III UNIT

FUNDAMENTALS OF VISCOSITY AND FLOW

Fundamentals of viscosity and flow – Petroff’s equation – Friction torque – Viscosity measurement,

factors affecting viscosity, Principle of Hydrostatic lubrication – Hydrostatic step bearing – Multi-

recess bearing – Design Problems – Different types of compensation and their effect on bearing

parameters – Hydrostatic lift, simple problems – Hydrostatic squeeze films

3.1 Fundamentals of viscosity

The parameter which plays a fundamental role in lubrication is oil viscosity. Different oils

exhibit different viscosities. In addition, oil viscosity changes with temperature, shear rate and

pressure and the thickness of the generated oil film is usually proportional to it. So, at first glance

it appears that the more viscous oils would give better performance, since the generated films

would be thicker and a better separation of the two surfaces in contact would be achieved.

This unfortunately is not always the case since more viscous oils require more power to be sheared.

Consequently the power losses are higher and more heat is generated resulting in a substantial

increase in the temperature of the contacting surfaces which may lead to the failure of the

component. For engineering applications the oil viscosity is usually chosen to give optimum

performance at the required temperature. Knowing the temperature at which the oil is expected to

operate is critical as oil viscosity is extremely temperature dependent.

The viscosity of different oils varies at different rates with temperature. It can also be affected by

the velocities of the operating surfaces (shear rates). The knowledge of the viscosity characteristics

of a lubricant is therefore very important in the design and in the prediction of the behavior of a

lubricated mechanical system. In this chapter a simplified concept of viscosity, sufficient for most

engineering applications, is considered. The refinements to this model, incorporating, for example,

transfer of momentum between the adjacent layers of lubricant and transient visco-elastic effects,

can be found in more specialized literature.

3.1.1Dynamic Viscosity

Dynamic Viscosity Consider two flat surfaces separated by a layer of fluid of thickness

‘h’ as shown in Figure 3.1. The force required to move the upper surface is proportional to the

wetted area ‘A’ and the velocity ‘u’, i.e.: F α A × u Assume that the fluid film separating the

surfaces is made up of a number of infinitely thin layers. Compare now two fluid films of different

thickness made up of equal spaced layers. If the surface velocity remains unchanged in these two

cases then a single layer in the thicker film will undergo less relative sliding than in the thinner

film. The velocity gradients for these two layers will be different. Since the thicker film contains

more single layers, less force will be needed to shear a single layer so the viscous resistance will

1

vary as the reciprocal of the film thickness ‘1/h’. The force needed to move the upper surface is

thus proportional to:

F α A × u/h (3.1)

Shear rate α shear stress

This relationship is maintained for most fluids. Different fluids will exhibit a different

proportionality constant ‘η’, called the ‘dynamic viscosity’. The relationship (3.1) can be written

as:

F = η × A × u/h (3.2)

Fig.3.1 Schematic representation of the fluid separating two surfaces

Rearranging gives:

η = (F/A) / (u/h) or

η = τ / (u/h) (3.3)

where: η is the dynamic viscosity [Pas];

τ is the shear stress acting on the fluid [Pa];

u/h is the shear rate, i.e. velocity gradient normal to the shear stress [s-1].

Before the introduction of the SI system the most commonly used dynamic viscosity unit was the

Poise. Incidentally this name originated not from an engineer but from a French medical doctor

Poiseuille, who studied the flow of blood. For practical applications the Poise [P] was far too large

thus a smaller unit, the centipoise [cP], was more commonly used. The SI unit for dynamic

viscosity is Pascal-second [Pas]. The relationship between Poise and Pascal second is as follows:

1 [P] = 100 [cP] ≈ 0.1 [Pas]

2

3.1.2 Kinematic Viscosity

Kinematic viscosity is defined as the ratio of dynamic viscosity to fluid density:

υ = η/ρ (3.4)

where: υ is the kinematic viscosity [m2 /s];

η is the dynamic viscosity [Pas];

ρ is the fluid density [kg/m3 ].

