Session Time & Room : 5B (Aug.25, 10:40-12:20) @G605
Type : Proposal of Minisymposium
Abstract : Since the beginning of continuum mechanics, the need to improve quantitative predictions of non-Newtonian flows continues.
The simulation of turbulence or of complex (non-homogeneous) fluids using good PDEs, in particular, remains an unsatisfied goal.
A major challenge is how to conciliate the conservation principles funding physics with quantitative observations.
A natural approach is to add dissipative relaxation terms in the hyperbolic PDEs resulting of conservation laws.
The goal of the minisymposium is to confront recent advances, with promising theoretical or numerical results, regarding hyperbolic PDEs plus relaxation sources for various non-Newtonian fluid flows.
[00332] Well-posedness and asymtotic behavior for hyperbolized compressible Navier-Stokes equations
Author(s) :
Yuxi Hu (China University of Mining and Technology, Beijing)
Abstract : We consider the non-isentropic compressible Navier-Stokes equations (CNS) for which the heat conduction of Fourier's law is replaced by Cattaneo's law and the classical Newtonian flow is replaced by a revised Maxwell flow. We shall present our recent results on global well-posedness, finite time blow-up and asymptotic behaviour of solutions. Some quality behaviour of solutions are shown to be changed, i.e., Global existence VS blowup in finite time, between the classical CNS and the studied hyperbolized model, although the solutions of two system are quite close to each other for small relaxation parameter.
[00503] Structure preserving finite element schemes for a non-Newtonian flow
Author(s) :
Gabriel R. Barrenechea (University of Strathclyde, Glasgow, UK)
Tristan Pryer (University of Bath)
Emmanuil Georgoulis (Heriot-Watt University)
Abstract : We propose a finite element discretisation of a three-dimensional non-Newtonian flow whose dynamics are described by an Upper Convected Maxwell model. The scheme preserves structure in the sense that the velocity is divergence-free and the overall discretisation is energy consistent with the underlying problem. We investigate the problem's complexity and devise relevant timestepping strategies for effcient solution realisation. We showcase the method with several numerical experiments, confirm the theory and demonstrate the efficiency of the scheme.
[00918] Temporal discretisation of non-Newtonian fluid flows
Author(s) :
Ben Ashby (Heriot-Watt University, Edinburgh, UK)
Tristan Pryer (University of Bath)
Gabriel Barrenechea (University of Strathclyde, Glasgow)
Emmanuil Georgoulis (Heriot-Watt University, UK & National Technical University of Athens, Greece)
Abstract : Choice of discretisation of the constitutive law for the stress in a non-Newtonian fluid is crucial for the success of any numerical method. We propose a new methodology for temporal discretisations of some non-Newtonian flows with the aim of preserving flow structure. We show that for models where both differential and integral constitutive laws are available, such as Oldroyd-B, a correspondence can be found between discretisations of both.
[04850] New symmetric-hyperbolic PDEs for viscoelastic fluid flows
Author(s) :
Sébastien Julien Boyaval (Ecole des Ponts ParisTech)
Abstract : Many Partial Differential Equations (PDEs) have been proposed to model viscoelastic flows, in between fluids and solids.
Seminal hyperbolic PDEs with stress relaxation have been proposed by Maxwell in 1867 to ensure propagation of 1D shear waves at finite-speed while capturing the viscosity of real fluid continua.
But actual computations of multi-dimensional viscoelastic flows using Maxwell's PDEs have remained limited,
at least without additional diffusion that blurs the hyperbolic character of Maxwell's PDEs.
We propose a new system of PDEs to model 3D viscoelastic flows of Maxwell fluids.
Our system, quasilinear and symmetric-hyperbolic, unequivocally models smooth flows on small times, while ensuring propagation of waves at finite-speed.
Our system rigorously unifies fluid models with elastodynamics for compressible solids,
and it can be extended for applications in environmental hydraulics (shallow-water flows) or materials engineering (non-isothermal flows).
