Flow inside a cylinder which is suddenly rotated Ω liquid of density ρ and viscosity µ Figure 1: Geometry of the problem A Newtonian liquid with density ρ and viscosity µ is initially at rest in a vertical, inﬁnitely long cylinder of radius R. At time 0, the cylinder starts to rotate with a constant rotational speed, Ω about its axis. This problem is a transient ﬂow like the Rayleigh plate problem (or Stokes’ 1st problem). The governing equation is the following: ∂Vθ =ν ∂t Vθ2 ∂ 2 Vθ 1 ∂Vθ + − r ∂r r2 ∂r2 The boundary/initial conditions are the following: ⎧ ⎫ = ΩR at r = R, 0 < t V ⎪ ⎪ θ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ Vθ = f inite at r = 0, 0 ≤ t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Vθ = 0 at t = 0, 0 ≤ r ≤ R (1) (2) This problem can be solved using separation of variables. The solution ﬁnally leads to Bessel equation and there is a bit of algebra in the process. The ﬁnal solution will be: ∞ Vθ (r, θ) = Ωr + 2ΩR 1 να2 J1 (αk r/R) exp − 2k t αk J0 (αk ) R 1 (3) in which J1 and J0 are respectively the ﬁrst and zeroth order Bessel functions of the ﬁrst kind and αk is the k − th root of J1 . 2