The most commonly used kinematic viscosity unit is the Stoke [S]. This unit, however, is often

too large for practical applications, thus a smaller unit, the centistoke [cS], is used. The SI unit for

kinematic viscosity is [m2 /s], i.e.: 1 [S] = 100 [cS] = 0.0001 [m2 /s] The densities of lubricating

oils are usually in the range between 700 - 1200 [kg/m3 ] (0.7 - 1.2 [g/cm3 ]).

The typical density of mineral oil is 850 [kg/m3 ] (0.85 [g/cm3 ] ). To find the dynamic viscosity

of any oil in [cP] the viscosity of this oil in [cS] is multiplied by its density in [g/cm3 ], hence for

a typical mineral oil: viscosity in [cP] = viscosity in [cS] × 0.85 [g/cm3 ]

3.2 Factor affecting viscosity

3.2.1 Temperature

In liquids, when the temperature increases (energy added) the particles move faster and begin

to move away from each other. Because the particles are moving around more they can flow

more; Their viscosity is lower.In gases, the particles are far apart so when energy is added the p

articles move faster and collide with each other more often causing an increase in viscosity.

3.2.2 Concentration

Concentration is the amount of substance that is dissolved in a specific volume.

An increase in concentration will usually result in an increase in viscosity.

3.2.1 Attractive force

Particles of the same substance have an attractive force on one another some substanc ha

ve a strong attraction while some substances have a weaker attraction.

The stronger the attraction of particles, the higher the viscosity

3.2.2 Particle size

The size of the particles of a substance will greatly affect its viscosity. S

mall particles can move more easily past each other and can therefore fl

ow faster, meaning they have a lower viscosity.

Large particles would mean a higher viscosity.

3

3.4 VISCOSITY MEASUREMENTS

Various viscosity measurement techniques and instruments have been developed over the years.

The most commonly used in engineering applications are capillary and rotational viscometers. In

general, capillary viscometers are suitable for fluids with negligible non-Newtonian effects and

rotational viscometers are suitable for fluids with significant non-Newtonian effects. Some of the

viscometers have a special heating bath built-in, in order to control and measure the temperature,

so that the viscosity-temperature characteristics can be obtained. In most cases water is used in the

heating bath. Water is suitable for the temperature range between 0° to 99°C. For higher

temperatures mineral oils are used and for low temperatures down to -54°C, ethyl alcohol or

acetone is used.

3.4.1 Capillary Viscometers

Capillary viscometers are based on the principle that a specific volume of fluid will flow

through the capillary (ASTM D445, ASTM D2161). The time necessary for this volume of fluid

to flow gives the ‘kinematic viscosity’. Flow through the capillary must be laminar and the

deductions are based on Poiseuille’s law for steady viscous flow in a pipe. There is a number of

such viscometers available and some of them are shown in Figure 3.2Assuming that the fluids are

Newtonian, and neglecting end effects, the kinematic viscosity can be calculated from the formula:

t is the flow time through the capillary, t = (t2 − t1), [s];

k is the capillary constant which has to be determined experimentally by applying a

reference fluid with known viscosity, e.g. by applying freshly distilled water. The

capillary constant is usually given by the manufacturer of the viscometer.

υ = πr4glt / 8LV = k(t2 − t1)

(3.5)

where:

υ is the kinematic viscosity [m2/s];

r is the capillary radius [m];

l is the mean hydrostatic head [m];

g is the earth acceleration [m/s2];

L is the capillary length [m];

V is the flow volume of the fluid [m3];

4

Fig. 3.2 Typical capillary viscometers

In order to measure the viscosity of the fluid by one of the viscometers shown in Figure 3.2, the

container is filled with oil between the etched lines. The measurement is then made by timing the

period required for the oil meniscus to flow from the first to the second timing mark. This is

measured with accuracy to within 0.1 [s].

