A Data-Driven Approach to Viscous Fluid Mechanics: The Stationary Case

We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead including constitutive laws for fluid's viscosity in mathematical model, we suggest directly use experimental data. Only set differential constraints, derived from first principles, boundary conditions are kept classical PDE model combined with data set. The framework builds on recently introduced solid-mechanics [KO16,CMO18]. construct optimal solutions that material free sense no assumptions rheological behaviour made or extrapolated constraints mechanics recast language constant rank operators. Adapting abstract results lower-semicontinuity $\mathscr{A}$-quasiconvexity, show $\Gamma$-convergence result functionals arising mechanical problem. theory is extended compact nonlinear perturbations, whence our apply both inertialess fluids flows finite Reynolds number. Data-driven provide new relaxed solution concept. prove constructed consistent PDEs if sets have form monotone relation.