MASTER’S THESIS2006:242 CIV

EVA WETTERHOLM

Simulation of Flowin a Radial Lip Seal UsingDifferent Viscosity Models

MASTER OF SCIENCE PROGRAMMEEngineering Physics

Luleå University of TechnologyDepartment of Applied Physics and Mechanical Engineering

Division of Machine Elements

2006:242 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 06/242 - - SE

Preface

This thesis is the final project for the Master of Science in Engineering Physics and apart of the Research Trainee program. This work has been carried out at Luleå Univer-sity of Technology at the Division of Machine Elements, in cooperation with SKF Engi-neering Research Centre, The Netherlands.

I would like to thank my supervisor and examiner Elisabet Kassfeldt for her help andsupport during the work. I would also like to thank COMSOL support in Stockholm fortheir help with the software problems and all in the Researchtrainee group 2005/2006.Finally I appreciate the time my supervisors at SKF Netherlands, Piet Lugt and JoopVree have spent in guiding me.

Luleå, May 2006

Eva Wetterholm

i

Abstract

Radial lip seals for rotating shaft have been used for a long time to keep lubricant withinbearings and air and dust particles outside. How they work isnot completely under-stood. This study examines operation of a plain radial lip seal lubricated with greaseand aims at developing a model that explains the lubricationand sealing behaviour ofthis type of seal.

One existing hypothesis how the seals function is the Weissenberg effect hypoth-esis. The hypothesis state that the lubricant is exposed to extreme shear load, whichwill contribute to the pumping effect and the maintenance ofthe lubricated film. Aninvestigation is therefore made to examine if the non Newtonian characteristics of thegrease prevents leakage.

Comparison between Newtonian and non Newtonian fluid calculated by using thefinite element software program COMSOL Multiphysics has been performed. Theresults show that there are differences between Newtonian and non Newtonian flow.The non Newtonian characteristics contribute to a larger flow in the axial direction ascompared to the corresponding Newtonian fluid. This may contribute to a pumpingeffect and transportation of heat near the minimum film thickness. At the edge ofthe contact, the non Newtonian fluid becomes more viscous than the Newtonian fluid.The magnitude of the velocity in the pumping direction for the non Newtonian fluiddecreases, which may contribute to the maintenance of the lubricated film.

iii

Contents

Nomenclature 1

1 Radial lip seals 31.1 Seal design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Elastomeric materials for radial lip seals . . . . . . . . .. . . 51.1.2 Garter spring . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Lubricating grease . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Non Newtonian fluids 72.1 The Weissenberg effect . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Method 93.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Course of action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Swirl flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 The geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 The boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 103.6 The mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 Solving the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.8 The accuracy of the results . . . . . . . . . . . . . . . . . . . . . . . 12

3.8.1 Numerical estimation of the shear rate . . . . . . . . . . . . .12

4 Results 134.1 Cursory investigation of the flow . . . . . . . . . . . . . . . . . . . .13

4.1.1 Non Newtonian fluid . . . . . . . . . . . . . . . . . . . . . . 134.1.2 Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Thorough investigation of the flow . . . . . . . . . . . . . . . . . . .154.2.1 Axial velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.2 Net flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.3 Effects of the centrifugal force . . . . . . . . . . . . . . . . . 15

5 Error analysis 19

6 Discussion 216.1 Accuracy of the result . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Future work 23

References 25

A The script 27

v

Nomenclature

F Body force [N]

h0 Minimum film thickness [m]

m Constant in the power law

n Constant in the power law

p Pressure [Pa]

u (ux,uy,uz) or(

ur ,uϕ,uz)

[m/s]

(ux,uy,uz) Velocity vector in Cartesian coordinates [m/s](

ur ,uϕ,uz)

Velocity vector in Cylindrical coordinates [m/s]

α lip angel air side [◦]

β lip angel seald side [◦]

γ̇ Total shear rate

η Dynamic viscosity [Pas]

ρ Density [kg/m3]

τyx Shear stress [kg/m3]

1

1 RADIAL LIP SEALS

1 Radial lip seals

The rotary lip seal, as shown in Figure 1, is the most common type of rotary shaft seals.They are used throughout the industry to withstand differences in pressure, containlubricant and exclude contaminants such as air, water and dust particles.

Figure 1: Radial lip seals are widely used to seal lubricant in many kinds ofindustrial compo-nents, for example rolling element bearings. The figure shows a schematic cross section of aradial lip seal.

The rotary lip seals has bean used for many decades but how they work is still notcompletely understood. Between the lip and the running shaft a 1-2µm thin lubricatedfilm is established [1]. Despite the presence of the film, a successful lip seal exhibits noleakage and a meniscus separates the sealed liquid from the surrounding atmosphere[2], see Figure 2. The meniscus is located on the air side of the lip, at the air side edgeof the sealing zone or at some location within the sealing zone. If the air side of asuccessful seal is flooded with liquid, then the liquid will be pumped to the liquid side.This reverse pumping is essential for the operation of the seal, if the reverse pumpingis very low or zero, the seal will not perform well under normal operating condition[3]. A balanced pumping without net flow in any direction is most favourable.

Figure 2: A schematic figure of the contact area between the shaft and the lip. If the lip issuccessful a lubricated film will separate the lip from the shaft and a meniscus will be established.The meniscus separates the lubricant from the surrounding media and the seal will not leak.

Several studies have been performed on oil lubricated radial lip seals. The mostcommon approximation in these studies is that the oil behavelike a Newtonian fluidbecause of the narrow gap between the lip and the shaft, see for example [4]. Other

3

1.1 Seal design 1 RADIAL LIP SEALS

studies have indicated that it is the non Newtonian behaviour of the lubricant due tothe extreme shear load that contributes to the maintenance of the film and the pumpingeffect [5]. This hypothesis is called the Weissenberg effect.

This study examines the grease lubricated radial lip seal where the non Newto-nian characteristics of the lubricant can not be neglected.The investigation is madeby using computer fluid dynamics (CFD) and the lubricating grease is assumed to be-have according to the power law. A comparison between non Newtonian fluids withpower law and Newtonian fluids will be performed, to investigate if the non Newtonianbehaviour prevents leakage.