Kinematic viscosity can also be measured by so called ‘short tube’ viscometers. In the literature

they are also known as efflux viscometers. As in the previously described viscometers, viscosity

is determined by measuring the time necessary for a given volume of fluid to discharge under

gravity through a short tube orifice in the base of the instrument. The most commonly used

viscometers are Redwood, Saybolt and Engler. The operation principle of these viscometers is the

same, and they only differ by the orifice dimensions and the volume of fluid discharged. Redwood

viscometers are used in the United Kingdom, Saybolt in Europe and Engler mainly in former

Eastern Europe. The viscosities measured by these viscometers are quoted in terms of the time

necessary for the discharge of a certain volume of fluid. Hence the viscosity is sometimes found

as being quoted in Redwood and Saybolt seconds.

3.4.2 Rotational Viscometers

Rotational viscometers are based on the principle that the fluid whose viscosity is being

measured is sheared between two surfaces (ASTM D2983). In these viscometers one of the

surfaces is stationary and the other is rotated by an external drive and the fluid fills the space in

between. The measurements are conducted by applying either a constant torque and measuring the

changes in the speed of rotation or applying a constant speed and measuring the changes in the

torque. These viscometers give the ‘dynamic viscosity’. There are two main types of these

viscometers: rotating cylinder and cone-on-plate viscometers.

5

Fig.3.3 Schematic diagram of a short tube viscometer.

3.4.3 Rotating Cylinder Viscometer

The rotating cylinder viscometer, also known as a ‘Couette viscometer’, consists of two

concentric cylinders with an annular clearance filled with fluid as shown in Figure 3.4. The inside

cylinder is stationary and the outside cylinder rotates at constant velocity. The force necessary to

shear the fluid between the cylinders is measured. The velocity of the cylinder can be varied so

that the changes in viscosity of the fluid with shear rate can be assessed. Care needs to be taken

with non-Newtonian fluids as these viscometers are calibrated for Newtonian fluids. Different

cylinders with a range of radial clearances are used for different fluids. For Newtonian fluids the

dynamic viscosity can be estimated from the formula:

Fig.3.4 Couette viscometer

η = M(1/rb2 − 1/rc

2) / 4πdω = kM / ω (3.6)

where:

η is the dynamic viscosity [Pas];

rb, rc are the radii of the inner and outer cylinders respectively [m]; M

is the shear torque on the inner cylinder [Nm];

ω is the angular velocity [rad/s];

6

d is the immersion depth of the inner cylinder [m];

k is the viscometer constant, supplied usually by the manufacturer for each pair of

cylinders [m-3].

When motor oils are used in European and North American conditions, the oil viscosity data at -

18°C is required in order to assess the ease with which the engine starts. A specially adapted

rotating cylinder viscometer, known in the literature as the ‘Cold Cranking Simulator’ (CCS), is

used for this purpose (ASTM D2602). The schematic diagram of this viscometer is shown in Figure

3.5.

Fig. 3.5 Cold Cranking Simulator’

The inner cylinder is rotated at constant power in the cooled lubricant sample of volume

about 5 [ml]. The viscosity of the oil sample tested is assessed by comparing the rotational speed

of the test oil with the rotational speed of the reference oil under the same conditions. The

measurements provide an indication of the ease with which the engine will turn at low temperatures

and with limited available starting power. In the case of very viscous fluids, two cylinder

arrangements with a small clearance might be impractical because of the very high viscous

resistance; thus a single cylinder is rotated in a fluid and measurements are calibrated against

measurements obtained with reference fluids.

3.4.4 Cone on Plate Viscometer

The cone on plate viscometer consists of a conical surface and a flat plate. Either of these

surfaces can be rotated. The clearance between the cone and the plate is filled with the fluid

and the cone angle ensures a constant shear rate in the clearance space. The advantage of this

viscometer is that a very small sample volume of fluid is required for the test. In some of these

viscometers, the temperature of the fluid sample is controlled during tests. This is achieved by

circulating pre-heated or cooled external fluid through the plate of the viscometer. These

viscometers can be used with both Newtonian and non-Newtonian fluids as the shear rate is

approximately constant across the gap. The schematic diagram of this viscometer is shown in

Figure 3.6.The dynamic viscosity can be estimated from the formula:

η = 3Mαcos2α(1 − α/2) / 2πωr3 = kM / ω where:

7

η is the dynamic viscosity [Pas];

r is the radius of the cone [m];

M is the shear torque on the cone [Nm]; ω is

the angular velocity [rad/s];

α is the cone angle [rad

k is the viscometer constant, usually supplied by the manufacturer [m-3].

