Flow through the Grooves of Journal Bearings

F. (Bill) HogeDesign Engineer

Advanced Transmission, Ford Motor Corporation

Livonia Michigan, 48150

W. W. SchultzAssociate Professor

Department of Mechanical Engineering and Applied Mechanics

University of Michigan, Ann Arbor, Michigan 48109-2125

S. Boedo�

Senior Technical Analyst

Federal-Mogul Corporation

Ann Arbor Michigan, 48108

C. Vanden BroekManager, Materials and Product Testing

Federal-Mogul Corporation

Ann Arbor Michigan, 48108

March 8, 1997

Abstract

A simple analysis of fully-developed Stokes ow in grooved bearings is performedthat does not use lubrication scaling arguments as in Muijderman (1966). When theinertial terms are ignored, the Navier{Stokes equations decouple. The ow through thegroove is then determined from a Poisson equation for the only component of velocitythat contributes to the ow through the bearing. This velocity is further decoupled intothat caused by shaft movement and that caused by the pressure di�erence across thebearing. Rectangular groove ow rate predictions are compared to those of Muijderman.Flow rates from semi-circular and irregular groove cross-sections are also presented.Comparison of calculated oil ow for a production-level grooved bearing shows goodagreement with experimental measurements from a bench test rig.

�Present Address: Sta� Engineer, Borg-Warner Automotive, Inc., Ithaca, New York 14850

1

Nomenclature

a = normal groove width, m (Fig. 1)ag = transverse groove width, m (Fig. 1)ap = transverse pad width, m (Fig. 1)A = cross-sectional area of groove, m2

c = average pad radial clearance, md = bearing width, m (Fig. 1)h = �lm thickness, m (Fig. 1)H = hp=hgL = d=sin� = length of grooveN = truncation parameterp = �lm pressure, N/m2

Q = volumetric ow through groove, m3/s(r; �; z) = polar coordinates for semicircular groove (App. B)R = journal radius, mRe = �Whg=� = Reynolds numberRh = hydraulic radius of groove, mt = time, sU = R! = shaft surface sliding velocity, m/sw = velocity �eld parallel with groove, m/sW = component of shaft velocity along groove(x; y; z) = normal coordinate system (Fig. 1)(x0; y0; z0) = transverse coordinate system (Fig. 1)� = groove angle to the � axis (Fig. 1) = ap=ag� = radius of semi-circular or partial-arc groove� = lubricant viscosity, N�s/m2

� = uid density, kg/m3

! = journal angular speed, rad/s

Subscripts

g groovep padp due to pressure drop across bearings due to shaft motion

1 Introduction

Journal bearings play a major role in automatic transmission design, as they are directlyintegrated into the bearing and support strategy of all major rotating components. At thesame time most journal bearings are directly in the path of the hydraulic circuitry of thetransmission. For this reason and also for bearing lubrication, grooves are designed into the

2

bearing to facilitate oil ow to rolling-element bearings and other parts of the lubricationcircuit. It is very desirable to be able to calculate the ow through a journal bearing basedon its geometry and operating characteristics.

The standard reference for calculating oil ow through bearing grooves is Muijderman(1966). This work describes how to calculate the ow rate through a spiral groove bearingbased on lubrication theory. The di�culty in using lubrication theory is that there areonly boundary conditions at the bearing and the journal surfaces. Therefore lubricationtheory cannot adequately account for the presence of the groove wall. We will show theimportance of this characteristic when we compare the results for the two methods of groove ow calculation.

Other studies of ow in bearing grooves concentrate on the spiraling ow in the groovesand the ow into the pad region (Ettles, 1967). As a result, the problem considered isconsiderably more complex than that considered here, without adding any new informationto determine ow rate though the groove.

