Hydrodynamic lubrication

In hydrodynamic lubrication a thin oil film constitutes the interface between a static part and a part rotating at high speed in all kinds of bearings. A quantity of major interest to engineers is the load bearing capacity of the film, expressed by the pressure. Integrating the hydrodynamic equations over the width of the thin film leads to the following equation [24]:

$$\nabla \cdot (- \frac{ h^3 \overline{ \rho }}{12 \mu_v} \boldsymb... ...ho }) - \frac{ \partial (h \overline{ \rho})}{\partial t} - \dot{ m}_{\Omega },$$ (99)

where $h$ is the film thickness, $\overline{ \rho }$ is the mean density across the thickness, $p$ is the pressure, $\mu_v$ is the dynamic viscosity of the fluid, $\boldsymbol{ v}_a$ is the velocity on one side of the film, $\boldsymbol{ v}_b$ is the velocity at the other side of the film and $\dot{ m}_{\Omega}$ is the resulting volumetric flux (volume per second and per unit of area) leaving the film through the porous walls (positive if leaving the fluid). This term is zero if the walls are not porous.

For practical calculations the density and thickness of the film is assumed to be known, the pressure is the unknown. By comparison with the heat equation, the correspondence in Table  (11) arises. $\overline{ \boldsymbol{ v}}$ is the mean velocity over the film, $\overline{ v}_n$ its component orthogonal to the boundary. Since the governing equation is the result of an integration across the film thickness, it is again two-dimensional and applies in the present form to a plane film. Furthermore, observe that it is a steady state equation (the time change of the density on the right hand side is assumed known) and as such it is a Poisson equation. Here too, just like for the shallow water equation, the heat transfer equivalent of a spatially varying layer thickness is a spatially varying conductivity coefficient.

Table 11: Correspondence between the heat equation and the dynamic lubrication equation.

heat quantity	dynamic lubrication quantity
T	$p$
$\boldsymbol{ q}$	$\overline{ \rho} h \overline{ \boldsymbol{ v}}$
$q_n$	$\overline{ \rho} h \overline{ v}_n$
$\boldsymbol{\kappa}$	$\frac{ h^3 \overline{ \rho}}{12 \mu _v} \boldsymbol{ I}$
$\rho h$	$- \left( \frac{ \boldsymbol{ v}_b+\boldsymbol{ v}_a}{2}\right) \cdot \nabla (h... ...rho }) - \frac{ \partial (h \overline{ \rho})}{\partial t} - \dot{ m}_{\Omega }$
$\rho c$	$-$