Surface wrinkling of a channelized flow

Abstract

We consider an inclined rectangular duct of constant cross-section conveying viscous fluid and covered by an elastic plate. The fluid is described by the Stokes equations and the plate by the Föppl-von Kármán equations. The equations admit an equilibrium solution in which the plate is flat and fluid flows underneath due to gravity. This base flow induces a varying traction across the plate, which can lead to out-of-plane buckling due to the associated in-plane shear. Linear stability analysis demonstrates that buckling occurs for sufficiently thin plates on steep slopes and deep channels. The most unstable modes take the form of either symmetric or antisymmetric downslope-directed chevron patterns that travel downstream at a fraction of the average speed of the base flow. An analogous analysis shows that similar buckling also occurs if the elastic plate is replaced by a thin skin of very viscous fluid. Our description provides a simple model for the formation of ropy pahoehoe lava.

Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations

“…Elastohydrodynamic interactions of this type arise in applications ranging from geophysical phenomena such as the formation of laccoliths (Michaut 2011;Lister et al 2013) and gravity-driven surface lava flows under solidified crusts (Slim et al 2009;Hewitt et al 2015) to physiological processes. For instance, the formation and growth of blisters (Chopin et al 2008) involves a competition between elastic and adhesive forces.…”

Viscoplastic plates

“…[2][3][4][5][6][7]). The link between Stokes flow and linear elasticity further led to development of theories of viscous plates and shells, together with investigations of the fluid analogues of classical bending problems from solid mechanics [8][9][10][11][12][13][14][15][16]. More recently, the bending dynamics of two-dimensional sheets of viscoplastic fluid have been considered [17][18][19][20], combining aspects of all three of these physical problems.…”

Viscoelastic ribbons

“…The similarity between the mathematical formulations of linear elasticity and slow viscous flow led Taylor (1969) to suggest the fluid mechanical analogies of various classical problems from solid mechanics. Continuing in this vein, a number of studies have explored the buckling and folding of viscous sheets and the coiling and 'sewing' of liquid threads (Buckmaster, Nachman & Ting 1975;Ribe 2002;Teichman & Mahadevan 2003;Chiu-Webster & Lister 2006;Slim et al 2008;Ribe, Habibi & Bonn 2012;Slim, Teichman & Mahadevan 2012). The implications of this work range from understanding the everyday observations of honey or syrup falling on toast, the manufacture of glass fibres and sheets, to inferences about the fate of subducting slabs in the Earth's mantle.…”