1:00 pm Monday, October 29, 2007
Analysis seminar: A Nonlinear Reynolds Equation for Thin Viscous Flows by
Juha Videman (Instituto Superior Técnico, Lisbon, Portugal) in RLM 10.176
Viscous flows in thin domains are encountered in a great number of practical situations. Think, for example, of flows in gas or pipelines, in fluid film bearings, in capillaries, or in oceans and in atmosphere. It is natural to expect that in these situations the governing equations of motion are approximated by simpler, lower-dimensional models leading to solutions which are precise only asymptotically as the width of the domain tends to zero. In his seminal paper in 1886, Osborne Reynolds proposed a model for the main pressure distribution in lubrication. This model, the classical Reynolds equation, even though derived under the assumption that the inertial and curvature effects are negligible, makes a reasonable (first-order) approximation for many, but not all, problems in hydrodynamic lubrication. In our work, we derive an equation, referred to as modified nonlinear Reynolds equation, for higher-order approximation of pressure (and consequently of velocity field) in hydrodynamic lubrication. This equation is a nonlinear second-order differential equation for a scalar unknown, which approximates the first two terms in the asymptotic expansion of the pressure field, and, in contrast to the classical Reynolds' equation, takes into account both the inertial and the curvature effects. The equation is justified with optimal integral and pointwise error estimates. Up to now, this kind of optimality has only been achieved for small fluxes and in simpler geometrical situations. Here, the flux, equivalently the Reynolds number, can be as large as possible within the limits of dimension reduction and the optimal estimates result from an accurate analysis of the auxiliary divergence equation. Submitted by
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