me 575 hydrodynamics of lubrication by parviz merati, professor and chair department of mechanical...

ME 575ME 575 Hydrodynamics of LubricationHydrodynamics of Lubrication

By

Parviz Merati, Professor and Chair

Department of Mechanical and Aeronautical Engineering

Western Michigan University

Kalamazoo, Michigan

ME 575ME 575 Hydrodynamics of LubricationHydrodynamics of Lubrication

Fall 2001Fall 2001 An overview of principles of lubrication

– Solid friction– Lubrication– Viscosity– Hydrodynamic lubrication of sliding surfaces– Bearing lubrication– Fluid friction– Bearing efficiency– Boundary lubrication– EHD lubrication

ME 575ME 575Hydrodynamics of LubricationHydrodynamics of Lubrication

Movie on “Lubrication Mechanics, an Inside Look” General Reynolds equation Hydrostatic bearings Thrust bearings Homework #1 Journal bearings Homework #2 Hydrodynamic instability Thermal effects on bearings

– Viscosity– Density

ME 575ME 575Hydrodynamics of LubricationHydrodynamics of Lubrication

Viscosity-pressure relationship Laminar flow between concentric cylinders

– Velocity profile– Pressure– Mechanical Seals– Moment of the fluid on the outer cylinder

Homework #3

Solid FrictionSolid Friction

Resistance force for sliding– Static– Kinetic

Causes– Surface roughness (asperities)– Adhesion (bonding between dissimilar materials)

Factors influencing friction– Frictional drag lower when body is in motion– Sliding friction depends on the normal force and frictional

coefficient, independent of the sliding speed and contact area

Solid FrictionSolid Friction

Effect of Friction– Frictional heat (burns out the bearings, ignites a match)– Wear (loss of material due to cutting action of opposing

Engineers control friction– Increase friction when needed (using rougher surfaces)– Reduce friction when not needed (lubrication)

LubricationLubrication

Lubrication– Prevention of metal to metal contact by means of an intervening

layer of fluid or fluid like material Lubricants

– Mercury, alcohol (not good lubricants)– Gas (better lubricant)– Petroleum lubricants or lubricating oil (best)

Viscosity– Resistance to flow – Lubricating oils have wide variety of viscosities– Varies with temperature


Hydrodynamic lubrication (more common)– A continuous fluid film exists between the surfaces

Boundary lubrication– The oil film is not sufficient to prevent metal-to-metal contact– Exists under extreme pressure

Hydrodynamic lubrication– The leading edge of the sliding surface must not be sharp, but must be

beveled or rounded to prevent scraping of the oil from the fixed surface– The block must have a small degree of free motion to allow it to tilt and to

lift slightly from the supporting surface– The bottom of the block must have sufficient area and width to float on

the oil


Fluid Wedge– The convergent flow of oil under the sliding block develops a pressure-

hydrodynamic pressure-that supports the block. The fluid film lubrication involves the ‘floating” of a sliding load on a body of oil created by the “pumping” action of the sliding motion.

Bearings– Shoe-type thrust bearings (carry axial loads imposed by vertically

mounted hydro-electric generators)– Journal bearings (carry radial load, plain-bearing railroad truck where the

journal is an extension of the axle, by means of the bearings, the journal carries its share of the load)

– In both cases, a tapered channel is formed to provide hydrodynamic lift for carrying the loads

Fluid FrictionFluid Friction

Fluid friction is due to viscosity and shear rate of the fluid– Generates heat due to viscous dissipation– Generates drag, use of energy– Engineers should work towards reducing fluid friction– Flow in thin layers between the moving and stationary surfaces of the

bearings is dominantly laminar

= shear stress

Z = viscosity

dU/dy = shear rate

dydUZ

Fluid FrictionFluid Friction

Unlike solid friction which is independent of the sliding velocity and the effective area of contact, fluid friction depends on both

Unlike solid friction, fluid friction is not affected by load

Partial Lubrication (combination of fluid and solid lubrication)– Insufficient viscosity– Journal speed too slow to provide the needed hydrodynamic pressure– Insufficient lubricant supply

Overall Bearing FrictionOverall Bearing Friction

A relationship can be developed between bearing friction and viscosity, journal rotational speed and load-carrying area of the bearing irrespective of the lubricating conditions

F = Frictional drag

N = Journal rotational speed (rpm)

A = Load-carrying area of the bearing

f = Proportionality coefficient

ZNAfF


Coefficient of friction (friction force divided by the load that presses the two surfaces together)

is the coefficient of friction and is equal to F/L.

