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W R r (A) e\ ´R(1 − ɛ), fluid X fluid (B) U h = ER
(35%) Steady flow in the annular gap between long, co-axial cylinders, one or both of which is rotated at a constant angular velocity, is termed Couette flow. In a Couette viscometer, as shown in the figure (A), the inner cylinder of radius R is rotating with an angular velocity w, and the outer cylinder of radius R(1 – ɛ), (given that 0 < ɛ < 1), is fixed. The annular gap of width h = ER is filled with a test fluid with unknown viscosity u. Noting the circular symmetry and neglecting end effects (cylinders length H » h), is it assumed that the fluid velocity and the dynamic pressure p do not depend on the polar angle and the distance along z-axis (perpendicular to the plane of the figure). a) Show that for incompressible fluid the solution for the velocity has the following form: V = Vė (r), Vr = 0; b) Using the flow equations of incompressible Newtonian fluid and the appropriate boundary conditions, show that the solution has the form: [/-/- (1 - 2)²] [1 − (1 − ɛ)²] Would this solution require an extra assumption of Stokes (viscous or creeping) flow? Ve(r) =wR. c) Measurements of the torque, Ỉ = ſƒÂ † × TdA, required to turn the outer cylinder provide an effective method for determining the viscosity μ of the fluid. Show that in the present case Lz = SS₁r TrødA and determine the expression for the torque using the result in (b) as a function of the geometry, w and µ (recall that dA = rd0dz); d) For a narrow gap, & < 1, the Couette flow can be approximated by the shear flow between two parallel planar surfaces as shown schematically in figure (B). Using this approximation obtain an approximate expression for the torque L₂ and show that it agrees with the exact result in (c) for ε → 0.