Transition to thermohydrodynamic lubrication problem

Abstract

We consider a non-isothermal Stokes equation used to calculate the pressure distribution in a thin layer of lubricant film between two surfaces. The problem is described in 2D and 3D settings by the Stokes and heat transfer equations. Under appropriate regularity assumptions on the data, existence results for the non-isothermal Stokes is recalled. Using a formal asymptotic expansion, we obtain a generalized Reynolds equation coupled with a limit energy equation, the so-called non-isothermal Reynolds system. Then existence and uniqueness are proved for this system by using a fixed-point argument. Finally, a rigorous justification of the convergence is established.