L is the force that presses the two surfaces together.

P is the pressure and is equal to L/A.

P

ZNf

L

F


ZN/P Curve– The relationship between and ZN/P depends on the lubrication

condition, i.e. region of partial lubrication or region of full fluid film lubrication. Starting of a journal deals with partial lubrication where as the ZN/P increases, drops until we reach a full fluid film lubrication region where there is a minimum for . Beyond this minimum if the viscosity, journal speed, or the bearing area increases, increases.

AnalysisAnalysis

Proper bearing size is needed for good lubrication. – For a given load and speed, the bearing should be large enough to operate

in the full fluid lubricating region. The bearing should not be too large to create excessive friction. An oil with the appropriate viscosity would allow for the operation in the low friction region. If speed is increased, a lighter oil may be used. If load is increased, a heavier oil is preferable.

Temperature-Viscosity Relationship– If speed increases, the oil’s temperature increases and viscosity drops,

thus making it better suited for the new condition.– An oil with high viscosity creates higher temperature and this in turn

reduces viscosity. This, however, generates an equilibrium condition that is not optimum. Thus, selection of the correct viscosity oil for the bearings is essential.

Boundary LubricationBoundary Lubrication

– Viscosity Index (V.I) is value representing the degree for which the oil viscosity changes with temperature. If this variation is small with temperature, the oil is said to have a high viscosity index. A good motor oil has a high V.I.

Boundary Lubrication– For mildly severe cases, additives known as oiliness agents or film-

strength additives is applicable– For moderately severe cases, anti-wear agents or mild Extreme Pressure

(EP) additives are used – For severe cases, EP agents will be used


Oiliness Agents– Increase the oil film’s resistance to rupture, usually made from oils of

animals or vegetables– The molecules of these oiliness agents have strong affinity for petroleum

oil and for metal surfaces that are not easily dislodged – Oiliness and lubricity (another term for oiliness), not related to viscosity,

manifest itself under boundary lubrication, reduce friction by preventing the oil film breakdown.

Anti-Wear Agents– Mild EP additives protect against wear under moderate loads for boundary

lubrications– Anti-wear agents react chemically with the metal to form a protective

coating that reduces friction, also called as anti-scuff additives.


Extreme-Pressure Agents– Scoring and pitting of metal surfaces might occur as a result of this case,

seizure is the primarily concern– Additives are derivatives of sulfur, phosphorous, or chlorine– These additives prevent the welding of mating surfaces under extreme

loads and temperatures

Stick-Slip Lubrication– A special case of boundary lubrication when a slow or reciprocating

action exists. This action is destructive to the full fluid film. Additives are added to prevent this phenomenon causing more drag force when the part is in motion relative to static friction. This prevents jumping ahead phenomenon.

EHD LubricationEHD Lubrication

In addition to full fluid film lubrication and boundary lubrication, there is an intermediate mode of lubrication called elaso-hydrodynamic (EHD) lubrication. This phenomenon primarily occurs on rolling-contact bearings and in gears where NON-CONFORMING surfaces are subjected to very high loads that must be borne by small areas.

-The surfaces of the materials in contact momentarily deform elastically under extreme pressure to spread the load.

-The viscosity of the lubricant momentarily increases drastically at high pressure, thus increasing the load-carrying ability of the film in the contact area.

Reynolds EquationReynolds Equation

In bearings, we like to support some kind of load. This load is taken by the pressure force generated in a thin layer of lubricant. A necessary condition for the pressure to develop in a thin film of fluid is that the gradient of the velocity profile must vary across the thickness of the film. Three methods are available.

– Hydrostatic Lubrication or an Externally Pressurized Lubrication- Fluid from a pump is directed to a space at the center of bearing, developing pressure and forcing fluid to flow outward.

– Squeeze Film Lubrication- One surface moves normal to the other, with viscous resistance to the displacement of oil.

– Thrust and Journal Bearing- By positioning one surface so it is slightly inclined to the other and then by relative sliding motion of the surfaces, lubricant is dragged into the converging space between them.


Use Navier-Stokes equation and make the following assumptions– The height of the fluid film h is very small compared with the length and

the span (x and z directions). This permits to ignore the curvature of the fluid film in the journal bearings and to replace the rotational with the transnational velocities.