A part of this project was to carry out a literature study and to determine howthe model should be simplified to make it solvable. The project considered differentdirections, but the deformation of the lip was the main concern. The investigation alsoconsidered as to how the anglesα and β in Figure 2 influenced the deformation ofthe lip due to the flow of grease, a fluid structure interactionmodel. The grease wasthen assumed to be a thick Newtonian fluid. After a literaturesurvey about radial lipseals and a course on introduction to COMSOL Multiphysics, the project went on withsimulation of the flow.

1.1 Seal design

The lip is a circular elastic part made of an elastomer bondedto a rigid case, see Figure3. The inner diameter of the lip is smaller than the shaft’s outer diameter so that whenthe seal is installed the lip is stretched outward and creates a force between the lip andshaft. As the shaft rotates, oscillates or reciprocates, the lip will flex and follow theshafts motion to prevent leakage. A garter spring is often added to compensate forchanges in the rubber properties that occur when the material is subjected to heat anddifferent kinds of chemicals.

Figure 3: A schematic figure of the sealed bearing unit where the lip seal isbounded to a rigidcase. A garter spring is added to compensate for the changes in the rubber properties that willoccur when the lip is subjected to chemicals.

4

1 RADIAL LIP SEALS 1.1 Seal design

An important factor of the lip is the cross section, for an oilseal the angle relativeto the shaft axis on the air side must be smaller than the corresponding angle on theliquid side, see Figure 2. The angles also must lie within a certain limit if the seal isto be successful [6]. If a successful seal is mounted by inverting the air and the liquidside, then the seal will leak excessively [7].

Since the seal behaviour is very complex, most of the development in both sealdesign and materials have been due to trial and error. That iswhy there are many typesof radial lip seal designs and material to choose from.

1.1.1 Elastomeric materials for radial lip seals

The elastomeric material in the lip consists mostly of a basepolymer mixed with fillers,antioxidants and curing agents. The material must be compatible with the sealed fluid,be abrasion and tear resistant, and maintain physical properties for an extended periodof time. The elastomer usually tends to swell in contact withoil and chemicals, whichcan influence the lip quality. Choice of the right material isessential for the sealslife and there are many different kind of materials to choosefrom. Two of them areFlouroelastomer (FKM) and Nitrile (NBR).

Fluoroelastomer

Fluoroelastomers is an expensive material but it is often used in rotating shaft sealapplications because it has a well oil and chemical resistance. It has good high temper-ature properties and will not fracture in low temperature. Low temperature can makethe elastomer stiff, and it may not follow the shaft wobble and vibrations until fric-tion heat warms up the elastomer. This could cause sporadic leakage if the lubricantremains at low temperatures. Fluoroelastomer compounds are mostly used for highspeed, high temperature operations with spars lubrications.

Nitrile

Nitrile is a low cost material relative to other polymers, with good oil and abrasionresistance. The biggest disadvantage of the compound is thelack of heat resistanceat elevated temperatures. The material will harden, crack and lose interference withthe shaft, which will result in failure. Nitrile compounds are mostly used when theoperating condition is mild.

1.1.2 Garter spring

The elastomer of the lip will absorb the lubricant when exposed to elevated tempera-tures. The seal lip swell and softens, which results in a decrease of lip interference andload between the lip and the shaft. In the worst scenario the lip expands completely ofthe shaft and results in a leakage. A garter spring is used to compensate for materialchanges and provide a uniform load which will extend the lifeof the seal. The garterspring is a helical coiled spring whose ends are connected sothat the spring forms acircle. The seal life and lip wear width is related to spring tension when the spring isused.

5

1.2 Lubricating grease 1 RADIAL LIP SEALS

1.2 Lubricating grease

Lubrication is used to minimise wear and help bearings and other mechanical applica-tions to run smoothly. The most common way is to lubricate with oil or grease.

The main difference between these two lubricants is their difference in consistency.Grease consistency is more semi-solid than oil and that is the key to many of its ad-vantages. For example, grease can prevent external contaminants from entering thebearing and it is less easily displaced from bearings surfaces than oil. It will thereforenot drain from the bearing under gravity. However, unlike oil, grease provides a verypoor means of transferring friction heat away from the sliding surface, which usuallyleads to higher friction due to viscous forces.

Lubricating grease consists of a mineral based oil and thickening agents. The ad-ditives can be of metallic soap, urea compound, carbon blackor other materials. Thecomponents of a grease make it very hard to describe its rheological properties. Thepolymeric substance has non Newtonian characteristics hence its resistance to shearand elastic properties. The apparent viscosity varies withtemperature, pressure, timeand shear. At very low shear rate the grease acts as a solid or semisolid. When theshear rate increases, the viscosity approaches that of the base fluid.

6

2 NON NEWTONIAN FLUIDS

2 Non Newtonian fluids

The mechanical behaviour of a fluid when it is subjected to forces depends on its rheo-logical properties. One way to define these properties is howthe shear strain respondsto shear stress. For Newtonian fluids the shear stress is proportional to the relative rateof movement

τyx = µγ̇yx (2.1)

whereµ is the dynamic viscosity, which is by definition only dependent on temperatureand pressure.̇γxy is the rate of shear strain or shear rate

γ̇yx =dγyx

dt=

dux

dy. (2.2)

When the stress is removed from the fluid, the shear rate goes tozero, i.e. the motionstops, but there is no tendency of the fluid to return to any past state. If the propertiesof the fluid are such that the shear stress and the shear rate are not directly proportionalbut are instead related by some more complex function, the fluid is said to be nonNewtonian. One approximation of this complex function is the power law where thedynamic viscosity depends on the shear rate

η = mγ̇n−1 (2.3)

wherem andn are scalars that characterize the non Newtonian fluid. The total shearrateγ̇ are in Cartesian respectively in cylindrical coordinate systems defined as

