Peter Stewart

University of Glasgow, Scotland

Lateral Strain and Stress Concentration in Liquid Foam Fracture

 

 

Joint work with Sascha Hilgenfeldt, University of Illinois

 

The rupture of successive films in a layer of liquid foam bubbles, resembling a brittle crack, has been understood theoretically as a self-similar dynamical feature growing out of fluid dynamical principles involving surface tension, nonlinear dissipation, and interfacial instability. The fracture mechanism is so robust that its features (such as a velocity gap) persist in reductive modeling including both 1D and 2D continuum limits. In solving this 2D continuum model it emerges that the stress and displacement fields ahead of and behind the crack tip can be expressed as an infinite series expansion of Fourier modes. These series expansions can be robustly matched together using orthogonality, facilitating an entirely analytical solution for the stress fields around the crack tip and elucidating a novel, width-dependent mechanism of lateral strain concentration. The resulting fracture dynamics and anisotropic stress fields are compared and contrasted with experiments, 2D simulations, and classical 2D continuum fracture mechanics.

 

Francesco De Anna

University of Würzburg, Germany

Quantitative Analysis for the Ill-Posedness of the Prandtl Equations 

 

The Prandtl theory of boundary layers has significantly impacted various scientific disciplines. However, despite its widespread use, the solutions to the underlying equations exhibit inherent instability. Recent research by Gérard-Varet and Dormy showed that the equations are ill-posed in Sobolev spaces (without making structural assumptions on the flow). Around a suitable shear flow, they determined a dispersion relation on the frequencies, which was interpreted later on as ill-posedness in a more general class of functions (the so-called Gevrey-classes m, m > 2).  

 

This talk intends to present a recent result, obtained in collaboration with S. Scrobogna (University of Trieste) and J. Kortum (University of Würzburg). Through a quantitative analysis of instabilities and norm inflations, we demonstrate that Gérard-Varet and Dormy's result only holds temporarily for ill-posedness in Sobolev spaces, leaving the ill-posedness in Gevrey classes as an open mathematical problem. We also explore related issues concerning the ill-posedness of a meaningful extension of the Prandtl Equations. 

Linda Cummings

New Jersey Institute of Technology, USA

Thermal effects and dewetting in nanoscale molten metal films