Thin-Film Flows with Thermoelastically Deformed Boundaries

Abstract

Equilibrium solutions are given for the thermoelastic displacements of an initially wavy, moving surface subjected to non-uniform viscous heating derived from a hydrodynamic lubricant film. The configuration studied is similar to a flexibly mounted face seal with one metallic face running against a thermal insulator. Changes in mean film-thickness with changing speed are discussed with reference to earlier analyses which predicted thermoelastic instability and to experiments which illustrated this. The operating conditions approach those where instability was predicted for conditions of fixed mean film thickness; however, no instability is predicted for present conditions where axial load is fixed. Thermoelastic effects upon growth of surface waviness become significant when the sliding speed exceeds u^*, given by u^* = h₁k √(K/µ), where h₁ is the initial waviness amplitude, K is the wave number (κ = π/Λ, where Λ is half the wavelength of a sinusoidal waviness), K is the thermal conductivity of the metal, μ is the fluid viscosity, and α is the coefficient of expansion. Past experience has shown that the product h₁k is such that long-wavelength waviness is associated with the lowest u^* and therefore magnified relative to shorter wavelength components of the surface topography. Thermal deformations appear to be favourable in their influence on film thickness—except where the unexplained but experimentally observed transition to point contact occurs.