The viscous curtain: General formulation and finite-element solution for the stability of flowing viscous sheets

Perdigou, Claude; Audoly, Basile

doi:10.1016/j.jmps.2016.07.015

Cited by 4 publications

(5 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The shape of the eigenmodes we obtain, and the transition from a static to a Hopf bifurcation, is analogous to the curtain modes seen in the work of Perdigou & Audoly [19] for a falling viscous sheet. Similar to their results, we observe that the wavelength of the instability does not span the entire width of the sheet, and is instead localized in the compressive zones.…”

Section: B Linear Stabilitysupporting

confidence: 77%

See 1 more Smart Citation

Wrinkling instability of an inhomogeneously stretched viscous sheet

2017

View full text Add to dashboard Cite

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.

show abstract

Section: B Linear Stabilitysupporting

confidence: 77%

“…However, they do not solve for the out-of-plane deformation to determine the shape of the unstable modes. Also related is the work of Perdigou & Audoly [19], who investigate the problem of a falling viscous sheet under the action of gravity, and determine the stability and out-of-plane modes for a constant thickness and viscosity.…”

Section: Introductionmentioning

confidence: 99%

Wrinkling instability of an inhomogeneously stretched viscous sheet

2017

View full text Add to dashboard Cite

show abstract

“…In this sense, the transient network theory (TNT) [35,36] enables to obtain a statistical knowledge of the molecular processes leading to viscoelasticity in active networks [37,38] and is often used to investigate the physics behind non-newtonian fluids [39] and the solid-fluid transition. Taken in the context of shells and membranes, it might, therefore, impact our understanding of lipids mono-and bi-layers [40], viscous sheets [41], or polymeric membranes.…”

Section: Introductionmentioning

confidence: 99%

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

2019

View full text Add to dashboard Cite

Pneumatic structures and actuators are found in a variety of natural and engineered systems such as dielectric actuators, soft robots, plants and fungi cells, or even the vocal sac of frogs. These structures are often subjected to mechanical instabilities arising from the thinning of their crosssection and that may be harvested to perform mechanical work at a low energetic cost. While most of our understanding of this unstable behavior is for purely elastic membranes, real materials including lipid bilayers, elastomers, and connective tissues typically display a time-dependent viscoelastic response. This paper thus explores the role of viscous effects on the nature of this elastic instability when such membranes are dynamically inflated. For this, we first introduce an extension of the transient network theory (TNT) to describe the finite strain viscoelastic response of membranes; enabling an elegant formulation while keeping a close connection with the dynamics of the underlying polymer network. We then combine experiments and simulations to analyze the viscoelastic behavior of an inflated blister made of a commercial adhesive tape (VHB 4905). Our results show that the viscous component induces a rich spectrum of behaviors bounded by two well-known elastic solutions corresponding to very high and very low inflation rates. We also show that membrane relaxation may induce unwanted buckling when it is subjected to cyclic inflations at certain frequencies. These results have clear implications for the inflation and mechanical work performed by time-dependent pneumatic structures and instability-based actuators.

show abstract

“…Next, we substitute the expansion (27) for the pressure, use (30) for the strain rate , and use the approximation for the contravariant metric tensor…”

Section: Asymptotic Expansion: Membrane Limitmentioning

confidence: 99%

“…For viscous materials, however, only a few thin sheet models have been derived. Membrane models (in which bending resistance is neglected) have been developed in arbitrary shapes [26,27] but models accounting for bending resistance were mostly derived for specific geometries [28,29,30], with one exception in general curvilinear coordinates [31]. This last study constitutes the starting point of our active shell theory of cell dynamics.…”

Section: Introductionmentioning

confidence: 99%

A viscous active shell theory of the cell cortex

da Rocha,

Bleyer,

Turlier

2021

Preprint

View full text Add to dashboard Cite

The cell cortex is a thin layer beneath the plasma membrane that gives animal cells mechanical resistance and drives most of their shape changes, from migration, division to multicellular morphogenesis. Constantly stirred by molecular motors and under fast renewal, this material is well described by viscous and contractile active-gel theories.Here, we assume that the cortex is a thin viscous shell with non-negligible curvature and use asymptotic expansions to find the leading-order equations describing its shape dynamics, starting from constitutive equations for an incompressible viscous active gel. Reducing the three-dimensional equations leads to a Koiter-like shell theory, where both resistance to stretching and bending rates are present. Constitutive equations are completed by a kinematical equation describing the evolution of the cortex thickness with turnover. We show that tension and moment resultants depend not only on the shell deformation rate and motor activity but also on the active turnover of the material, which may also exert either contractile or extensile stress. Using the finite-element method, we implement our theory numerically to study two biological examples of drastic cell shape changes: cell division and osmotic shocks. Our work provides a numerical implementation of thin active viscous layers and a generic theoretical framework to develop shell theories for slender active biological structures.

show abstract

The viscous curtain: General formulation and finite-element solution for the stability of flowing viscous sheets

Cited by 4 publications

References 48 publications

Wrinkling instability of an inhomogeneously stretched viscous sheet

Wrinkling instability of an inhomogeneously stretched viscous sheet

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

A viscous active shell theory of the cell cortex

Contact Info

Product

Resources

About