Linear and nonlinear instabilities of sliding Couette flow

We consider an incompressible viscous fluid with the kinematic viscosity v between two infinitely long concentric cylinders with radii a and b (b > a). The fluid experiences a shear motion produced by pulling the inner cylinder with the axial speed U while keeping the outer cylinder at rest (see Fig.1). The axial basis flow at the radius r, $$U_{B}(r) = R\frac{\ln(r(1 - \eta)/2)}{\ln(\eta)},$$ can be obtained as an exact solution of the Navier-Stokes equation, where \(R = U(b - a)/2v\) is the Reynolds number and \(\eta = a/b\) is the radius ratio.