– Since the fluid layer is thin, we can assume that the pressure gradient in the y direction is negligible and the pressure gradients in the x and z directions are independent of y

– Fluid inertia is small compared to the viscous shear– No external forces act on the fluid film– No slip at the bearing surfaces– Compared with u/y and w/y, other velocity gradient terms are

negligible

0.0

y

p

)(yfnz

pand

x

p


2

21

y

u

x

p

2

21

y

w

z

p

B.C.

y = 0.0, u = U1 , v = V1 , w = W1

y = h, u = U2 , v = V2 , w = W2

Integrating the x component of the above equations would result in the following equation.


Integrating the z-component

)()(2

1121

2 UUh

yUyhy

x

pu

)()(2

1121

2 WWh

yWyhy

z

pw


u and w have two portions;– A linear portion– A parabolic portion


Using continuity principal for a fluid element of dx, dz, and h, and using incompressible flow, we can write the following relationship

Where,

21qdz

z

qqdx

x

qqqqq z

zx

xzx

h

x dzdyuq0

.0h

z dxdywq


dzhUU

dzx

phq

x 21221

3

dzdxVq11

dxhWW

dxz

phqz 212

213

Fluid moving into the fluid element in the Y direction is q1


dzdxz

hWdzdx

x

hUdzdxVq

2222

z

hWWVV

x

hUU

z

ph

zx

ph

x

)()(2)()()(6

1212121

33

)()( 2121 WWz

hUUx

h

The last two terms are nearly always zero, since there is rarely a change in the surface velocities U and W.

Reynolds Equation in Cylindrical Coordinate Reynolds Equation in Cylindrical Coordinate SystemSystem

h

rTTVV

r

hRR

ph

r

phr

rr r

1)()(2)()()(

1

6

1212121

31

3

2

)()( 2121 TTRRrrr

h

R1 and R2 are the radial velocity of the two surfaces

T1 and T2 are the tangential velocity of the two surfaces

V1 and V2 are the axial velocity of the two surfaces

Hydrostatic BearingsHydrostatic Bearings

Lubricant from a constant displacement pump is forced into a central recess and then flows outward between bearing surfaces. The surfaces may be cylindrical, spherical, or flat with circular or rectangular boundaries.

If the pad is circular as shown in the following figure,


0)(

r

pr

r

dDr

D

p

p

ln

2ln

0

2

20

..

0

dratpp

Dratp

CB

0

22

24

)2( pd

drrpP

D

d

Total Load P

)(

)/(ln8220 dD

dDPp

The hydrostatic pressure required to carry this load is p0.


What is the volumetric flow rate of the oil delivery system?

h

r dyurQ0

2

Using Reynolds Equation for rectangular system, and substituting x with r, and considering that U1 and U2 are zero, the following relationship can be obtained for radial component of the flow velocity ur.

)(

)(422

2

dDr

yhyPur


)(3

422

3

dD

hPQ

What is the power required for the bearing operation?

A = Cross sectional area of the pump delivery line

V = Average flow velocity in the line

= Mechanical efficiency

QpVAp

quiredPower 00 )(Re


What is the required torque T if the circular pad is rotated with speed n about its axis ?

The tangential component of the velocity is represented by Wt and the shear stress is shown by

h

ynrWt 2

h

nr

y

wt 2

drrh

nrrdArdFrT

D

d

222

2

)(16

442

dDh

nT

Thrust BearingsThrust Bearings There should be a converging gap between specially shaped pad or

tilted pad and a supporting flat surface of a collar. The relative sliding motion forces oil between the surfaces and develop a load-supporting pressure as shown in the following figure.

– Using the Reynolds Equation and using h/z = 0, for a constant viscosity flow, the following equation is obtained

x

hUU

z

ph

zx

ph

x

)(6)()( 21

33

Thrust BearingsThrust Bearings

This equation can be solved numerically. However if we assume that the side leakage w is negligible, thus p/z is negligible, then the equation can be solved analytically

x

hUU

x

ph

x

)(6)( 213

bxathh

xathh

CB

2

1 0

..

bhh

Defining 21


222

21

)()2(

)()(6

xbhhh

xbxUUp

Total load can be found by integrating over the surface area of the bearing.

Flat Pivot

Flat pivot is the simplest form of the thrust bearing where the fluid film thickness is constant and the pressure at any given radius is constant. There is a pressure gradient in the radial direction. The oil flows on spiral path as it leaves the flat pivot.


What is the torque T required to rotate the shaft?

Shear stress is represented by

1

0

22r

drrT

hr

2

2

1 ,2

rAwhereh

rAT


What is the pressure in the lubricant layer?