γ̇ =

√

2

(

∂ux

∂x

)2

+2

(

∂uy

∂y

)2

+2

(

∂uz

∂z

)2

+

+

(

∂ux

∂y+

∂uy

∂x

)2

+

(

∂uy

∂z+

∂uz

∂y

)2

+

(

∂ux

∂z+

∂uz

∂x

)2

, (2.4)

γ̇ =

√

2

(

∂ur

∂r

)2

+2

(

1r

∂uϕ

∂ϕ+

ur

r

)2

+2

(

∂uz

∂z

)2

+

+

(

r∂∂r

(uϕ

r

)

+1r

∂ur

∂ϕ

)2

+

(

1r

∂uz

∂ϕ+

∂uϕ

∂z

)2

+

(

∂ur

∂z+

∂uz

∂r

)2

. (2.5)

To solve the velocity profile needed to calculate 2.4 and 2.5 Navier-Stokes equationis used, with the assumption that the fluid is incompressible

ρ∂u∂t

−∇η(∇u+(∇u)T)+ρ(u ·∇)u+∇p = F (2.6)

∇ ·u = 0. (2.7)

In the equations above isu the velocity vector,(ux,uy,uz) in Cartesian coordinates and(

ur ,uϕ,uz)

in cylindrical coordinates,ρ is the density of the fluid,p is the pressure andF is the body force term.

7

2.1 The Weissenberg effect 2 NON NEWTONIAN FLUIDS

2.1 The Weissenberg effect

There are many ways the human eye can see differences betweenthe Newtonian andnon Newtonian fluid. One example is the Weissenberg rod climbing effect, which willoccur when a vertical rod is rotating in a cup of non Newtonianfluid. If the fluid is ofNewtonian character the centrifugal force causes the fluid to move radially towards thecups wall, see Figure 4(a). For a non Newtonian or polymeric liquid the fluid moves tothe rotating rod, see Figure 4(b). This is caused by the influence of normal stresses onflow properties. These normal stresses create tension alongthe circular lines of flowand generate a pressure towards the centre, which drives thefluid up the rod. Thisphenomenon was first described by Garner and Nissan [8], the experiment was thenanalyzed by Weissenberg [9].

In the radial lip seal should it therefore be, caused by the normal stresses, a flow inthe axial direction. That flow could be a reason why the lubricated film is establishedand why it is remained.

(a) In Newtonian fluids,centrifugal forces generatedby the rotation push the fluidaway from the rod.

(b) In non Newtonian fluids,normal forces are strongerthan centrifugal force andforce the fluid inward towardthe rod.

Figure 4: Weissenberg effect, also called the rod climbing effect.

8

3 METHOD

3 Method

Radial lip seals are very complex to describe mathematically; to make it solvable sev-eral assumptions and approximations had to be made. The equations, boundary- andsurface-settings that should be used were also needed to be determined. The geometrywas also modified to minimise the singularities to make it solvable.

3.1 Assumptions

• The model is axial symmetric, the gradient in the rotational direction is zero.

• Run-in lip, the examined lip is worn and has a smooth curvature.

• Smooth surfaces, there is no roughness on either the lip or the shaft.

• There will be no deformation on the lip due to shaft motion orflow of the lubri-cant.

• Fully flooded contacts, there is no contact between the lip and the shaft.

• No external forces. The gravitational force, the radial force from the garter springand forces from the lip are assumed to be zero.

• Isothermal conditions, there is no changes in temperature.

• The Newtonian fluid is incompressible, with constant density and viscosity.

• The non Newtonian fluid is incompressible with constant density, and the dy-namic viscosity is according to the power lawη = mγ̇n−1.

3.2 Course of action

In this investigation, a 2D axial symmetry model with swirl flow was studied. To makethe comparison between Newtonian and non Newtonian, a non Newtonian model wascalculated first. It had the propertiesm= 1000 andn = 0.2 in the power law describedin chapter 2 on page 7 and the density was set to 960 kg/m3. The properties werechosen in collaboration with SKF ERC, the Netherlands.

Two Newtonian fluids were then chosen, based on the results from the non New-tonian model. The first one was chosen to have the same viscosity as the non Newtonianat the area of highest shear i.e. the area of the thinnest film thickness. The second onewas chosen to have the same viscosity as the non Newtonian hadat the edge of thearea of interest. The velocity profiles in the z direction, see Figure 2 on page 3, for thedifferent models were then compared to investigate if the non Newtonian characteristicsprevents leakage.

The models were calculated with a radius of 0.05 m. The radiuswas then increasedfor the non Newtonian model. It was interesting to see how theflow in the z directionwould change when the centrifugal forces decreased. The velocity on the shaft periph-ery was kept at the same speed, i.e. the angular velocity of the shaft was decreased sothat the velocity on the boundary was the same for every case.

9

3.3 Swirl flow 3 METHOD

The models were solved with aid of COMSOL Multiphysics and the post process-ing was modified in an m-file in MATLAB. Comparisons between the maximum andminimum velocity in the z direction, net flow in the z direction and the velocity profileat the area of interests were made. See appendix A for the m-file that were used for the2D axial symmetric model.

3.3 Swirl flow

Swirl flow is a swirling flow in 2D axial symmetric coordinate systems. This meansthat the rotational speed is important, although constant,so that the gradient in thisdirection can be considered to be zero. This application take the volume force in therotational direction,Fϕ into consideration. The rotational velocityuϕ will still remainto be solved in the Navier Stokes equation, see equation 2.6.The equation of motioncan therefore be simplified as

ρ(

ur∂ur

∂r−

uϕ2

r+uz

∂ur

∂z

)

+∂p∂r

= η(

1r

∂∂r

(

r∂ur

∂r

)

−ur

r2 +∂2ur

∂r2

)

ρ(

ur∂uϕ

∂r−

uruϕ

r+uz

∂ur

∂z

)

+∂p∂r

= η(

1r

∂∂r

(

r∂uϕ

∂r

)

−uϕ

r2 +∂2uϕ

∂r2

)

(3.1)

ρ(

ur∂uz

∂r+uz

∂uz

∂z

)

+∂p∂r

= η(

1r

∂∂r

(

r∂uz

∂r

)

+∂2ur

∂z2

)

,

which will be used when solving the models.