Fig.3.6 Cone on plate viscometer.

3.4.5 Falling Ball Viscometer

The Most commonly used in many laboratories is the ‘Falling Ball Viscometer’. A glass

tube is filled with the fluid to be tested and then a steel ball is dropped into the tube. The

measurement is then made by timing the period required for the ball to fall from the first to the

second timing mark, etched on the tube. The time is measured with accuracy to within 0.1 [s]. This

viscometer can also be used for the determination of viscosity changes under pressure and its

schematic diagram is shown in Figure 3.7

The dynamic viscosity can be estimate d from the formula:

η = 2r2(ρb − ρ)gF / 9v (3.8 )

where:

η is the dynamic viscosity [Pas];

r is the radius of the ball [m];

ρb is the density of the ball [kg/m3];

ρ is the density of the fluid [kg/m3];

g is the gravitational constant [m/s2];

v is the velocity of the ball [m/s];

F is the correction factor.

8

9

Fig.3.7 Falling Ball Viscometer

3.5 Principle of Hydrostatic lubrication

The term 'Hydrostatic Lubrication' was coined arid introduced by D. D. Fuller in

1947..In case of hydrodynamic bearings, the pressure is developed by relative motion of the

mating surfaces. So, unlike the hydrodynamic bearings, hydrostatic bearings do not require

motion of one surface relative to another. Thus hydrostatic bearing can be defined as one in

which the loaded surfaces are separated by to fluid film which is forced between them by an

externally generated pressure thus formation of fluid film and the successful operation of the

bearing requires to supply pump which can operate continuously. In hydrostatic bearings, high

pressure lubricating fluid film is created by external source as the lubricating fluid is supplied

between two surfaces under pressure; these bearings are often called as externally pressurized

bearings.

In operation the loaded member as shown in Fig. 3.8 is raised by the pressure in the recess acting

on the bearing surface until flow through the restriction is equal to flow from the recess. The

integrated product of pressure and bearing area is then equal to the applied load. Thus, the bearing

clearance changes will accommodate load changes. As the fluid is supplied to the recess at certain

high pressure, a particular pressure profile exists over the area of bearing. This pressure

9

10

distribution can be maintained only if fluid is supplied to the recess at a rate equal to the rate at

which it escapes over the lands of bearings. The film thickness and lubricating fluid pressure

profile are fairly uniform across the interface. Usually, more than one bearing is supplied with

fluid by the same pump. These bearings are designed for use with both incompressible and

compressible fluids. Hydrostatic bearing can fulfill some of the extreme requirements:

(I) Extreme low frictional resistance.

(II) Heavy loadings at low speeds.

(III) High positional accuracy e.g. in machine tool Spindles

Fig. 3.8 Working principle of Hydrostatic lubrication

3.5.1 Compensator or Restrictor

The pads and flow restrictors are designed to adjust as needed to balance a load that is

unbalanced or uneven. The flow restrictors are necessary to make sure that the stiffness, pumping

power, pressure and lubricant flow are maintained at the necessary levels. In order to ensure that

a pressure drop between the manifold and the pad recess is steadily correct and never exceed the

supply pressure a flow restrictor is necessary.

A hydrostatic bearing system is a compensated bearing system. It is a compensated bearing system

because the hydrostatic bearing has restrictors that react to the balance of the load. If the load is

even across all pads, then the pressure is equal and lower then the designated pump pressure.

However, if the load is unbalanced then the restrictors adjust the pad pressure allowing the load to

automatically stay balanced.

Hydrostatic bearings have three different styles of compensators that are used. These three

different styles are

a) Orifice,

b) Capillary and

10

11

c) Variable-flow restrictors.