Pressure varies linearly from the center value of p0 to zero at the outer edge of the flat pivot.

If we define an average pressure as pav

)1(1

0 rr

pp

1

0

2r

av drrpAp

30p

pav


What is the viscous friction coefficient?

RPMshafttheisNwherehr

pN

fav

12

hrA

drrpfTr

22

2

1

0

21


Pressure Variation in the Direction of Motion

B

xeh

breadththeof

centerthefromxcedisaatXXatThickness

2

tan,

0,max'

121

21 3

dxdp

pressureimumtoingcorrespondx

dxdp

VXXacrossFlow


Integrating and using the following boundary condition

)'(12

12

1)

2(

2

1)'2

(2

1

3

3

xxB

Ve

dx

dp

dx

dp

B

exhV

B

exhV

Continuity

0,2

1 pBx

surfacepadorbearingtheofattitudeh

eawhere

hh

a

ehVBp

,

2

11

)(

1

2

1

)3

( 2

2


As the attitude of the bearing surface a is reduced, pressure magnitude decreases in the fluid film and the point of maximum pressure approaches the middle of the bearing surface. For a = 0, the pressure remains constant.

)1(2)3

(2

2

a

a

ehVB

p

ppressureMaximum

m

m


What are the total load and frictional force on the slider?

Define P and F' as the load and drag force per unit length perpendicular to the direction of motion.

q is the shear stress and is defined by the following equation

2/

2/

'

2/

2/

B

B

B

B

dxqF

dxpP

dx

dpVq

2

1


Coefficient of friction f is defined by the following relationship.

)2

3

1

1(ln

2

)(

'

)21

1(ln

2

3

)3

(2

2

2

a

a

a

ahVBF

aa

a

ahVBP

PfF '

If is the angle in radians between the slider and the bearing pad surface, then the following equations based on the equilibrium conditions of the film layer exist.

Since is very small, film layer thickness h and e are small relative to the bearing length B, sin , and cos 1. It is also safe to assume that Fr is small compared with Q.


cossin

sincos'

r

r

FQF

FQP

Thrust BearingsThrust BearingsCritical value of occurs when Fr =0. This will result in

is the angle of friction for the slider. When > , Fr becomes negative. This is caused by reversal in the direction of flow of the oil film . The critical value of a is thus obtained by using the following relationship.

Thus the range of acceptable variation for a is 0 < a <0.86

tan'

P

F

86.0

2

h

ea

B

ef

Homework 1Homework 1

For a thrust bearing, plot non-dimensionalized pressure along the breath of the bearing for several values of the bearing attitude defined by a=e/h, ( 0 a 0.86). In addition, plot non-dimensionalized maximum pressure, load per unit length measured perpendicular to the direction of motion, tangential pulling force, and virtual friction coefficient versus the bearing attitude. For each plot, please discuss your findings and provide conclusions.

Note:

Please refer to figure 5.11 and sections 5.4.2, 5.4.3, and 5.4.4 of your notes for additional information.

Journal BearingsJournal Bearings

In a plain journal bearing, the position of the journal is directly related to the external load. When the bearing is sufficiently supplied with oil and external load is zero, the journal will rotate concentrically within the bearing. However, when the load is applied, the journal moves to an increasingly eccentric position, thus forming a wedge-shaped oil film where load-supporting pressure is generated.


Oj = Journal or the shaft center

Ob = Bearing centere = Eccentricity

The radial clearance or half of the initial difference in diameters is represented by c which is in the order of 1/1000 of the journal diameter. = e/c, and is defined as eccentricity ratio

If = 0, then there is no load, if = 1, then the shaft touches the bearing surface under externally large loads.

ce 0

10


What is the lubricant’s film thickness h?Using the above figure, the following relationship can be obtained for h

The maximum and minimum values for h are

r = Journal radiusr+c = Bearing radius

)cos1( ch

)1(

)1(

min

max

cech

cech


Using Reynolds equation and assuming an infinite length for the bearing, i.e., p/ z = 0, and U = U1+U2 , the following differential equation is obtained.

Reynolds found a series solution in 1886 and Sommerfeld found a closed form solution in 1904 which is widely used.

x

hU

x

ph

x

6)( 3

222 )cos1()2(

)cos2(sin6

c

Urp


Modern bearings are usually shorter, the length to diameter ratio is often shorter than 1. Thus, the z component cannot be neglected. Ocvirk in 1952 showed that he could safely neglect the parabolic pressure induced part of the U component of the velocity and take into account the z variation of pressure. Thus, the following simplified equation can be obtained.

