Abstract
Motivated by evaluating coating oil films within bearing chambers in an aero-engine application, an analysis is presented for the fluid dynamics relevant in their dual capacity as both the coolant and lubricant in highly sheared flows that may approach microscale thickness. An extended model is developed for isothermal rimming flow driven by substantial surface shear within a stationary cylinder. In particular, a partial slip condition replaces the no-slip condition at the wall whilst retaining inertial effects relevant to an intrinsic high speed operation. A depth-averaged formulation is presented that includes appropriate inertial effects at leading-order within a thin film approximation that encompasses a more general model of assessing the impact of surface slip. Non-dimensional mass and momentum equations are integrated across the film depth yielding a one dimensional problem with the a priori assumption of local velocity profiles. The film flow solutions for rimming flow with wall slip are modeled to a higher order than classical lubrication theory. We investigate the impact of wall slip on the transition from pooling to uniform films. Numerical solutions of film profiles are provided for the progressively increased Reynolds number, within a moderate inertia regime, offering evaluation into the effect of film slippage on the dynamics of rimming flow. We find that slip allows non-unique solution regions and existence of multiple possible steady state solutions evaluated in transforming from smooth to pooling film solutions. Additionally, boundary slip is shown to enhance the development of recirculation regions within the film which are detrimental to bearing chamber flows.
Highlights
Thin film flows are widely used in engineering applications, notably as a lubricant between moving surfaces and in providing a thermal conduit for heat exchange
This paper investigates the effects of the slip boundary condition for thin fluid films driven by surface shear
Upon balancing the inertia and surface tension, instabilities arising from inertia are suppressed by surface tension leading to a smoothed solution for short wave disturbances, whereas the opposite is true for long scale disturbances which are counteracted by viscosity, see the work of Benilov32 or Benilov and O’Brien31 Recently, Kay5 evaluated a thin film approximation for the effects of inertia, where low Reynolds numbers exist within the realm of lubrication theory and high Reynolds numbers corresponded to uniform film profiles
Summary
Thin film flows are widely used in engineering applications, notably as a lubricant between moving surfaces and in providing a thermal conduit for heat exchange. A model to the first order of the viscous, gravitational, inertial, and capillarity effects was proposed by Pougatch and Frigaard, noting minor changes to the location and amplitude of the film thickness profile in two and three dimensional flow They suggest that gravity provides a stabilising force, whereas surface tension and inertia tend to destabilise the film for lower levels of inertia. Upon balancing the inertia and surface tension, instabilities arising from inertia are suppressed by surface tension leading to a smoothed solution for short wave disturbances, whereas the opposite is true for long scale disturbances which are counteracted by viscosity, see the work of Benilov or Benilov and O’Brien Recently, Kay evaluated a thin film approximation for the effects of inertia, where low Reynolds numbers exist within the realm of lubrication theory and high Reynolds numbers corresponded to uniform film profiles. Kay et al. covered the depth-averaging method in a rimming flow context, with a priori use of quadratic profile or cubic velocity profile when incorporating cylinder roughness for reducing the dimensionality of the Navier-Stokes equations
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