2 Summary of Muijderman's Method

The governing equation used by Muijderman (1966) is the Reynolds equation:

@

@x0

h3

12�

@p

@x0

!+

@

@z0

h3

12�

@p

@z0

!=U

2

@h

@x0+@h

@t; (2.1)

where p is the �lm pressure, h is the �lm thickness, � is viscosity, t is time, and we use a localCartesian coordinate system: z0 is the axial coordinate, x0 = R� is in the circumferentialdirection in the direction of the shaft movement at U = R! (see Fig. 1) and y0 is the directionnormal to the shaft surface (in the radial direction). We use primed coordinates to indicatethose of Muijderman. The two terms on the left-hand side of (2.1) are the Poiseuille termsand describe the net ow rates due to pressure gradients within the lubricated area. The�rst term on the right-hand side is the Couette term or the physical wedge mentioned andthe second term on the right-hand side is the squeeze e�ect.

We assume a steady, end fed concentric bearing so that the pressure never drops belowthe cavitation pressure and the bearing is fully wetted. We further assume, as in Muijder-man, that there are only two uid clearances, one for the pad and one for the groove. Withthe constant clearances, the Reynolds equation becomes the Laplace equation

@2p

@x02+

@2p

@z02= 0: (2.2)

If the groove width ag is much smaller than the groove length L, then we may assumesmall pressure end e�ects and obtain a bilinear pressure pro�le as given in Muijderman(1966) equations (2.33{2.36). However, the groove-to-bearing width ratio is 0.31 for aproduction bearing as given in Table 1 | not very low to make this approximation. Sincethe di�erential equation and end boundary conditions are homogeneous, the solution isforced by the matching condition at the pad/groove interface. A di�culty also occurs there

3

Journal

Bearingh y

x h=h g

p

U cos α

Section A-A

a

Pad Groove

x’

z

x

z’α

a ag p

A

A

d

p

∆p

Figure 1: Two coordinate systems for a grooved journal bearing in \unwrapped" form.The primed coordinates are those of Muijderman, while the new unprimed coordinates arealigned with the groove.

4

Table 1: Dimensions of typical bearing: F2TP-7025-AA

Bearing width d = 14.5 mmGroove width ag =4.44 mmJournal radius R = 18.66 mm

Arc radius � = 4.32 mm

Pad

Groove

Figure 2: Flow Mismatch at Groove Boundary. The velocity pro�les shown are assumed topersist all the way to the pad/groove interface.

since the discontinuity in gap thickness causes a ow incompatibility as shown in Fig. 2.Ignoring this di�culty, Muijderman gives the equation for pressure di�erence as:

�ph2g6�Ud

= cot�(1�H)(1�H3)� 2Q

aghgU(1 + cot2 �)( +H3)

(1 + H3)( +H3) +H3 cot2 �(1 + )2: (2.3)

where Q is the volumetric ow rate through one groove and one pad, H = hp=hg; = ap=ag,and � is the groove angle.

For the case of interest, H ! 0 and then Q becomes

Q

aghgU=

1

2(1 + cot2 �)

��ph

2g

6d�U+ cot�

!: (2.4)

This gives a nondimensional ow rate (using this scaling) ofp38

for � = 30o when there isno pressure gradient. Since W = U cos�, d = L sin�, and a = ag sin �, this gives (droppingthe subscript on h)

Q sin�

ahW=

1

2(1 + cot2 �)

� �ph2

6d�W+ csc�

!: (2.5)

5

The �rst term is due to the pressure gradient and the second is due to the shaft motion. Ifwe break these two into separate components we obtain Q = Qs +Qp such that

Qs

ahW=

csc2 �

2(1 + cot2 �)=

1

2(2.6a)

Qp�L

ah3�p=

�112(1 + cot2 �)

: (2.6b)

The ow rate due to shaft motion (Qs) turns out to be that for simple Couette ow inan in�nitely wide groove. Hence the average ow rate is just an average of the shaft andgroove bottom surfaces. The e�ect of the groove aspect ratio a=h does not appear. The ow rate due to pressure gradient (Qp) is also independent of the a=h aspect ratio, butsurprisingly is dependent on the groove angle � in a nontrivial (and incorrect) way.

