For a steady flow between a fixed & moving plate as shown in Fig 3-1 velocity distribution is u=(U/2)*(1+(y/h)) and friction coefficient is Cf=1/Re(h).
Assumption: Viscosity is constant.
first of all, the Couette flow in your document is laminar flow. So the Poiseuille no in Churchill's comment is one. Physically, Re in Couette flow is to measure the turbulence intensity. For laminar flow, since it is not turbulent, Reynolds number should not play any role for Couette flow.
The Reynolds number measures the importance of the inertial forces relative to viscous forces in the studied flow. In laminar if the Reynolds number is high inertial force is the right scale to make dimensionless stresses. If the Reynolds number is small, choose the viscous forces for the friction coefficient.
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