The form of the Reynold's equation [433] [434] used to describe hydrodynamic lubrication is deduced from the Navier-Stokes equations considering the following assumptions:
The flow is laminar and continuous.
The flow is isoviscous (see note below).
The flow is incompressible (see note below).
The fluid inertia effects are negligible.
The thickness of the fluid is very small when compared to other dimensions of the fluid domain.
There is no slipping between the fluid film and the shaft and bearing journal walls.
It can be written in Cartesian coordinates as:
 | (7–27) |
where
 and  = axes of the film plane.  is tangent to the bearing and  is along the length of the bearing. |
 = fluid thickness |
 = fluid viscosity |
 = fluid pressure |
 = wall (corresponding to the shaft journal) velocity along  . The bearing sleeve is fixed. |
Note: In most cases, the lubricant is considered isoviscous and incompressible. However, the effect of variable viscosity and density can be included using the 3-D formulation and tabular material properties.
The first term on the right-hand side of Equation 7–27 represents the shear (or slide) effect, while the second term represents the squeeze effect.
The equation is solved using a finite-length formulation for 2-D elements. A negative pressure reflects the fluid film rupture or cavitation. Since it has no physical meaning, it is set to zero to calculate related output results, for example pressure forces.
Assuming zero pressure at both ends, a solution of Equation 7–27 can be found using a combination of short and modified long bearing pressure expressions [435]. In cylindrical coordinates using non-dimensional quantities, the pressure is given by:
 | (7–28) |
where
 = bearing length (real constant L) |
 = bearing radial clearance (real constant C) |
 = non-dimensional coordinate along the bearing length |
 = non-dimensional eccentricity ratio (eccentricity  divided by clearance) |
 = time derivative of the eccentricity ratio |
 = time derivative of the attitude angle. The attitude angle is the angle of
the line of centers (shaft center to bearing center). |
 = average rotational velocity. Because the bearing is fixed, it is equal to
half of the shaft rotational velocity  . |
 = cylindrical coordinate with respect to the line of centers |
 = non-dimensional film thickness (film thickness  divided by clearance) |
|
 = bearing radius (real constant R) |
The pressure is integrated over the shaft journal to obtain the pressure forces. Spatial
increment in 
defaults to 2 degrees. It can be modified using the ThetaInc real constant
(see COMBI214 element).
The stiffness and damping coefficients are obtained from the derivative of the pressure forces based on a small perturbation increment (real constant PertInc) in displacement and velocity.
 | (7–29) |
where
 = stiffness coefficients (see Table 214.2: COMBI214 Element Output Definitions for KEYOPT(1) = 1 or 2) |
 = damping coefficients (see Table 214.2: COMBI214 Element Output Definitions for KEYOPT(1) = 1 or 2) |
 = pressure force along direction (1), (2) |
 = displacement in direction (1), (2) |
 = velocity in direction (1), (2) |
In a 3-D approach, the fluid domain is meshed using surface elements FLUID218. The derivation of the finite element matrices is similar to the derivation of the acoustic fluid matrices.
For the pressure-only element, the pressure shape functions are used to determine the fluid
stiffness matrix from the left-hand side of Equation 7–27,
giving 
similar to Equation 8–33. The film thickness h is
approximated with the same shape functions to obtain the element internal fluid forces from the
right hand side of Equation 7–27.
For the coupled-field element (PRES and U degrees of freedom), the fluid-structure
interaction matrix 
(Equation 8–136) is introduced. The assembled stiffness
matrix is then unsymmetric and expressed as:
 | (7–30) |
where
 = structural stiffness matrix (Equation 2–58) |
 = interaction matrix (Equation 8–136) |
 = fluid stiffness matrix (Equation 8–33) |