Quasi-stability of the primary flow in a cone and plate viscometer

Abstract

We investigate the flow between a shallow rotating cone and a stationary plate. This cone and plate device is used in rheometry, haemostasis as well as in food industry to study the properties of the flow w.r.t. shear stress. Physical experiments and formal computations show that close to the apex the flow is approximately azimuthal and the shear-stress is constant within the device, the quality of the approximation being controlled, essentially by the single parameterRe ∈2, whereRe is the Reynolds number and e the thinness of the cone-plate gap. We establish this fact by means of rigorous energy estimates and numerical simulations. Surprisingly enough, this approximation is valid though the primary flow is not itself a solution of the Navier-Stokes equations, and it does not even, fulfill the correct boundary conditions, which are in this particular case discontinuous along a line, thus not allowing for a usual Leray solutin. To overcome this difficulty we construct a suitablecorrector.