Hamid A. Said | Searcharxiv

arXiv2025

Application of Onsager's Variational Principle to the Navier-Stokes Equations

In this note we propose a basic $L^2$-based approach for studying the global and local energy equalities of the incompressible 3D Navier-Stokes equations in the standard energy class on $\mathbb{T}^3 \times (0,T]$. Motivated by L. Onsager's principle of least dissipation of energy (1931), we give a new sufficient condition for the energy equalities in terms of the limit of a sequence of minimizers. In particular, we show that the equalities are attained if this limit is non-vanishing. We observe that the indeterminacy in the vanishing case is reminiscent of turbulence-driven energy transfer dominating other transport processes.

math.AP↗

arXiv2017

A Continuum Description of Failure Waves

Shattering of a brittle material such as glass occurs dynamically through a propagating failure wave, which however, can not be assigned to any of the classical waves of the elasto-plastic theories of materials. Such failure waves have been a topic of research for decades. In this paper, we build a thermodynamically consistent theory based on the idea that a failure wave is analogous to a deflagration wave. Our theory admits, as special cases, the classical models of Feng and Clifton. Two fundamental thermodynamic functions (the free energy and the entropy production rate) form the basis of our theory. Such a two-function approach allows for the construction of a new variational principle and a new Lagrangian formulation that produce the equations of motion. Finally, a linearization of this theory is examined to gain insight into the coupling between the diffusive and elastic wave phenomena.

math.AP↗