A micro-electro-mechanical system vibrating in its shear mode was used to study the viscosity of normal liquid $^3$He from 20mK to 770mK at 3bar, 21bar, and 29bar. The damping coefficient of the oscillator was determined by frequency sweeps through its resonance at each temperature. Using a slide film damping model, the viscosity of the fluid was obtained. Our viscosity values are compared with previous measurements and with calculated values from Fermi liquid theory. The crossover from the classical to the Fermi liquid regime is manifest in the temperature dependence of viscosity. In the Fermi liquid regime, the temperature dependence of viscosity changes from $T^{-1}$ to $T^{-2}$ on cooling, indicating a transition from the Stokes flow to the Couette flow regime.
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