the redrawing of hot glass into thin sheets, we investigate the shape and stability of a thin viscous sheet that is inhomogeneously stretched in an imposed non-uniform temperature field. We first determine the associated base flow by solving the long-timescale stretching flow of a flat sheet as a function of two dimensionless parameters: the normalized stretching velocity α, and a dimensionless width of the heating zone β. This allows us to determine the conditions for the onset of an out-of-plane wrinkling instability stated in terms of an eigenvalue problem for a linear partial differential equation governing the displacement of the midsurface of the sheet. We show the sheet can become unstable in two regions that are upstream and downstream of the heating zone where the minimum in-plane stress is negative. This yields the shape and growth rates of the most unstable buckling mode in both regions for various values of the stretching velocity and heating zone width. A transition from stationary to oscillatory unstable modes is found in the upstream region with increasing β while the downstream region is always stationary. We show that the wrinkling instability can be entirely suppressed when the surface tension is large enough relative to the magnitude of the in-plane stress. Finally, we present an operating diagram that indicates regions of the parameter space that result in a required outlet sheet thickness upon stretching, while simultaneously minimizing or suppressing the out-of-plane buckling; a result that is relevant for the glass redraw method used to create ultrathin glass sheets.

Interplay of elastic instabilities and viscoelasticity in the finite deformation of thin membranes

“…All three terms on the right-hand side can be readily identified with those appearing in Eq. (7). In other words, in this appropriate curvilinear system, the evolution of the distribution tensor degenerates to the simple component equation:…”

“…For example, the thinning of a stretched rubber membrane affects its internal pressure [5] and the electric field across it [6]. Surface wrinkles appear due to a competition between curvature and compressive strains [7], and cellular blebbing [8] is caused due to a competition between adhesion and internal pressure. These and other instabilities are the keys to understand the mechanics of greater problems such as the vesicle transport in porous media [9,10], or the electroporation in animal cells [11].…”

How viscous bubbles collapse: topological and symmetry-breaking instabilities in curvature-driven hydrodynamics

“…( 3), is absent from the linear stress-strain relation of elastic sheets, reflecting the lack of target metric of liquid films and the associated role of isotropic tensile stress in retaining their thermodynamic stability [2]. This realization of SRA underlies the viscous hydrodynamics of planar extensional flows [8,[22][23][24] as well as the Buckminster-Nachman-Ting equations for planar and non-planar viscous flow of one-dimensional threads [25,26]. Furthermore, SRA remains valid for an adiabatic depressurization process, T 1, where the RHS of Eq.…”

Symmetrization of Thin Free-Standing Liquid Films via Capillary-Driven Flow

“…At macroscopic scales, a viscous sheet experiences bending instabilities such as wrinkling [17][18][19][20], and folding [21] when submitted to compressive forces. Such viscous buckling phenomena occur in various contexts, like tectonic-plate dynamics [22,23] and industrial float-glass processes [24][25][26].…”

“…Then, the mid-plane line H = h anti /2 follows the equation 1 3 ηh 3 0 ∇ 4 ∂ t H = δP . This equation corresponds to the torque balance in the liquid film [15,25,26], and is the viscous analogue of the Föppl-von Kármán equation for an incompressible elastic membrane in pure bending, where the bending modulus is replaced by ηh 3 0 /3, and the deflection field is replaced by the deflection rate ∂ t H.…”