3.4 The geometry

The lubricated geometry was modelled in drawing mode in COMSOL Multiphysics.The geometry was chosen to have a smooth curvature with a minimum film thicknessof 4 µm, because the geometry should have as few discontinuities as possible. Thecontact width was 1.1 mm and extended a bit longer on the lubricated side and theposition of minimum film thickness was located atz= 0. The curvature near the airside had a smaller angle than the lubricated side, all this because the geometry shouldbe as similar to the reality as possible. For the same reason,the outflow on the air sidewas chosen to have smaller width than the outflow on the lubricated side, see Figure 5.The outflows on both sides were then extended, so the boundarycondition should haveas little affect on the flow near the area of interest as possible. The shaft radius waschosen to be 0.05 m. All the parameters above were chosen in collaboration with SKFERC, the Netherlands.

3.5 The boundary conditions

At the inlet and outlet the boundary conditions were set to pressure zero so the flow inthe z direction should be affected as little as possible. Theextension on the inlet andoutlet were chosen to be as long as it were needed so the pressure distribution were setto normal when entering the lip sealing zone.

10

3 METHOD 3.6 The mesh

The lip side of the geometry had a no slip condition, i.e. there was no movementon the boundary. On the shaft side, a velocity field with a constant velocity of 5.24 m/sin theϕ direction were implemented. This corresponded to an angular velocity of 1000rpm when the shaft had a radius of 0.05 m.

3.6 The mesh

For this application, a structured mesh was the best choice and received the best con-vergence but because of some problem in the post processing,caused by the largedifference between the film thickness and the shaft radius, an unstructured mesh hadto be chosen. After convergence tests of the mesh were made, amesh consisted ofelement with the size of 0.5µm at the area of contact and 4µm at the extended inletand outlet were implemented.

3.7 Solving the model

COMSOL Multiphysics solves all physics with finite element method (FEM). Math-ematically this method is used for finding approximations ofpartial differential equa-tions (PDE), where the domain is broken into a set of discretevolumes or finite ele-ments. The equations are multiplied by a weight function before they are integratedover the domain. In the simplest finite element method, the solution is approximatedby a linear shape function within each element in a way that guarantees continuity ofthe solution across element boundaries. In these calculations, a second order Lagrangeelements are used for the velocity components and linear elements for the pressure,which are the default elements for the incompressible Navier-Stokes equation mode.

The solvers in COMSOL Multiphysics are either direct or iterative. Both type ofsolvers break down the problem into the solution of one or several linear systems. Thedirect solvers solve linear systems with Gaussian elimination, which is a stable processwell suited for ill-conditioned systems according to COMSOL Multiphysics guide. Theproblem with direct solvers is that they are memory consuming. The iterative solversare less memory consuming but they are not as fast as the direct solvers. Both types

Figure 5: The geometry used in the calculation, it was chosen to be asymmetric to resemble thereal seal lip. The whole geometry is lubricated.

11

3.8 The accuracy of the results 3 METHOD

of solvers were used to solve this model. Because the model had to be solved severaltimes for obtaining accurate result, the mesh was refined each time.

For this particular problem a stationary non linear solver was needed. One of theparameters in this mode is the relative tolerance, which give the criterion for conver-gence. The software stops calculating when the relative error, err is less than the relativetolerance. The relative error is given by

err =

(

1N ∑

i(|Ei |/Wi)

2

)1/2

. (3.2)

whereN is the number of degrees of freedom,Wi is a weight factor and E is an estimatederror in to a current approximation to the true vector.

3.8 The accuracy of the results

The software checked for convergence of iterations when therelative tolerance was setto 10−9. A convergence test of the mesh was also made to investigate if the solution ata certain point will converge to a specific value, when the number of elements in thegeometry goes to infinity. A Richardson interpolation test was to be constructed for anestimation of the error.

3.8.1 Numerical estimation of the shear rate

One more way to check the accuracy of the model is to compare the simulations witha numerical estimation of the shear rate. Equation 2.5 on page 7 which describes thetotal shear rate was used. After terms that were zero or closeto zero were cancelled,the shear rate could be approximated as

γ̇ ≈

√

(

r∂∂r

(uϕ

r

)

)2

+

(

∂uϕ

∂z

)2

. (3.3)

Because the contact width is much longer than the film thickness, equation 3.3 can beshortened and simplified as

γ̇ ≈ r∂∂r

(uϕ

r

)

=∂uϕ

∂r−

uϕ

r≈

uϕ

h0−

uϕ

r≈

uϕ

h0. (3.4)

The numerical values ofuϕ = 5.24 m/s andh0 = 4 µm gives a shear rate of 1.31 Ms−1.

12

4 RESULTS

4 Results

The results from the simulations are presented below. To investigate the pumping abil-ity of a seal, the flow in the axial direction is in the main concern.

The non Newtonian fluid was first calculated and the two Newtonian fluids werechosen based on that result. The simulations were computed for the geometry in Figure5 on page 11 but only the contact area was analysed. The contact area is locatedbetween the z coordinates -50µm and 60µm. The extended inlet and outlet are onlyused to minimize the effects of the boundary conditions and are therefore not examined.

4.1 Cursory investigation of the flow

A cursory investigation of the non Newtonian flow and the Newtonian flows were firstmade to see if there are any distinct differences.

4.1.1 Non Newtonian fluid

The pressure distribution, shear rate, dynamic viscosity and the velocity profile of thenon Newtonian fluid were calculated and results are shown in Figure 6. The Reynoldsnumber is smaller than 0.7 for the whole domain, the flow is therefore laminar with noturbulent tendency.

