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