Bending the edge of a thin elastic material promotes rigidity far from its clamped boundary. However, this curvature-induced rigidity can be overwhelmed by gravity or other external loading, resulting in elastic buckling and large deformations. We consider the role of body geometry on this competition using experiments, numerical simulations, and reduced-order models. Finite element simulations are performed using a model nonlinear hyperelastic material, and a theoretical framework is proposed that incorporates small lateral curvatures, large longitudinal rotations, and a varying cross-sectional width. A particular focus is on the comparison between rectangular and triangular sheets, and trapezoidal sheets in between. Sheet geometry affects downward tip deflection by changing the relative importance of the sheet's weight and the rigidity provided by curvature, often in subtle ways. In extreme cases, non-monotonic deflection is observed with increasing sheet length, and a region of hysteretic bistability emerges, becoming more pronounced with rectangular sheets and large imposed curvatures. These findings demonstrate the profound impact of geometry on the competition between curvature-induced rigidity and gravity-induced deformation in thin elastic materials.
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Y. Sun, J.W.M. Bush, S.E. Spagnolie, and C.H. Rycroft, The hydrodynamics of marbling art, Physical Review Fluids (2024).
A body immersed in a supersaturated fluid like carbonated water can accumulate a dynamic field of bubbles upon its surface. The bubbles grow and coalesce as gas is continually pulled from the environment. If the body is mobile, the attached bubbles can lift it upward against gravity, but arrival at a free surface can clean the body of these lifting agents and the body may plummet. The process then begins anew, and continues for as long as the concentration of gas in the fluid supports it. In this work, experiments using fixed and free immersed bodies reveal fundamental features of force development. A continuum model which incorporates the dynamics of a surface buoyancy field is used to predict the ranges of body mass and size, and fluid properties, for which the system is most dynamic, and those for which body excursions are suppressed. And simulations are used to probe systems which are dominated by a small number of large bubbles. Body rotations at the surface are found to be critical for driving periodic vertical motions of large bodies, which in turn can produce body wobbling, rolling, and damped surface 'bouncing' dynamics. The body affects the environment as well, modifying gas transport, as evidenced by a much longer fluid degassing time when the body is present.
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S. E. Spagnolie, S. Christianson and C. Grote, Levitation and dynamics of bodies in supersaturated fluids, Nature Communications (2024).
Many fundamental questions remain unanswered about the sedimentation of bodies in viscous fluids. For instance, even the dynamics of a single flexible filament have only recently been analyzed, and the interactions of viscous and elastic stresses can lead to slow shape changes or rapid buckling dynamics, as characterized by a dimensionless elasto-gravitation number. The dynamics of suspensions of flexible bodies has only just begun to receive mathematical attention, and even then only for weakly flexible filaments. Even two rigid sedimenting particles can undergo complex periodic sedimentation dynamics, so the general case is far from being completely characterized. The dynamics of flexible bodies in viscous flows remains both beautiful and analytically challenging, and is a topic of considerable practical interest.
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