Figure 6(a) shows the pressure distribution, which has fluctuations in the extendedinlet and outlet. As desired, the distribution has stabilized when entering the contactarea. Figure 6(b) shows the shear rate, which has the maximumvalue of 1.41· 106

located atz= 0. The shear rate is decreasing towards the edges.The dynamic viscosity in Figure 6(c), has the smallest valueof 0.012 Pas at the

point of highest shear, i.e. at the minimum film thickness. Near z= 30 µm and at theedge of the contact area at the air side are the dynamic viscosity 0.03 Pas. Those valueswill be used in the following calculations of the Newtonian fluids.

In Figure 6(d), a pumping effect is visible, from both sides near the lip is the fluidpumping towards the point of minimum film thickness. But it isnot certain if it is dueto the non Newtonian characteristics together with the asymmetric geometry or due tothe centrifugal force.

4.1.2 Newtonian fluids

Calculations of the Newtonian fluids with viscosity 0.012 Pas and 0.03 Pas shows sim-ilar surface velocity profile as the non Newtonian fluid in Figure 6(d). The normalizedvelocity arrows for the three fluids can not be distinguishedfrom each other. Thethicker fluid has the same magnitude of the z velocity as the non Newtonian fluid andthe thinner fluid has a slightly larger.

13

4.1 Cursory investigation of the flow 4 RESULTS

(a) Surface plot of the pressure distribution. The lowest pressure occurs near the shaft.

(b) Surface plot of the shear rate. Maximum shear is at the thinnest film thickness, which isindicated.

(c) Surface plot of the dynamic viscosity. The minimum viscosity is 0.012, at the edge of area ofinterest is the viscosity 0.03.

(d) Surface plot of the velocities profile in the z direction,i.e. the axial direction. The arrowsindicate the normalized velocity in the r and z direction.

Figure 6: Results from the calculation of the non Newtonian fluid

14

4 RESULTS 4.2 Thorough investigation of the flow

4.2 Thorough investigation of the flow

Because the velocity profiles of the non Newtonian and Newtonian fluids were alikea more thorough investigation was needed. Comparisons between the cross section ofthe flow in the axial direction and net flow in the axial direction were performed. Aninvestigation of the centrifugal effects for the non Newtonian fluid was also made.

4.2.1 Axial velocity

Cross section plots in Figure 7 show that the non Newtonian fluid is more alike thethicker Newtonian fluid. However the non Newtonian fluid has larger axial velocitynearz= 0, i.e. at the minimum film thickness. The curves of the Newtonian fluids arevery similar to each other; the only difference is the magnitude of the z velocity. Thecross sections are taken at the same positions for all three models, where the z was keptconstant and r value is taken from the lip to the shaft.

The cross section plots, Figure 7 and the surface velocity plot in Figure 6(d) indicatethat the flow in z direction will go towards the point of minimum film thickness at thelip side and to the other direction at the shaft side for the same z value. The maximumz velocity will therefore lie near the shaft for positive z and the minimum near the lip.The opposite occurs for negative z positions. A simple summary of Figure 6(d) and 7is therefore to plot the maximum and minimum z velocities as afunction of the axialposition. The results are shown in Figure 8.

The two Newtonian fluids in Figure 8(b) and 8(c) are comparable, the only differ-ence is the power of the velocity. Figure 8 also shows that thenon Newtonian fluid ismore alike to the thicker Newtonian fluid, regarding the magnitude of the axial velocity.

In Figure 8 the positions of minimum film thickness are marked. For the threefluids, the axial velocity is slightly increased at the position of minimum film thicknessas compared to that in the nearest neighbourhoods. A stirring effect occurs whenz<−30µm andz> 20µm. The effect is much larger for the Newtonian fluids than for thenon Newtonian fluid.

4.2.2 Net flow

The integral of the z velocity profiles in Figure 7, i.e. the net flows, are constant throughthe contact area for all three models. The non Newtonian has acalculated net flow of8.2 ·10−12 s−1 throughout the contact in the z direction. The Newtonian fluids has aslightly larger net flow of 1.2·10−11 s−1 for the thinner fluid and a slightly smaller netflow of 5.0·10−12 s−1 for the thicker.

4.2.3 Effects of the centrifugal force

The shaft radius was then increased while the velocity at theperiphery of the shaftwas kept at the same speed. The test was only performed for thenon Newtonian fluid.The results show similar velocity profiles for the lubricantin the axial direction for alldifferent shaft radiuses. The only difference is the magnitudes of the axial velocitiesand they are exponentially decreased when the radius is increased by a factor 10. Figure9 shows the maximum velocities in axial direction atz= 0 for different shaft radius.

15

4.2 Thorough investigation of the flow 4 RESULTS

(a) Velocity in the z direction at different cross sectionfor the non Newtonian fluid.

(b) Velocity in the z direction at different cross sectionfor the thicker Newtonian fluid.

(c) Velocity in the z direction at different cross sectionfor the thinner Newtonian fluid.

Figure 7: Cross section plots of the velocity in the z direction for the three different models.

16

4 RESULTS 4.2 Thorough investigation of the flow

−5 −4 −3 −2 −1 0 1 2 3 4 55

x 10−5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

11x 10

−5

z [m]

velo

city

z−

dire

ctio

n [m

/s]

maxmin

(a) Maximum and minimum velocities for the nonNewtonian fluid.

(b) Maximum and minimum velocities for thethicker Newtonian fluid.

−5 −4 −3 −2 −1 0 1 2 3 4 55

x 10−5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

11x 10

−5

z [m]

velo

city

z−

dire

ctio

n [m

/s]

maxmin

(c) Maximum and minimum velocities for the thin-ner Newtonian fluid.

Figure 8: Maximum and minimum velocities trough the contact area in the z direction. Theposition of minimum film thickness is indicated with a circle.

10−2

10−1

100

101

10−8

10−7

10−6

10−5

Shaft radius [m]

Vel

ocity

[m/s

]

Figure 9: Maximum axial velocity atz= 0 with different shaft radius when the periphery speedof the shaft is constant.

17

4.2 Thorough investigation of the flow 4 RESULTS

18

5 ERROR ANALYSIS

5 Error analysis

Structured mesh gave the best convergence but because of problem in the post pro-cessing an unstructured mesh had to be used. The problem was caused by the largedifference between the shaft radius and the film thickness.

