Viscosity and Stokes-Einstein relation in deeply supercooled water under pressure

We report measurements of the shear viscosity $\eta$ in water up to $150\,\mathrm{MPa}$ and down to $229.5\,\mathrm{K}$. This corresponds to more than $30\,\mathrm{K}$ supercooling below the melting line. The temperature dependence is non-Arrhenius at all pressures, but its functional form at $0.1\,\mathrm{MPa}$ is qualitatively different from that at all pressures above $20\,\mathrm{MPa}$. The pressure dependence is non-monotonic, with a pressure-induced decrease of viscosity by more than 50 % at low temperature. Combining our data with literature data on the self-diffusion coefficient $D_\mathrm{s}$ of water, we check the Stokes-Einstein relation which, based on hydrodynamics, predicts constancy of $D_\mathrm{s} \eta/T$, where $T$ is the temperature. The observed temperature and pressure dependence of $D_\mathrm{s} \eta/T$ is analogous to that obtained in simulations of a realistic water model. This analogy suggests that our data are compatible with the existence of a liquid-liquid critical point at positive pressure in water.