High shear helical flow of a Sisko fluid

The Sisko fluid model was proposed in response to observed failure of the well-known Bingham model to represent the flows of viscoplastic materials in high shear regimes, such failure increasing with shear rate. We consider the flow of a Sisko fluid between infinitely long coaxial cylinders, when the inner cylinder rotates with constant angular speed and a constant axial fluid flow is maintained. High shear rates are ensured by assuming an inter-cylindrical gap small in relation to the overall geometry. Such a helical flow and geometry is of relevance to a number of applications, including rheometry. We assume a laminar flow and apply a perturbation procedure based on a scaled form of the inter-cylindrical gap dimension as a perturbation parameter to obtain simple explicit approximations for the fluid velocity field as well as the fluid viscosity variation in this gap. We also derive an approximate form of the Reiner--Riwlin equation, linking the defining fluid parameters to measurable quantities in the flow.