The use of the vane to measure the shear modulus of linear elastic solids

The vane rheometer may be used to obtain an estimate of the shear modulus G of a gelled suspension. Complex variable techniques are used to determine the torque required to turn a two-bladed vane (i.e. a single plate) of diameter 2a through an angle θ when immersed in a linear elastic material with shear modulus G and Poisson's ratio σ. The torque per unit length is πa2Gθ(1 + k−1), where k = 3 – 4σ. When σ = 12 (a reasonable approximation when studying weak gels) the shear stress τxy over the surface of the vane is zero, and there is no tendency for slip to occur at the walls of the vane. The more general case of a two-bladed vane in the form of an ellipse is studied, with either a no-slip boundary condition, or a stress-free (slip) boundary condition. This latter case is resolved by representing the complex deformation potentials as infinite series of exponentials. These series are truncated, and the coefficients of each term are determined numerically. Boundary element methods are then used to compute the torque required to turn finite vanes, of length l = la, in a fluid of viscosity μ, at low Reynolds numbers. There is a direct analogy between the governing equations for low Reynolds number flow and the equations governing linear elastic or viscoelastic deformation when σ = 12, with G replaced by μ and θ by the vane's angular velocity Ω. A no-slip boundary condition is assumed over the surface of the vane. The torque is found to be M = 2πa3μΩ(l + 0.66) for the two-bladed vane, and M = 2πa3μΩ(1.5l + 1.20) for a four-bladed vane, when l ⩾ 3. Numerical results on vanes with 3, 6 and 8 blades indicate that the torque per unit length on an n-bladed vane of infinite length is 2πa2μΩ(2 - 2n−1). These results were tested experimentally, and good agreement was found between theory and experiment.