Abstract

In this talk, I introduce a data-driven approach to viscous fluid mechanics, in particular to the stationary Navier-Stokes equation. The essential idea is to replace the constitutive law by experimental data. More precisely, usually one takes experimental data and then extrapolates a relation (the viscosity) between the deviatoric stress $\sigma$ and the strain $\epsilon$, for example $\sigma(\epsilon) = \mu_0 \epsilon$ (Newtonian fluid) or $\sigma(\epsilon)= \mu_0 |\epsilon|^{\alpha-1} \epsilon$ (power-law fluid). This relation is then used to obtain the Navier-Stokes equation.

Instead of using a constitutive relation, we introduce a data-driven formulation that has previously been examined in the context of solid mechanics. The idea is to find a solution that satisfies the differential constraints, derived from first principles, and is as close as possible to the experimental data. We obtain a variational formulation which we analyse under the aspects of weak lower-semicontinuity, coercivity and relaxation/$\Gamma$-convergence.

This talk is based on joint work with Christina Lienstromberg (Stuttgart) and Richard Schubert (Bonn).