Orifice and capillary restrictors are fixed flow restrictors, meaning that the flow of fluid does not

automatically adjust to balance the pads. The variable flow restrictor is designed to adjust as

needed automatically, without manual adjustment.

Hydrostatic bearing systems require a number of specialized adjustments to ensure optimal

performance. The adjustments can be time consuming and have to be precise in order to guarantee

a balanced load. Many factors must be taken into account. But when done properly the hydrostatic

bearing design system assures a smooth and level ride.

3.6 Petroff’s Equation

The Petroff equation gives the coefficient of friction in journal bearings. It is based on the

assumption that the shaft is concentric. Though the shaft is not concentric but the coefficient of

friction predicted by this equation turns out to be quite good.Consider a shaft of radius “ r ” rotating

inside a bearing with rotational speed “ ”, and the clearance between the shaft and sleeve “c ” is

filled with oil (leakage is negligible).

From Newton’s viscosity equation we get

𝜏 = µ𝑈

ℎ

= µ2 𝜋 𝑟 𝑁

𝑐

The force needed to shear the film is

F = 𝜏 A

A = 2 π r l

11

12

Torque Fr = 𝜏 A r

Thus the torque can be written as

𝑇 =4 𝜋2𝑟3µ 𝑁

𝑐

The pressure on the projected area is P = 𝑊

2 𝑟 𝑙

W= 2 r l p and created frictional force “f w” is

T = f W r

= f x 2 r l p x r

=2 r2 f l p

Equating two frictional equation is “ f ” gives:

f = 2 π2 µ 𝑁

𝑝 𝑟

𝑐

Note: the quantities µ 𝑁

𝑝 and

𝑟

𝑐 are non-dimensional, and they are very important parameters in

lubrication.

12

3.6 Hydrostatic circular step Bearing

Hydrostatic, also called externally pressurized, lubricated bearings can

operate at little or no relative tangential motion with a large film thickness. The bearing

surfaces are separated by supplying a fluid (liquid or gaseous) under pressure at the interface

using an external pressure source, providing a high bearing stiffness and damping. There is

no physical contact during start-up and shut-down as in hydrodynamic lubrication.

Hydrostatic bearings provide high load-carrying capacity at low speeds, and therefore are

used in applications requiring operation at high loads and low speeds such as in large telescopes

and radar tracking units. High stiffness and damping of these bearings also provide high

positioning accuracy in high-speed, light- load applications, such as bearings in machine

tools, high-speed dental drills, gyroscopes, and ultracentrifuges. However, the lubricating

system in hydrostatic bearings is more complicated than that in a hydrodynamic bearing.

Hydrostatic bearings require high-pressure pumps and equipment for fluid cleaning which

adds to space and cost.

By supplying high-pressure fluid at a constant pressure or volume to a recess relief or

pocket area at the bearing interface, the two surfaces can be separated and the frictional force

reduced to a small viscous force, Figure 3.9. By proper proportioning of the recess area to

the cross-sectional (land) area of the bearing surface, the appropriate bearing load capacity

can be achieved (Wilcock, 1972; Gross et al., 1980; Fuller, 1984; Bhushan, 2001; Hamrock

et al., 2004; Williams, 2005).

Figure 3.9 a shows the essential features of a typical hydrostatic thrust bearing with a

circular step pad, designed to carry thrust load (Williams, 2005). A pump is used to draw

fluid from a reservoir to the bearing through a line filter. The fluid under pressure ps supplied

to the bearing before entering the central recess or pocket, passes through a compensating

or restrictor element in which its pressure is dropped to some low value pr. The fluid then

passes out of the bearing through the narrow gap of thickness, h, between the bearing land and

the opposing bearing surface, also known as the slider or runner. The depth of the recess is

much larger than the gap. The purpose of the compensating element is to bring a pressurized

fluid from the supply tank to the recess. The compensating element allows the pocket pressure

pr to be different from the supply pressure ps; this difference between pr and ps depends on

the applied load W. Three common types of compensating elements for hydrostatic bearings

include capillary tube, the sharp-edge orifice and constant-flow-valve compensation.

