Davim J.P. Tribology for Engineers: A Practical Guide

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Lubrication and roughness

height R

. A commonly accepted illustration of the regimes

is the Stribeck curve (Fig. 3.1) which represents the

dependence of the friction coefﬁ cient on the so-called bearing

ratio

u/p

–

where

is the kinematic viscosity, u the velocity

of the moving part, and p

–

the mean pressure in the ﬂ uid ﬁ lm).

3.3.1 Hydrodynamic lubrication regime

The hydrodynamic lubrication regime, h>> R

, occurs when

the lubricant completely separates the surfaces (see Fig. 3.1);

it is mostly associated with ﬁ lm thicknesses near or more

than 1

Friction losses under hydrodynamic lubrication are very

small – less than in the other lubrication regimes. The

hydrodynamic pressure generated in the lubricant ﬁ lm due to

the relative motion and inner ﬁ lm friction is too low to cause

surface deformation. The ﬂ ow of the lubricant ﬁ lm is laminar,

but at thicknesses above 20

m it becomes turbulent and that

leads to undesirable friction losses.

Hydrodynamic lubrication

Stribeck curve and lubrication regimes

Figure 3.1

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Tribology for Engineers

3.3.2 Boundary lubrication regime

The boundary regime, h< R

, occurs when the ﬂ uid ﬁ lm

is discontinuous and permits direct contact between high

points (known as asperities) of the opposite surfaces (Fig.

3.1). It is characterized by ﬁ lm thicknesses less than 70 nm

and higher friction losses than under the other regimes.

Examples when this may occur are during equipment startup

or shutdown, when the bearing may operate in boundary

rather than in unbroken ﬂ uid ﬁ lm conditions, or in toothed

gear contact, or in reciprocating motion (e.g. car valve on

valve seat). In this mode the ﬁ lm has less carrying capacity

than with contacted asperities.

3.3.3 Mixed lubrication regime

The mixed regime, h ~ R

, is transitional between the

boundary and hydrodynamic, when the fully-lubricated

(separated) and contacted (unseparated) surface areas

equally inﬂ uence the friction and the ﬁ lm parameters. Film

thicknesses range from above 70 nm to 1 μm (Fig. 3.1).

Friction losses vary over a wide range according to the two

limiting regimes involved.

3.3.4 Elastohydrodynamic lubrication

regime

The elastohydrodynamic (EHD) regime is a particular case

of the hydrodynamic with high hydrodynamic pressures,

sufﬁ cient to impact one or two of the opposite surfaces.

The latter are separated, but there is some interaction

between the asperities, resulting in elastic deformation and

enlargement of the contactless area (Fig. 3.1), whereby the

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Lubrication and roughness

viscous resistance of the lubricant becomes capable of

supporting the load. Film thicknesses usually range from 10

to 70 nm; the minimal thickness is often connected to the

ﬁ lm’s carrying capacity by the relation h

min

~ w

–0.073

. The

losses are the same as under hydrodynamic lubrication, as

the surfaces are completely separated. The main assumption

for the theoretical analysis is immediate elastic deformation

of the contacting materials and immediate increase of the

lubricant viscosity.

3.4 Reynolds’ equation

The ﬂ ow of the lubricant obeys the classical hydrodynamic

theory with some simpliﬁ cations speciﬁ c to thin ﬂ uids. The

surfaces, the ﬂ uid ﬁ lm thickness, and the ﬂ uid- and moving-

surface velocities are shown schematically in Fig. 3.2. Here

we proceed to derive an expression that describes the

hydrodynamic pressure in lubrication, making use of the

momentum-transport (Navier-Stokes) and continuity

equation.

U, V

h(x,z,t)

, V

h<<I

Derivation scheme for Reynolds’ equation.

Surfaces, ﬂ uid ﬁ lm thickness, velocities and

coordinates

Figure 3.2

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Tribology for Engineers

The ﬁ rst of these in vector form reads:

[3.1]

where p is hydrodynamic pressure, v is ﬂ ow velocity, t

is time, f are forces acting on the ﬂ uid, ρ is ﬂ uid density,

kinematic viscosity, Δ is Lapltacian

grad is

gradient

and i, j, k are unit vectors of the x–,

y–, and z directions respectively.

For an incompressible, isoviscous Newtonian ﬂ uid in laminar

ﬂ ow, its components can be described in full format as:

[3.2]

where μ is the dynamic viscosity.