If there is no misalignment of the shaft and bearing, h and h/ x are independent of z, then the above equation can be easily integrated with the following boundary conditions for a journal of length l.

x

hU

z

ph

z

6)( 3


Ocvirk Solution of the Short Bearing Approximation

Thus, axial pressure distribution is parabolic.

0,2

0,0

..

pl

zAt

z

pzAt

CB

32

2

2 )cos1(

sin3)

4(

z

l

rc

Up


At which angle the maximum pressure occur? m=?

To find m’ p/ =0.

What is the total load that is developed within the bearing?

The oil film experiences two forces, one from the bearing, the other from the journal. The bearing force P passes through the center point of the bearing, the journal force P passes through the journal center.

)4

2411(cos

21

m


The hydrodynamic pressure force is always normal to the bearing and journal surfaces. In order to find the total load, the pressure force over the bearing surface must be integrated. Since the oil film is stationary, the resultant of the external forces and moments, i.e. bearing and journal forces and moments exerted on the oil film, must be zero. The total load P carried by the bearing is calculated by the following equation.

Where is defined as the attitude angle and is the angle between the line of force and the line of centers. The two components of the load normal and parallel to the line of centers are represented by P sin and P cos .

22 )sin()cos( PPP


l

l

dzdrpP

dzdrpP

0 0

0 0

sin)(2sin

cos)(2cos

22

2/1222

2

3

)1(4

16)1(

c

lUP

41

tan)cossin

(tan2

11

PP

Journal Load and the Attitude Angle


With an increasing load, will vary from 0 to 1 and the attitude angle vary from 90 degrees to zero. The path of the journal center Oj as the load and eccentricity are increased is shown in the following figure.

HomeworkHomework 2 2

Non-dimensionalize the hydrodynamic pressure and load of equations 5.48 and 5.51 of your notes, respectively. These are the Ocvirk equations for short journal bearings. Plot this non-dimensionalized pressure versus at z = 0.0 for eccentricity ratios = 0.1, 0.3, 0.5, 0.7, and 0.9. Plot the location and magnitude of the maximum pressure with respect to at

z = 0.0. Plot the non-dimensionalized load P and the attitude angle versus . For each plot, please discuss your findings and provide conclusions.

Hydrodynamic InstabilityHydrodynamic Instability

Synchronous whirl– Caused by periodic disturbances outside the bearing such that the bearing

system is excited into resonance. Shaft inertia and flexibility, stiffness and damping characteristics of the bearing films, and other factors affect this instability. The locus of the shaft center called the whirl orbit increases at the critical shaft speed where there is resonance. It is usual procedure to make the bearings such that the critical speeds do not coincide with the most commonly used running speeds. This may be done either by increasing the bearing stiffness so that the critical speeds are very high, or reducing the stiffness so that the critical speeds are quickly passed through and normal operation takes place where the attenuation is large. Stiffness can be increased by reducing the bearing clearance. Introduction of extra damping by mounting the bearing housings in rubber “O” rings or metal diaphragms are other methods to suppress the synchronous whirl.


Half-Speed whirl– This is induced in the lubricant film itself and is called “half-speed whirl”.

This is because due to existence of the attitude angle , the reaction force from the lubricant on the shaft has a component normal to the line connecting the centers of the shaft and the bearing. This component causes the shaft to move in a circumferential direction, i.e., at the same time as the shaft moves around its center, the shaft center rotates about the bearing center. If the whirl takes place at the half the rotational speed of the shaft, this will coincide with the mean rotational speed of the lubricant. Because, the lubricant, on the average, does not have a relative velocity with respect to the shaft, the hydrodynamic lubrication fails. Extra damping, axial groves on the bearing housing, partial bearing are some of the techniques to get rid of this instability.

Thermal Effects on BearingsThermal Effects on Bearings

We have assumed that fluid viscosity and density remains constant in deriving the Reynolds equations. In reality due to viscous dissipation because of the large existing shear stress, the lubricant’s temperature rises and thus the fluid density and viscosity change. Since the fluid is unable to expand due to restriction, fluid pressure increases as the temperature increases. This is called Thermal Wedge. Consider the General R.E. with the viscosity and density variation in the sliding direction.

)2

()12

(3 hU

dx

d

dx

dph

dx

d


After integrating the above equation,

In this equation, A and B are constants. The variation of density with temperature can be approximated by the following relationships.

