Quasi-Stability of the Primary Flow in a Cone and Plate Viscometer
Copyright policy )Quasi-Stability of the Primary Flow in a Cone and Plate Viscometer
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 parameter Re ε<sup>2</sup>, where Re is the Reynolds number and ε 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 solution. To overcome this difficulty we construct a suitable corrector.
- University of Montpellier France
- Leibniz Association Germany
- Weierstrass Institute Germany
- University of Montpellier France
- Weierstrass Institute for Applied Analysis and Stochastics Germany
Navier-Stokes equations -- shallow domains -- rotating fluids -- nonlinear stability -- asymptotic analysis -- haemodynamics -- flow chamber -- haemostasis -- rheometry -- CFD -- finite elements, rheometry, Navier-Stokes model, 510, flow chamber, Parallel shear flows in hydrodynamic stability, ddc:510, asymptotic stability, haemodynamics, 35B40, article, 76D05, Lubrication theory, asymptotic analysis, numerical simulation, 35Q30, rotating fluids, haemostasis, finite elements, singular part, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, nonlinear stability, Navier-Stokes equations, shallow domains, CFD, Stability in context of PDEs, approximate solution, Finite element methods applied to problems in fluid mechanics, 65M60
Navier-Stokes equations -- shallow domains -- rotating fluids -- nonlinear stability -- asymptotic analysis -- haemodynamics -- flow chamber -- haemostasis -- rheometry -- CFD -- finite elements, rheometry, Navier-Stokes model, 510, flow chamber, Parallel shear flows in hydrodynamic stability, ddc:510, asymptotic stability, haemodynamics, 35B40, article, 76D05, Lubrication theory, asymptotic analysis, numerical simulation, 35Q30, rotating fluids, haemostasis, finite elements, singular part, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, nonlinear stability, Navier-Stokes equations, shallow domains, CFD, Stability in context of PDEs, approximate solution, Finite element methods applied to problems in fluid mechanics, 65M60
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