As regards the continuity equation, for laminar inertialess

ﬂ ow it reads:

[3.3]

Because of the smallness of the gap-to-length ratio

(h/l) << 1 (the ratio of ﬁ lm thickness to its length) the ﬂ ow

terms of order (h/l)

and higher can be disregarded in the

eqs. [3.2], presented previously in dimensionless form. Thus,

eqs. [3.2], rewritten for inertialess

ﬂ ow,

reduce to:

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Lubrication and roughness

[3.4]

with the boundary conditions

[3.5]

Integrating the continuity equation with respect to y over the

thickness limits 0 . . . h (the pressure and

do not change

in that direction) and applying Leibnitz’s integration

formulae

, we obtain

[3.6]

Here the velocity v

y = h

is the total derivative, or in partial

form

[3.7]

Further, the integrals in eq. [3.6] represent the mass ﬂ ow

rates (lube consumption) in the x and z directions. Substituting

the velocities (determined from eqs. [3.4]) and considering

the boundary conditions [3.5], we can obtain

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Tribology for Engineers

[3.8]

Note now that

, the law of mass

conservation. Considering this in conjunction with eqs. [3.7]

and [3.8], eq. [3.6] transforms into the elliptical differential

equation obtained by Reynolds, named after him, and known

as the central expression of hydrodynamic lubrication theory:

[3.9]

The pressures on the left-hand side generate the ﬂ ow through

their gradients; the right-hand terms represent the ﬂ ow

generated by a possible shift in the y-direction (squeeze) and

by the moving- surface velocity u. The ﬂ uid velocities

[3.10]

reﬂ ect the superposition of two ﬂ ow modes: the Poiseuille,

due to imposed pressure gradients; and the Couette, a shear-

driven effect due to motion of the upper surface. In the v

curve the Poiseuille ﬂ ow is associated with a parabolic

velocity distribution across the ﬁ lm, and the Couette ﬂ ow

with a linear one.

Solution of Reynolds’ equation yields the hydrodynamic

pressure distribution which in turn, combined with eqs.

[3.10], yields the ﬁ lm’s load capacity

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Lubrication and roughness

[3.11]

as well as the shear stress and the friction force at the

lubricant-surface contact:

[3.12]

Solution of Reynolds’ equation involves boundary conditions

speciﬁ c to each practical case. How this is done is shown

in the subsequent sections, on the examples of slider and

cylindrical journal bearings.

3.5 Applications of hydrodynamic

lubrication theory

In view of its key role in the pressure distribution

determination, Reynolds’ equation is instrumental in

designing surfaces with optimal geometries. Such is the case

with bearings, mechanical seals, gears, jet engines (e.g.

turbine blades), internal combustion engines (e.g. piston-

cylinder and ring-cylinder couples, crank mechanisms),

nanotechnology, and also in biomedicine (e.g. artiﬁ cial

joints, synovial ﬂ uid), and many other areas.

Below, a concise analysis of 1D ﬂ uid ﬂ ow in slider bearings is

presented. These parts appear in a variety of conﬁ gurations, e.g.

plain and step bearings, plain journal and partial arc bearings,

axial groove, offset half, four-lobe, tilting pad and others. In all

of them, only one member of the couple is in motion.

3.5.1 Slider bearings

Figure 3.3 shows the geometry of a 1D slider (thrust) bearing,

with the coordinate system used for solving Reynolds’ equation.

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Tribology for Engineers

The ﬁ lm thickness (h) has a linear taper codirectional with

the surface velocity u

[3.12]

Note that the exit ﬁ lm thickness h

is unknown and should be

determined as part of the bearing design. The bearing taper

-h

) is also a design parameter (determined via analysis).

Reynolds’ equation for this 1D case has the form

[3.13]

Transformation of eqs. [3.12] and [3.13] with X = x/L, H =

h/h

, P = ph

/(6

uL), k = (h

–h

)/h

leads to the dimensionless

expressions

[3.14]

and

[3.15]

Substituting the derivative of the thickness and passing

to those with respect to H according to the chain rule

, we obtain

Slider geometry and coordinates

Figure 3.3

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Lubrication and roughness

[3.16]

And after double integration with respect to H

[3.17]

The constants of integration should be determined from the

boundary conditions P = 0 at both X = 0 and X = 1 (no

hydrodynamic pressure outside the slider wedge). Applying

this to eq. [3.15], we obtain C

= –2(1 + k)/[k(2 + k)] and

= –1/[k(2 + k)]. Thus

[3.18]

The hydrodynamic pressures for several p calculated by

eq. [3.18] are presented in Fig. 3.4 for several slider inlet/

outlet ratios H

= k + 1. Increase of the maximal pressure

= 1.5

= 2

= 3

= 4

Dimensionless pressure and inlet-to-outlet ratio

Figure 3.4

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Tribology for Engineers

with the ratio H

occurs at small wedge angles (ratio less

than 2.2), above which it drops – see curve for H

= 4.

3.5.2 Cylindrical journal bearings

Cylindrical journal bearings are common in rotating

machinery such as generators, motors, compressors, turbines,

etc. This bearing comprises of an inner rotating cylinder

(journal) of radius R and an outer immobile cylinder

(bearing) of radius R

> R. The geometry and coordinate

system are shown in Fig. 3.5.

In most ﬂ uid-ﬁ lm bearings with incompressible liquid

lubricant, the radial clearance ratio (R

– R)/R

is typically

0.001. When x = R

, z = ZL, t = TR/u, h = Hc, and P = pc

uR), Reynolds’ equation [3.9] can be rewritten in

dimensionless form as:

[3.19]

Cylindrical journal bearing geometry and

coordinate system

Figure 3.5

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