Section: New Results

Complex fluids

Participants : Sébastien Boyaval, Claude Le Bris, Tony Lelièvre.

In this field, the numerical analysis of discretizations of different models for non-Newtonian fluids has been pursued. We have considered:

• models using constitutive relations , like the Oldroyd-B model, in which the non-Newtonian behaviour of the fluid is accounted for at the same macroscopic scale as the flow by constitutive equations coupled to the Navier-Stokes equations, and

• micro-macro models, like dumbbell models, in which a kinetic description of particles diluted in the fluid explains the non-Newtonian behaviour of the suspension as a coupling between the Navier-Stokes equations and the averages of stochastic quantities computed from the solution to a Fokker-Planck equation.

An equivalence between the two modellings is only possible under very stringent assumptions. The former models typically involve nonlinear equations in a low-dimensional space, whereas the latter ones involve a linear Fokker-Planck equation for a probability density functional in a high-dimensional space.

(i) Models using constitutive relations

The issue we have dealt concerns free energy dissipative schemes for the Oldroyd-B model.

In [16] , the stability of various finite element schemes is analyzed regarding free energy dissipation. More precisely, under no-flow boundary conditions, some criteria have been identified in order for a finite element scheme to reproduce at the discrete level the dissipation of free-energy. In particular, discretizations of a reformulation of the Oldroyd-B model using the logarithm of the stress have also been analyzed. This reformulation had been introduced recently by R. Fattal and R. Kupferman, and numerically observed to be more stable. The work has been conducted in collaboration with C. Mangoubi (The Hebrew University of Jerusalem, IL).

In [10] , the previous analysis has been complemented. It is shown that there exist solutions to free-energy-dissipative schemes regardless of the formulation (with or without logarithm), whatever the time-step in the backward Euler time discretization. However, the uniqueness of the solutions is not ensured anymore for too large time steps. In [16] , only the schemes for the logarithm formulation are shown to dissipate the free energy for all solutions.

In [10] , it is also shown that provided a particular discretization of the advection term is used for the constitutive equations, continuous finite element spaces may also be used in order to obtain a discrete scheme for the Oldroyd-B model that also dissipates a discrete free energy. Then, the convergence of the discretized solutions for a regularized Oldroyd-B model can be shown. This work is in collaboration with J.W. Barrett (Imperial College, London, UK).

The stability of finite element discretizations for the constitutive equations is still being investigated:

• a collaboration involves T. Lelièvre and S. Boyaval with R. Kupferman (The Hebrew University of Jerusalem, IL) and M. Hulsen (Technische Universiteit Eindhoven, NL) regarding numerical simulations using new schemes for benchmark flows,

• a collaboration involves S. Boyaval with J.W. Barrett (Imperial College, London, UK) regarding the extension of the work [10] to other constitutive equations (the FENE-P model).

(ii) Micro-macro models

The work [35] analyzes a numerical method recently proposed by Ammar et al. to solve the Fokker-Planck equation for micro-macro models for complex fluids. This method is based on a representation of the solution as a sum of tensor products of one-dimensional functions, and a greedy algorithm to sequentially compute the terms of the sum (see also Section 5.9). Using known results from approximation theory, a variational formulation of the numerical method (arising from the minimization of some functional) is proved to actually converge to the solution. Many questions remain open concerning the original algorithms proposed (based on the Euler-Lagrange equations associated to the minimization problem), in particular in the case of non self-adjoint operators. The work is joint work with Y. Maday (University Paris 6 and Brown University) and has opened up many new directions in the team as a generic numerical method for high dimensional PDEs.

The work [17] addresses the numerical simulation of the micro-macro models using numerous stochastic processes describing the time evolution of the particles diluted in the fluid, rather than computing the probability density functional solution to a Fokker-Planck equation. Two numerical methods are proposed in this work to reduce the variance in the Monte-Carlo evaluation of such expected values. The approaches are based on the Reduced-Basis method, which is used for speeding up the computation of many solutions to a parametrized partial differential equation at many parameter values. It also has many possible extensions to other applications such as the calibration of the volatility in finance (see [17] ), or Bayesian statistics (work in progress).

Another work in progress, by T. Lelièvre, G. Samaey and V. Legat (Université catholique de Louvain, BE), aims at testing a numerical method based on coarse-time stepping schemes for the FENE model. The objective is twofold: discuss the existence and accuracy of closure approximations for the FENE model (a widely used micro-macro model for dumbbells), and compare this numerical approach with standard ones in terms of computational efficiency.

On a general level, a pedagogical mathematical introduction [48] to the subject of complex fluid modelling and simulation has been written by C. Le Bris and T. Lelièvre. Also, as a follow up to theoretical contributions on the theory of ordinary and stochastic differential equations originally motivated by the complex fluids context, C. Le Bris and M. Hauray(Paris 6) have pursued the analysis of differential equations with irregular coefficient fields in [31] .