Convergence tests show that the structured mesh is in the asymptotic area, see Fig-ure 10(a), the unstructured is not, see Figure 10(b). The tests were made for differentnumbers of element with respective values for the shear rateat the point(0.05,0). Itis clear that the unstructured mesh need more elements, but because of the computeravailable was not as powerful, a finer mesh could not be constructed. A Richarsonsextrapolation test could not therefore be performed to get an estimation of the errorof discretization. However the values were in a small range of variation and compar-isons between the surface plots of the different element sizes of the triangular meshand comparisons between the structured mesh do not show any visible differences.

The error of iteration should also be insignificant because the software is pro-grammed to check for convergence.

The estimated value of the shear rate on page 12 was calculated to 1.31·106. Acomparison with the calculated value in Figure 10 of 1.25·106 shows that calculationsseems to be in order.

0 0.4 0.8 1.2 1.61.6

x 10−3

1.253

1.2532

1.2534

1.2536

1.2538

1.254x 10

6

Elements−1

She

ar r

ate

(a) Test of convergence for the structured mesh

0 1 2 3 4

x 10−4

1.253

1.2532

1.2534

1.2536

1.2538

1.254x 10

6

Elements−1

She

ar r

ate

(b) Test of convergence for the unstructured mesh

Figure 10: Test of convergence, the shear rate is calculated at(0.05,0)

19

5 ERROR ANALYSIS

20

6 DISCUSSION

6 Discussion

The pumping ability is essential for a radial lip seal; how the flow appears beneath thelip affect that phenomenon. The calculations show that there are differences betweenthe axial flow for the non Newtonian and Newtonian fluids.

The velocity profiles in Figures 7 and 8 show that the non Newtonian fluid is almostsimilar to the thicker Newtonian fluid than to the thinner. But the non Newtonian fluidhad a larger net flow through the contact compared to the thicker Newtonian fluid.Even the maximum velocity of the non Newtonian fluid was larger at the position ofminimum film thickness. This may indicate that the non Newtonian fluid has a largerpumping ability.

Lubricating grease provides a poor mean of transporting friction heat away. Butat the small scale, near the location of minimum film thickness, the larger pumpingmay contribute to transportation of heat away from that area. This will influence theviscosity of the lubricant.

Near the edge of the contact, at aboutz= 40µm andz= −40µm there is a smallerdifference between the maximum and minimum z velocity for the non Newtonian thanfor the Newtonian fluids, see Figure 8. This may contribute tothe maintained lubricatedfilm because the magnitudes of the velocity in the axial direction are smaller for thenon Newtonian fluid. The non Newtonian fluid may be compared toas a plug, themovement is getting smaller where the shear rate is decreasing and the fluid is gettingmore viscous.

A Weissenberg effect can not be separated from the results. The velocity in the axialdirection seems to go towards zero when the centrifugal effect is minimized. However,the Weissenberg effect can not be discarded as a hypothesis.The power law is a roughapproximation of the grease rheology and it is only a shear thinning formula. Theformula does not take the established normal forces into consideration, as described inchapter 2.1. A more complex approximation of the grease rheology is needed if thenormal forces are to be considered.

6.1 Accuracy of the result

One essential question in this model is if the axial velocities arise only because of nu-merical errors. Test has been made when the speed of shaft have been decreased andincreased. The investigation showed that the velocity in the axial direction decreaseswhen the speed of the shaft decreased and increases when the speed of the shaft in-creased, which is a normal behaviour. The convergence test of the structured mesh,which was in the asymptotic area, also indicates that the velocity in the axial directionwas due to the asymmetric geometry and due to the curvature ofthe shaft.

21

6.1 Accuracy of the result 6 DISCUSSION

22

7 FUTURE WORK

7 Future work

This is a simplified model of the radial lip seal and the power law is a rough approxima-tion of grease rheology. To have a more accurate investigation, a better approximationis needed to examine the importance of the non Newtonian characteristics. It is alsointeresting to take the meniscus into consideration, how itbehaves and influences theflow of the lubrication.

Because the viscosity of the grease is strongly temperaturedependent, it would alsobe interesting to have heat transfer within the model.

A different direction of the project would also to have a fluidstructure interaction.The lip is made of rubber and should be deformed under operating condition, but thatwould require a lot of computer capacity.

23

7 FUTURE WORK

24

REFERENCES REFERENCES

References

[1] E.T. Jagger. Study of the lubrication of synthetic rubber rotary shaft seal.Proc.Conf. Lubric. Wear, pages 409–415, 1957.

[2] M. J. L. Stakenborg. On the sealing mechanism of radial lip seal. Tribol. Int.,21(6):335–340, Dec 1988.

[3] L. Horve. The correlation of rotary shaft radial lip sealservice reliability and pump-ing ability to wear track roughness and microasperities formation. SAE, 100(sect6):620–627, 1991.

[4] R. Salant. Modelling rotary lip seals.Wear, 207(1-2):92–99, Jun 1997.

[5] F. Schulz, K. Wiehler, V. M. Wollesen, and M. Voetter. A molecular-scale view onrotary lip sealing phenomena.Leed- Lyon Symposium on Tribology, (25):457–466,1998.

[6] F. Hirano and H. Ischiwata. The lubricating condition ofa lip seal. Proc. Inst.Mech. Engrs., 180(Pt. 3B):138–147, 1965-66.

[7] Y. Kawahara, M. Abe, and H. Hirabayashi. An analysis of sealing characteristicsof oil seals.ASLE Trans., 23:93–102, 1980.

[8] F. H. Garner and A. H. Nissan. Rheological properties of high viscosity solutionof long molecules.Nature, 158:634–635, 1946.

[9] K. Weissenberg. A continuum theory of rheological phenomena.Nature, 159:310–311, 1947.

25

REFERENCES REFERENCES

26

A THE SCRIPT

A The script

This is a shorten version of the script that was used to solve the model. In reality themodel had to be solved repeatedly to get a fine mesh. The solution was saved for everytime it was solved and the solution was used as an initial value for the next time. In thisextraction the subdomain settings are not included.