3 Derivation of the Exact Stokes Equations

Our alternate method of solving this problem is to recognize that if we align the coordi-nate frame with the groove, we e�ectively decouple the ow �elds when inertial terms areneglected. The Navier{Stokes equations for incompressible ow are:

�

�@u

@t+ u

@u

@x+ v

@u

@y+ w

@u

@z

�= �@p

@x+ �

@2u

@x2+@2u

@y2+@2u

@z2

!(3.1a)

�

�@v

@t+ u

@v

@x+ v

@v

@y+ w

@v

@z

�= �@p

@y+ �

@2v

@x2+@2v

@y2+@2v

@z2

!(3.1b)

�

�@w

@t+ u

@w

@x+ v

@w

@y+ w

@w

@z

�= �@p

@z+ �

@2w

@x2+@2w

@y2+@2w

@z2

!: (3.1c)

Here, we have ignored body forces (gravity), and the velocity components u; v; w are inthe x; y; z directions as shown in Fig. 1. Since the grooves we examine here are symmetric,we place the origin at the base of the symmetry plane of the groove. Simplifying further, weassume that the ow is steady and that inertial terms can be neglected (Reynolds numbersare typically Re � 1). We assume the groove is long compared to its width (again asin Muijderman); hence the end e�ects are small and the velocity is fully-developed (i.e.@w=@z � 0). Then it is easily shown that the ow through the groove is determined onlyby the w velocity component. (This assumption will have a slight correction when thegroove depth is not signi�cantly deeper than the pad). Now we only need to solve

@p

@z=

�p

L= �

@2w

@x2+@2w

@y2

!: (3.2)

This is the equation for unidirectional ow down a duct of constant cross-sectional geometryas solved in White (1974), for example. The di�erence in this case is that the u and vcomponents are not negligible or even small. Nonetheless, when Re is su�ciently small, theequations decouple and (3.1c) is the only Navier-Stokes equation that need be solved.

6

Simplifying further, we assume that the groove height is much larger than the padclearance as in most production journal bearings and hence, the majority of the uid owsthrough the bearing grooves. The problem thus reduces to a Poisson equation with inho-mogeneous boundary conditions.

3.1 Solution For Rectangular Geometry

This problem is solved analytically by a superposition solution of a Laplace equation withone inhomogeneous boundary condition and a Poisson equation with homogeneous boundaryconditions. Hence, the velocity w is written as w = wp+ws where wp is the Poisson problemand ws is the solution to the Laplace equation. The details are presented in Appendix A.The total velocity solution can be written as:

w =1Xn=1

1Xm=1

�bmn�mn sinn�x

asin

m�y

h+

1Xn=1

An sinn�x

asinh

n�y

a: (3.3)

The ow rate in the groove is now a straightforward integration of the velocity over thegroove area and gives

Qs

ahW= 8

1Xn=1;3;5:::

a

h(n�)3 sinh n�ha

�cosh

hn�

a� 1

�: (3.4a)

Qp�L

ah3�p= �64

�6

1Xn=1;3;5:::

1Xm=1;3;5:::

1�n2 + (ma

h )2�m2n2

: (3.4b)

3.2 Solution For Semi{Circular Geometry

A similar separation of variables solution exists for a semi{circular groove of radius �.Again, decomposing the velocity into a superposition solution (w = wp + ws) leads to aLaplace equation with one inhomogeneous boundary condition and a Poisson equation withhomogeneous boundary conditions. The details are presented in Appendix B. Again, thevelocity �eld is a superposition of w = wp + ws resulting in:

w(r; �) =1Xm=1

1Xn=1

AmnJm�p

�mnr�sin n� +

1Xn=1

Bnrn sinn�: (3.5)

Groove ow, again, is a straightforward integration of the velocity �eld over the area of thegroove.

Q =Z �

0

Z �

0

w r dr d�: (3.6)

In this case, however the integration of Bessel functions is most easily performed numericallysince the sums converge slowly for the roots �mn with large indices.

