The torque on a rotating n -bladed vane in a newtonian fluid or linear elastic medium

The vane is an n -bladed paddle which rotates with angular velocity Ω in a linear viscoelastic fluid. The blades, of zero thickness, are equally spaced around the axis r = 0, and extend from r = 0 to r = a . The problem is assumed two-dimensional. The stress and fluid velocity (or material displacements) are obtained by a Wiener–Hopf technique for the case of a no-slip boundary condition on the surface of the blades, and for the case of zero shear stress on the blades. The torque M (per unit length) required to rotate the vane in an incompressible newtonian fluid of viscosity μ may be approximated as M ≈ 4π μa 2 Ω (1 – n –1 ) to within 1% for the no-slip boundary condition; with the slip boundary condition the same expression is accurate to within 4%. Results are also given for the angular dependence and strength of the stress singularity at the tip of each blade.