On the Influence of the Convective Term in the Navier–Stokes Equation on the Forces in Hydrodynamic Bearings

Abstract

Many theories describing the flow of viscous fluids in thin lubrication layers during rotor motion inside a stator, including the influence of the convective term in the Navier–Stokes equation, are known and widely used. However, the results of individual studies show some inconsistencies in evaluating the influence of the convective term on the force occurring in the lubrication layer. Here, the effect of the convective term on the force acting on an arbitrarily moving rotor is explained based on a theoretical analysis of the Navier–Stokes equation. It is shown that for a constant fluid density in the case of an arbitrary trajectory of the centre of a non-rotating rotor, the convective term has zero effect on the force on the rotor. A non-zero effect of the convective term may only arise as a result of the spatial distribution of the momentum density at the inlet and outlet surfaces of the lubricating layer or as a result of variable fluid density due to cavitation or the compressibility of the fluid. Thus, the theoretical discussion presented here clarifies the numerical solutions obtained by researchers in the field of hydrodynamic lubrication and allows us to understand the reasons for the numerical behaviour of some simplified models.

Many theories describing the flow of viscous fluids in thin lubrication layers during rotor motion inside a stator, including the influence of the convective term in the Navier–Stokes equation, are known and widely used. However, the results of individual studies show some inconsistencies in evaluating the influence of the convective term on the force occurring in the lubrication layer. Here, the effect of the convective term on the force acting on an arbitrarily moving rotor is explained based on a theoretical analysis of the Navier–Stokes equation. It is shown that for a constant fluid density in the case of an arbitrary trajectory of the centre of a non-rotating rotor, the convective term has zero effect on the force on the rotor. A non-zero effect of the convective term may only arise as a result of the spatial distribution of the momentum density at the inlet and outlet surfaces of the lubricating layer or as a result of variable fluid density due to cavitation or the compressibility of the fluid. Thus, the theoretical discussion presented here clarifies the numerical solutions obtained by researchers in the field of hydrodynamic lubrication and allows us to understand the reasons for the numerical behaviour of some simplified models.