We now analyze the bearing performance. The bearing has an outer radius r0 and the central

recess of ri with slider at rest, Figure 8.4.2a. The film thickness is the same in radial or angular

positions and the pressure does not vary in the θ direction. We assume an incompressible fluid.

In the land region, ri < r < r0, the simplified Reynolds equation in the polar coordinates is

given as

∂

∂r ∂r

∂p

r = 0 (3.9)

Integrating we get

∂p= C

∂r1

p = C1 ℓn r +C2 (3.10)

13

Figure 3.8 Schematics of (a) a hydrostatic thrust bearing with circular step pad, and (b) a fluid

supply system. Reproduced with permission from Williams, J.A. (2005), Engineering Tribology, Second

edition, Cambridge University Press, Cambridge. Copyright 2005. Cambridge University Press.

We solve for constants C1 and C2 by using the boundary conditions that p = pr at r = ri and

p = 0 at r = r0. We get

p =

ℓn (r0/r)

pr ℓn (r0/ri )(3.11)

and

dp= −

pr

dr r ℓn (r0/ri )(3.12)

14

Figure 3.10 (a) Geometry and (b) pressure distribution for a circular step hydrostatic thrust bearing.

The radial volumetric flow rate per unit circumference in polar coordinates is given as

q =h

−dp3

12 η0 dr

h pr=3

12 η0 r ℓn (r0/ri )(3.13a)

and the total volumetric flow rate is

Q = 2 π r q (3.13b)

Combining Equations. 3. 11and 3.13we obtain an expression for p in terms of Q:

p =6 η Q0

π h3ℓn (r /r)0 (3.14)

3.6.1 Multi Recess pad Bearing

15

The drop in fluid pressure across the land is shown in Figure 3.10b. It is generally assumed

that the pressure of the fluid is uniform over the whole area of recess because the depth of the

recess in a hydrostatic bearing is on the order of one hundred times greater than the mean film

thickness of the fluid over its lands.

The normal load carried by the bearing, load-carrying capacity, is given as

W = π r p + prz2i r

r0

ri

ℓn (r0/r)

ℓn (r0/ri ) 2 ℓn (r0/ri )2 π r dr =

π pr r2 − r20 i

(3.15)

For a given bearing geometry, the load capacity linearly increases with an increase of fluid

pressure. Note that the load capacity is not a function of the viscosity. Therefore, any fluid that

does not damage the bearing materials can be used.

The load capacity in terms of Q can be obtained by combining an expression for Q from

Equations. 8.4.5a and b,

Wz =3 η0 Q

h3

r2 − r 2

0i

(3.16)

Fluid film bearings of the hydrostatic or hydrodynamic types have a stiffness characteristic

and will act like a spring. In conjunction with the supported mass, they will have a natural

frequency of vibration for the bearing. The frequency is of interest in the dynamic behavior

of rotating machinery. To calculate the film stiffness of the bearing, we take the derivative

of Equation 3.16 with respect to h, where h is considered a variable. For a bearing with a

constant flow-valve compensation (constant feed rate of Q, and not a function of h), the film

stiffness is given as

k f ≡dWz

dh h= −

3 3 η0 Q r2 − r20 i

h3

= −3Wz

h(3.17)

The negative sign indicates that kf decreases as h increases. The stiffness of the films in a

hydrostatic bearing with capillary tube and orifice compensation is lower than for a bearing

with a constant feed rate. Expressions for bearing stiffness for these bearings are presented by

Fuller (1984). The oil film stiffness in a hydrostatic bearing can be extremely high, comparable

to metal structures.

