Basic Hydrodynamic Equations
5.1 NAVIER^STOKES EQUATIONS
The pressure distribution and load capacity of a hydrodynamic bearing are
analyzed and solved by using classical fluid dynamics equations. In a thin fluid
film, the viscosity is the most important fluid property determining the magnitude
of the pressure wave, while the effect of the fluid inertia (ma) is relatively small
and negligible. Reynolds (1894) introduced classical hydrodynamic lubrication
theory. Although a lot of subsequent research has been devoted to this discipline,
Reynolds’ equation still forms the basis of most analytical research in hydro-
dynamic lubrication. The Reynolds equation can be derived from the Navier–
Stokes equations, which are the fundamental equations of fluid motion.
The derivation of the Navier–Stokes equations is based on several assump-
tions, which are included in the list of assumptions (Sec. 4.2) that forms the basis
of the theory of hydrodynamic lubrication. An important assumption for the
derivation of the Navier–Stokes equations is that there is a linear relationship
between the respective components of stress and strain rate in the fluid.
In the general case of three-dimensional flow, there are nine stress
components referred to as components of the stress tensor. The directions of
the stress components are shown in Fig 5-1.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved.
The stress components sx, sy, sz are of tension or compression (if the sign
is negative), as shown in Fig. 5-1. However, the mixed components txy, tzy, txz are
shear stresses parallel to the surfaces.
is possible to show by equilibrium considerations that the shear
components are symmetrical:
txy ¼ tyx;
tyz ¼ tzy;
txz ¼ tzx
Due to symmetry, the number of stress components is reduced from nine to six.
In rectangular coordinates the six stress components are
sx ¼ p þ 2m
sy ¼ p þ 2m
sz ¼ p þ 2m
txy ¼ tyx ¼ m
tyz ¼ tzy ¼ m
tzx ¼ txz ¼ m
A fluid that