Symmetric-hyperbolic PDEs for viscoelastic Maxwell flows
par
Sébastien Boyaval
→
Europe/Paris
3L15 (Laboratoire de Mathématiques d'Orsay)
3L15
Laboratoire de Mathématiques d'Orsay
Description
Many materials exhibit mechanical behaviours in between elastic solids and viscous fluids, in particular the various polymer suspensions that are often encountered in civil engineering, food inustry or biological applications. To model the (large) deformations of such non-Newtonian materials, i.e. to close the momentum balance equations (Navier-Stokes) with relevant stresses, various rheological laws have been proposed following Maxwell relaxation approach [4], which imply the use of upper-convected/lower-convected derivatives for an "extra" (viscoelastic) stress term [5]. But the numerical simulation of the resulting viscoelastic flows remains unreliable, in particular in the high-Weissenberg (i.e. purely elastic) limit [6] where the models are suspected unstable.
As a remedy, starting from the now well-established symmetric-hyperbolic formulation of elastodynamics, we showed in [1] that standard multidimensional relaxation laws (of Maxwell-type) could be (re)formulated in a mathematically-sound way. On enriching the well-established symmetric-hyperbolic formulation of elastodynamics with adequate internal variables, one can then define unequivocally some smooth solutions to multidimensional PDEs of Maxwell-type – see [1] for upper-convected Maxwell, and [2] for lower-convected derivatives. In comparison with the Godunov-Peshkov-Romenski (GPR) reformulation [3], ours is similar in spirit, but different in the choice of the internal variable.
In the talk, we will describe and discuss our reformulation of the upper- and lower-convected Maxwell laws.
[1] Sébastien Boyaval. Viscoelastic flows of Maxwell fluids with conservation laws. ESAIM Math. Model. Numer. Anal., 55(3):807–831, 2021.
[2] Sébastien Boyaval and Mark Dostalík. Non-isothermal viscoelastic flows with conservation laws and relaxation. J. Hyperbolic Differ. Equ., 19(2):337–364, 2022.
[3] Michael Dumbser, Ilya Peshkov, Evgeniy Romenski, and Olindo Zanotti. High order {ADER} schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids. Journal of Computational Physics, 314:824 – 862, 2016.
[4] James Clerk Maxwell. IV. On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London, 157:49–88, 1867.
[5] J. G. Oldroyd. On the formulation of rheological equations of state. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 200(1063):523–541, 1950.
[6] R. G. Owens and T. N. Philips. Computational rheology. Imperial College Press / World Scientific, 2002.