Next we calculate the frictional torque. Assume that the circumferential component of the

fluid velocity varies linearly across the film and that viscous friction within the recess is

negligible. From Equation 8.3.1, the shear force on a fluid element of area dA is written as

f = η0 d Au

h

= η0 (r dθ dr)rω

h

=η ωr drdθ0

2

h(3.18)

16

The friction torque is given by integrating over the entire land outside the recess area,

T =η0 ω

h

2π r0

0 ri

r3 dr dθ

=π η ω0

2 h r4

0—r 4

i

(3.19)

The total power loss consists of viscous dissipation, Hv , and pumping loss, Hp, which are

given as

Hv = T ω

and

Hp = pr Q

Therefore, the total power loss

Ht = Hv + Hp

=πη ω0

2

2 h 6 η ℓn (r /r )r i

4 40—r +

π h pr3 2

0 0 i

Note that Hv is inversely proportional to h and proportional to the square of the sliding velocity,

and Hp is proportional to h3 and independent of velocity. Generally, the bearing velocities are

low and only pumping power is significant.

It is generally assumed that total power loss is dissipated as heat. Further assuming that all

of the heat appears in the fluid, then the temperature rise, t, is given as

Ht = Q ρ cp t

or

t =Ht

Q ρ cp

(3.20)

The load-carrying capacity, associated flow rate and pumping loss are often expressed in

nondimensional terms by defining a normalized or nondimensional load W z , nondimensional

flow rate Q and nondimensional pumping loss H p , known as bearing pad coefficients. These

are given as,

W z =Wz 1−(ri /r0)2

Ap pr 2 ℓn (r0/ri )= (3.21)

Q =Q π

(W/Ap) (h3/η)

Hp

= (3.22)3 1−(ri /r0)2

2π ℓn (r0/ri )H p =

(W/Ap)2(h3/η)=

3 1−(r /r )2 2

i 0

(3.23)

where Ap is the total projected pad area = π r2.0

17

Figure 3.11 Bearing pad coefficients as a function of bearing geometry for circular step hydrostatic

thrust bearing (Source: Rippel, 1963).

Figure 3.11 shows the three bearing pad coefficients for various ratios of recess radius to

bearing radius. W∗ is a measure of how efficiently the bearing uses the recess pressure to

support the applied load. It varies from zero for relatively small recesses to unity for bearings

with large recesses with respect to pad dimensions. Q∗ varies from unity for relatively small

recesses to a value approaching infinity for bearings with large recesses. H p approaches

infinity for extremely small recesses, decreases to a minimum as the recess size increases(ri /r0 = 0.53) then approaches to infinity again for large recesses.

Example Problem 3.1

A hydrostatic thrust bearing with a circular step pad has an outside diameter of 400 mm and

recess diameter of 250 mm. (a) Calculate the recess pressure for a thrust load of 100,000 N,

(b) calculate the volumetric flow rate of the oil which will be pumped to maintain the filmthickness of 150 µm with an oil viscosity of 30 cP, (c) calculate the film stiffness for an applied

load of 100,000 N and operating film thickness of 150 µm, and (d) calculate the pumping loss

and the oil temperature rise. The mass density of the oil is 880 kg/m3 and its specific heat is

1.88 J/g K.

18

Solution

(a) Givenr0 = 200 mm

ri = 125 mm

Wz = 100,000 N

pr =2Wℓ n (r0/ri )

π r2 −r 20

i

=2× 10 × ℓn (200/125)

Pa5

π 0.22 −0.1252

= 1.23 MPa

(b)

η0 = 30 mPa s

h = 150 µm

Q =π h p3

r

6 η0 ℓn (r0/ri )

=π 1.5× 10 × 1.23× 10−4 3 6

6× 30× 10−3 ℓn (200/125)

= 154.1× 103 mm3/s

m /s3

(c)

k f =−3Wz

h

=−3× 10 5

150× 10−6N/m

= −2× 109 N/m

(d) Given

ρ = 880 kg/m3

cp = 1.88 J/g K

Hp = pr Q

= 1.23× 106 × 154.1× 10−6 N m/s

= 189.5 W

t =Hp

Q ρ cp

=189.5 ◦C

154.1× 10−6 × 880× 103 × 1.88

= 0.74 ◦C

19

Figure 3.12 Schematics of a hydrostatic multi recess bearing (a) with annular recess, and (b) four

recess segments.