% COMSOL Multiphysics Model M-file% Generated by COMSOL 3.2 (COMSOL 3.2.0.222, $Date: 2005/09/01 18:02:30 $)

flclear fem

% COMSOL versionclear vrsnvrsn.name = ’COMSOL 3.2’;vrsn.ext = ’’;vrsn.major = 0;vrsn.build = 222;vrsn.rcs = ’$Name: $’;vrsn.date = ’$Date: 2005/09/01 18:02:30 $’;fem.version = vrsn;

% Constantsr_shaft=0.05; % Radius of the shaftr_nere=0.00003; % Constant to construct the geometryz_nere=-5e-5; % Constant to construct the geometryz_uppe=5e-5+1e-5; % Constant to construct the geometryr_uppe=(z_uppe+5e-6)/5.5; % Constant to construct the geometryz_langd_in=2e-5; % Length of the extended outlet air sidez_langd_ut=3e-5; % Length of the extended outlet lubricated side

% Boundary constantsP1=0; % Pressure at air sidePs=0; % Pressure at lubricated sideomega_shaft=1000*2*pi/60; % Speed of shaft% Properties for greaserho_grease=960; % Density of greasem_grease=1000;n_grease=0.2;m_Newmin=0.0121;m_Newmax=0.03;n_New=1;

m=m_grease;%m=m_Newmin;%m=m_Newmax;

27

A THE SCRIPT

n=n_grease;%n=n_New;

fem.const = {’r_shaft’,r_shaft,...’P1’,P1,...’Ps’,Ps,...’omega_shaft’,omega_shaft,...’m_grease’,m_grease,...’n_grease’,n_grease,...’rho_grease’,rho_grease,...’m’,m,...’n’,n};

% Descriptionsclear descrdescr.const= {’r_shaft’,’radius of shaft [m]’,...’P1’,’pressure [Pa]’,...’Ps’,’pressure [Pa]’,...’omega_shaft’,’speed of shaft [rad/s]’,...’m_grease’,’m’,...’n_grease’,’n’,...’rho_grease’,’density of grease [Kg/m^3]’};

fem.descr = descr;

% Drawing the geometrycarr={curve2([0,0],[5.0E-5,z_nere],[1,1]), ...curve2([0,r_nere],[z_nere,z_nere],[1,1]), ...curve2([r_nere,0,0,1.0E-5],[z_nere,-1.5E-5,5.0E-6,5.0E-5],[1,1,1,1]),...curve2([1.0E-5,0],[5.0E-5,5.0E-5],[1,1])};

g5=geomcoerce(’solid’,carr);

g1=rect2(r_nere,z_langd_ut,’base’,’corner’,’pos’,[0,z_nere-z_langd_ut]);

carr={curve2([0.00001,r_uppe],[5.0E-5,z_uppe],[1,1]), ...curve2([r_uppe,0],[z_uppe,z_uppe],[1,1]), ...curve2([0,0],[z_uppe,5.0E-5],[1,1]), ...curve2([0,0.00001],[5.0E-5,5.0E-5],[1,1])};

g3=geomcoerce(’solid’,carr);g2=geomcomp({g5,g3},’ns’,{’CO1’,’CO3’},’sf’,’CO1+CO3’,’edge’,’all’);

g6=rect2(r_uppe,z_langd_in,’base’,’corner’,’pos’,[0,z_uppe]);

g3=mirror(g1,[0,0],[0,1]);g4=mirror(g2,[0,0],[0,1]);g5=mirror(g6,[0,0],[0,1]);

28

A THE SCRIPT

clear g1clear g2clear g6

g3=move(g3,[r_shaft,8.7e-6]);g4=move(g4,[r_shaft,8.7e-6]);g5=move(g5,[r_shaft,8.7e-6]);

clear ss.objs={g3,g4,g5};s.name={’CO1’,’CO3’,’CO5’};s.tags={’g3’,’g4’,’g5’};

fem.draw=struct(’s’,s);fem.geom=geomcsg(fem);

% Initialize meshfem.mesh=meshinit(fem, ...

’hmax’,[], ...’hmaxfact’,1, ...’hgrad’,1.3, ...’hcurve’,0.3, ...’hcutoff’,0.001, ...’hnarrow’,1, ...’hpnt’,10, ...’xscale’,1.0, ...’yscale’,1.0, ...’mlevel’,’sub’, ...’hmaxsub’,[1,4e-6,2,0.5e-6,3,4e-6]);

% Application mode 1clear applappl.mode.class = ’NonNewtonian’;appl.mode.type = ’axi’;appl.dim = {’u’,’v’,’w’,’p’};appl.sdim = {’r’,’phi’,’z’};appl.name = ’chnn’;appl.module = ’CHEM’;appl.shape = {’shlag(2,’’u’’)’,’shlag(2,’’v’’)’,...

’shlag(2,’’w’’)’,’shlag(1,’’p’’)’};appl.gporder = {4,2};appl.cporder = {2,1};appl.sshape = 2;appl.border = ’off’;appl.assignsuffix = ’_chnn’;clear prop

29

A THE SCRIPT

prop.elemdefault=’Lagp2p1’;prop.analysis=’static’;prop.stensor=’full’;prop.nisot=’Off’;prop.swirl=’On’;prop.frame=’rz’;clear weakconstrweakconstr.value = ’off’;weakconstr.dim = {’lm1’,’lm2’,’lm3’};prop.weakconstr = weakconstr;appl.prop = prop;clear pntpnt.pnton = 0;pnt.p0 = 0;pnt.ind = [1,1,1,1,1,1,1,1,1,1];appl.pnt = pnt;clear bndbnd.u0 = 0;bnd.type = {’noslip’,’neutral’,’uv’,’out’};bnd.p0 = 0;bnd.w0 = {0,0,’omega_shaft*0.05’,0};bnd.v0 = 0;bnd.name = ’’;bnd.ind = [3,4,3,2,3,3,2,4,1,1,1,1];appl.bnd = bnd;clear equequ.eta_inf = 0;equ.delcd = 0.35;equ.init = 0;equ.F_phi = 0;equ.shape = [1;2;3;4];equ.F_z = 0;equ.cporder = {{1;1;1;2}};equ.delid = 0.5;equ.sdtype = ’pgc’;equ.cdon = 0;equ.usage = 1;equ.type_visc = ’power’;equ.delsd = 0.25;equ.kappadv = 0;equ.delps = 1;equ.rho = ’rho_grease’;equ.cdtype = ’sc’;equ.eta0 = 1;equ.gporder = {{1;1;1;2}};equ.m = ’m’;