Bdxh

Adx

h

Up

32

126

)( iti TT

)( ioi TTL

xTT

)( ioi L

x


Contribution of viscosity variation for liquids compared with density variation is negligible since viscosity increases with pressure and decreases with temperature. Thus, we can assume that viscosity remains constant in the sliding direction. Using the following boundary conditions, the pressure variation due to temperature variation for a parallel bearing can be obtained.

Lxatp

xatp

0

00

'

'

2 ln

)1(1ln6

L

x

L

x

h

ULp


Where,

For mineral oil,

)(1'

'

ioi

t

i

o

TT

3

3

9.0

00065.0

cm

gr

Ccm

gr

i

t


Thus, for a rise in temperature of 100 C, ' =0.93. The dimensionless pressure p' is

L

x

L

x

LU

hpp 07.01ln78.13

6

2'


p´max is about 0.011 and for a plane-inclined slider, p´max is about 0.042. The parallel surface bearing has a load capacity approximately 1/3.5 that of the corresponding inclined slider. It is rare that the temperature rise is 100 C, usually the temperature rise due to viscous dissipation is in the order of 2-20 C and under these conditions, it is safe to assume that the effect of temperature is negligible.

Viscosity-Pressure RelationshipViscosity-Pressure Relationship

In some situations where extreme pressures can occur such as in the restricted contacts between gear teeth and between rolling elements and their tracks, viscosity relationship with pressure is represented by the following equation.

Where 0 and are reference viscosity and the pressure exponent of the viscosity, respectively. In order to integrate R.E., we have to introduce parameter q defined as

pe 0

)1(1 peq

Viscosity-Pressure RelationshipViscosity-Pressure Relationship

The differential equation that is obtained as the result of this substitution, looks like a normal R.E. with viscosity term being 0. This equation can then be integrated and pressure can be obtained from the following relationship.

Although load remains finite, pressure is tending to approach an infinite value between two disks rolling with some degree of sliding as shown in the following figure. This does not happen in reality. In reality, large pressures produce deformation of the bodies which distribute the pressure over a finite area. This is called “Elasto-hydrodynamic” lubrication or EHD.

)1(ln1

qp

Laminar Flow Between Concentric CylindersLaminar Flow Between Concentric Cylinders

Using Navier-Stokes equations in cylindrical systems and the following simplifications,

Vr =0

Vz = 0

v = uThe r-component is

The component is

r

u

dr

dp 2

0)(1

rudr

d

rdr

d


B.C. for velocity

B.C. for pressure

222

111

,

,

RuRr

RuRr

)()(

121

22

21

12

122

221

22

r

RRrRR

RRu

11, ppRr


)

11()(

2

1ln))((2

2

)()(

)( 221

221

42

41

11

212

2221

22

21

21

22

12

122

2221

22

1 rRRR

R

rRRRR

RrRR

RRpp

For the case of mechanical seals where the inner cylinder is rotating and the outer cylinder is stationary, i.e. 2 = 0

2

22

21

22

12

2

21

22

2

2

22

21

2

2

121

21

11 2

1ln2)(

2

1

1)(

11

r

R

R

R

R

r

R

R

R

r

R

R

RR

R

pp

p


If the inner cylinder is at rest , 1 = 0, the moment of the fluid on a length L of the outer cylinder is described by

2

1

2

1

2

2

111 )(1

1

R

R

R

r

r

R

RRR

u

2)(

2

2

2222

Rrr

u

rr

RLRM


Viscosity can be calculated from this equation if the moment on the outer cylinder is measured.

222

21

22

21

2 4 RR

RRLM

HomeworkHomework 3 3

Calculate and plot pressure ratio p/p1, and velocity ratio u/(R11) versus the radial location (r-R1)/(R2-R1) for the flow between concentric cylinders for water, oil, and sodium iodide solution. The radii of the inner and outer cylinders are R1 = 0.031 m and R2 = 0.046 m, respectively. p1 is the pressure at the inner cylinder surface and r is the radial location. The outer cylinder is stationary and the inner cylinder is rotating at 1,200 rpm. Density of water, oil and sodium iodide solution (67% by volume) are 1,000, 880, and 1,840 Kg/m3, respectively. Assume that p1 is atmospheric pressure. Although the flow at this rotational speed is turbulent, the time average of the flow velocity and pressure are close to the laminar flow values. For each plot, please discuss your findings and provide conclusions.

Thank YouThank You

me 575 hydrodynamics of lubrication by parviz merati, professor and chair department of mechanical...

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