Hydrostatic bearings can have single or multiple recesses that are circular, or annular or

rectangular in shape. Schematics of thrust bearings with annular recess, four recess segments

and a rectangular recess are shown in Figures. 3.12a, b and 3.11. A schematic of a journal

bearing with four rectangular recesses is shown in Figure 3.13

In the case of a rectangular recess without the essential degree of symmetry of the circular

pads, there are pressure gradients and so fluid flow in both the x and y directions in the bearing

plane. For a bearing with constant film thickness along the x and y axes in the land region and

for an incompressible fluid, the modified Reynolds equation is given as

∂2 p ∂2 p

∂x2 ∂y2+ = 0 (3.24)

This is known as the Laplace equation in two dimensions. For the case of a bearing with

the length much greater than the width of the lands, i.e. ℓ >> b, most of the fluid which is

supplied to the bearing by the pump leaves by flowing from the recess over the lands in the

20

Figure 3.13 Schematics of (a) a hydrostatic thrust bearing with rectangular recess, and (b) pressure

distribution within the recess along the horizontal axis, at the bearing interface.

Figure3.14 Schematic of a hydrostatic journal bearing with four rectangular recesses.

21

direction of the y axis. It is then evident that there is a negligible change of pressure over the

lands in the direction of the x axis at the ends of the recess. Therefore, the Reynolds equation

reduces to

d2 p

dy2= 0 (3.25a)

The Laplace equation can be solved for the rectangular recess by analytical methods. For

complex geometries this should be solved by numerical methods. Integrating Equation 8.4.18a

we get

p = C1 y +C2 (3.25b)

Integrating constants C1 and C2 are calculated by considering the boundary conditions thatp = pr at y = 0 and p = 0 at y = c (Figure 8.4.5). Therefore

p = pr 1− y

c(3.26a)

or

dp=− pr

dy c(3.26 b)

The pressure gradient is linear. From Equation 3.26, the volumetric flow rate along the y axis

for uniform pressure along the x axis is h3 ℓ pr /12 η0 c. Doubling this quantity must equal the

total flow rate of flow of fluid into the bearing from the pump:

Q =h ℓ p3

r

6 η0 c(3.27)

The load capacity of the bearing is given as

Wz = pr bℓ+ 2ℓ0

= pr ℓ (b+ c)

cpr 1− y

cdy

(3.28)

=6 η0 c Q

[ℓ (b + c)]

h3 ℓ(3.29)

The film stiffness is given as

k f ≡∂Wz

∂h=−

18 η0 c Q

h4 ℓ[ℓ (b+ c)] (3.30)

22

3.7 Hydrostatic lift

A rigorous analysis of hydrostatic journal bearings involves consideration of pad curvature, non-

uniformity of film thickness and effects of shaft rotation. Sometimes other factors like dynamic

response, cavitation, energy dissipation, active compensation, etc. become important in hydrostatic

bearings. Such analyses are performed by numerical methods and available in open literature.

The hydrostatic lubrication can maintain the fluid film at very low speed of shaft or even at zero speed

and this is done by external pressurization. Raising the shaft at low or zero speed is known as hydrostatic

lift or oil lift. The hydraulic lifts are used in heavy machinery such as hydraulic turbines, large

alternators, etc. the hydrostatic lift as shown in fig.3.15

Figure 3.15 Hydrostatic lift

3.8 Squeeze Film Journal Bearing

In a wedge film journal bearing, the bearing carries a steady load and the journal rotates relative

to the bearing. But in certain cases, the bearings oscillate or rotate so slowly that the wedge film cannot

provide a satisfactory film thickness. If the load is uniform or varying in magnitude while acting in a

constant direction, this becomes a thin film or possibly a zero film problem. But if the load reverses its

23

direction, the squeeze film may develop sufficient capacity to carry the dynamic loads without contact

between the journal and the bearing. Such bearings are known as squeeze film journal bearing. Different

between three lubrication as shown in fig 3.16

Fig.3.16 Different between three lubrication

Hydrodynamic

lubrication

Squeeze Film

lubrication

Hydro static

lubrication

24