30

A THE SCRIPT

equ.sdon = 0;equ.lambda = 0;equ.pson = 0;equ.n = ’n’;equ.F_r = 0;equ.idon = 0;equ.ind = [1,1,1];appl.equ = equ;fem.appl{1} = appl;fem.sdim = {’r’,’z’};fem.frame = {’rz’};

% Shape functionsfem.shape = {’shlag(2,’’u’’)’,’shlag(2,’’v’’)’,...

’shlag(2,’’w’’)’,’shlag(1,’’p’’)’};

% Integration orderfem.gporder = {4,2};

% Constraint orderfem.cporder = {2,1};

% Geometry shape orderfem.sshape = 2;

% Simplify expressionsfem.simplify = ’on’;fem.border = 1;

% Equation formfem.form = ’general’;fem.units = ’SI’;

% Global expressionsfem.expr = {};

% Functionsclear fcnsfem.functions = {};

% Descriptionsclear descrdescr.const= {’r_shaft’,’radius of shaft [m]’,...’P1’,’pressure [Pa]’,...’Ps’,’pressure [Pa]’,...’omega_shaft’,’speed of shaft [rad/s]’,...

31

A THE SCRIPT

’m_grease’,’m’,...’n_grease’,’n’,...’rho_grease’,’density of grease [Kg/m^3]’};

fem.descr = descr;

% Solution formfem.solform = ’weak’;

% Multiphysicsfem=multiphysics(fem, ...’sdl’,[]);

% Extend meshfem.xmesh=meshextend(fem,’geoms’,[1],’eqvars’,’on’,...

’cplbndeq’,’on’,’cplbndsh’,’off’);

% Solve problemfem.sol=femnlin(fem, ...

’method’,’eliminate’, ...’nullfun’,’auto’, ...’blocksize’,5000, ...’complexfun’,’off’, ...’solfile’,’off’, ...’conjugate’,’off’, ...’symmetric’,’off’, ...’solcomp’,{’w’,’u’,’p’,’v’}, ...’outcomp’,{’w’,’u’,’p’,’v’}, ...’rowscale’,’on’, ...’ntol’,1.0E-6, ...’maxiter’,25, ...’hnlin’,’off’, ...’linsolver’,’umfpack’, ...’thresh’,0.1, ...’umfalloc’,0.7, ...’uscale’,’auto’, ...’mcase’,0);

% Save current fem structure for restart purposesfem0=fem;

% Plot solution, surface plot of the z velocityfigure(1)postplot(fem, ...

’tridata’,{’v’,’cont’,’internal’}, ...’triedgestyle’,’none’, ...’trifacestyle’,’interp’, ...

32

A THE SCRIPT

’tribar’,’on’, ...’trimap’,’jet(1024)’, ...’solnum’,’end’, ...’phase’,(0)*pi/180, ...’title’,’Surface: Velocity field [m/s]’, ...’refine’,3, ...’geom’,’on’, ...’geomnum’,[1], ...’sdl’,{[1,2,3]}, ...’axisvisible’,’on’, ...’axisequal’,’on’, ...’grid’,’off’);

% Cross-section plotfigure(2)postcrossplot(fem,1,[r_shaft (r_shaft+0.00003);0 0], ...

’lindata’,’v’, ...’linstyle’,’-’, ...’lincolor’,’cycle’, ...’linmarker’,’none’, ...’npoints’,1000, ...’spacing’,[3e-5 2e-5 0 -2e-5], ...’solnum’,’all’, ...’phase’,(0)*pi/180, ...’axislabel’,{’Arc-length’,’z-velocity [m/s]’}, ...’axistype’,{’lin’,’lin’}, ...’geomnum’,[1], ...’transparency’,1.0);

legend(’z=3e-5’,’z=2e-5 m’,’z=1e-5 m’,’z=0 m’,’z=-1e-5 m’,...’z=-2e-5 m’,’z=-1e-5 m’,’z=-2e-5 m’,’Location’,’SouthEast’);

title([’z-velocity [m/s] m=’,num2str(m),’, n=’,...num2str(n) ’, r_{shaft}=’,num2str(r_shaft)])

% Construction of the max and min velocity in the z direction% and the plot of the netflow thought the whole domain

i=1;zi=[];zmax=[];zmin=[];I=[];z_mini=-6e-5-z_langd_in;z_maxi=6e-5+z_langd_ut;for zii=z_mini:0.5e-6:z_maxi;

zi(i)=zii;pd = postcrossplot(fem,1,[r_shaft (r_shaft+0.00003);zii zii], ...

33

A THE SCRIPT

’lindata’,’v’,’npoints’,1000,’outtype’,’postdata’);I(i) = meshintegrate(pd.p);zmax(i)=max(pd.p(2,:));zmin(i)=min(pd.p(2,:));i=i+1;

endfigure(3)plot(zi,I)title([’The integral over z-velocities m=’,num2str(m),...

’, n=’,num2str(n) ’, r_{shaft}=’,num2str(r_shaft)])xlabel(’z [m]’)ylabel(’netflow z-direction’)

figure(4)plot(zi,zmax,’b’, zi,zmin,’r’)title([’max min velocities m=’,num2str(m),’, n=’,...

num2str(n) ’, r_{shaft}=’,num2str(r_shaft)])xlabel(’z [m]’)ylabel(’velocity z-direction [m/s]’)legend(’max’,’min’);

34