= Solution
Interpreting the nominal size as diameter $d=10\,\mathrm{nm}$, fluid viscous dominance requires
$$
\boxed{U\ll\frac{\eta}{\rho_f d}=\frac{6.0\times10^4}{\rho_f}\,\mathrm{m\,s^{-1}},}
$$
with $\rho_f$ in $\mathrm{kg\,m^{-3}}$. \b[The liquid <density> is not supplied], so the conventional fluid-Reynolds crossover is not numerically fixed. As an explicitly additional illustrative assumption, $\rho_f=870\,\mathrm{kg\,m^{-3}}$ gives $\mathrm{Re}=1$ near $69\,\mathrm{m\,s^{-1}}$; ordinary much slower <nanoparticle> motion is well within the viscous regime.
If instead the supplied gold <density> is inserted into the dimensionless ratio $\rho_pUd/\eta$, that ratio reaches one at \b[$3.0\,\mathrm{m\,s^{-1}}$]. It is not the fluid <Reynolds number>. Including the spherical Stokes prefactor, the particle relaxation criterion $\tau_pU/d=1$ occurs at $54\,\mathrm{m\,s^{-1}}$. These quantify different inertia comparisons. The wording's speed direction is reversed: viscous dominance is obtained by making speed sufficiently \b[small]. If 10 nm denotes radius, the length-based thresholds halve.
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