7

4 Comparison of Results with Muijderman

The series expansions for the ow rates (3.4a,b) were summed for various aspect ratios.We truncated the double series at di�erent m = n = N to determine solution convergence.The rectangular groove results converged rapidly for both the Poisson and Laplace ows.For N = 5, both converged to the second signi�cant digit. For N = 19, the results areconverged in the third and fourth signi�cant digits. The results shown here were computedwith N = 51.

We compare groove ow rates based on Muijderman (2.6a,b) with various groove aspectratios a=h. The ow rates due to shaft motion are shown in Fig. 3 and those due to thepressure gradient in Fig. 4.

Several observations can be made from these results. The �rst is that the ow calculatedusing the Muijderman method does not change as a function of aspect ratio. The owcomponent due to shaft velocity using the analytical method only agrees in the limit ofa=h ! 1 as an obvious consequence of the absence of sidewall e�ects on Muijderman'smethod. From the mean value theorem and symmetry arguments, it can be shown thatQs

ahW = 0:25 for a=h = 1 and Qs

ahW = 0:5 for a=h!1, in agreement with the solid curve inFig. 3. The Muijderman prediction for Qp (2.6a) also does not depend on the aspect ratio.It can be shown that, as expected, our prediction of Qp is the same when a is interchangedwith h in contrast to (2.6a). The ow rate per unit area is shown to be maximized whenthe a=h aspect ratio is unity, by multiplying the calculated result in Fig. 4 by the aspectratio.

It should also be noted that the hydraulic radius approximation for Qp is fairly accuratefor modest aspect ratios. The Hagen-Poiseuille formula for laminar ow in a circular tubegives (Fox & McDonald, 1992)

Qp�L

AR2h�p

=1

8; (4.1)

where A is the cross-sectional area. Employing equation (4.1) to the square cross-section,the hydraulic radius Rh (twice the cross-sectional area A divided by the perimeter) is equal

to a=2 or h=2 and Qp�Lah3�p

= 1=32 = 0:03125, or 10 percent lower than theQp�Lah3�p

= 0:035144

computed from (3.4b).

5 Calculation for Non-rectangular Groove Geometry

Since Bessel functions converge slowly for large arguments, we used the numerical integra-tion capability of MathCad to evaluate the Bessel integrals involved in semi-circular grooves.For the semi-circular case, the Poisson ow also converges very rapidly. For N = 5, thesum converged in the second signi�cant digit. For N = 9, the sum converged in the thirdand fourth signi�cant digits. The sum for the Laplace problem converges much more slowlybecause the sum for evaluating the Bessel function converges rather slowly. Convergence inthe fourth signi�cant digit required at least 1000 terms.

For the semi-circular groove, we �nd that Qp�LA2�p = 0:0298 and Qs

AW = 0:33, where the

cross-sectional area is given by A = ��2=2. For this case, the hydraulic radius is Rh =

8

Figure 3: Dimensionless Flow Rate Due to Shaft Motion. The solid line is the exact Stokesresult for Qs from (3.4a) while the dashed line is the approximate result of Muijderman(2.6a). The results agree in the limit a=h!1

9

Figure 4: Dimensionless Flow Rate Due to Pressure. The solid line is the exact Stokesresult for Qp from (3.4b) while the dashed line is the approximate result of Muijderman(2.6b). The results agree only near a � 1:7. For purposes of comparison � = 300 in (2.6b).

10

�1+2=� = :611 �; and (4.1) gives Qp�L

A2�p=

R2

h

8A = 0:030; within one-half percent of the exactStokes value.

We use the commercial software package PDEase for solving partial di�erential equationsusing �nite elements for arbitrary geometry. An example of these numerical solutions areshown for an a=h aspect ratio of unity in Fig. 5. The �gure shows isovelocity contour linesover the cross section of the groove. This particular case shows reverse ow in the uppercorners (the ow is seen to be symmetric about x = 0) because an adverse dimensionlesspressure gradient of 1 was applied in addition to the Then the numerical results calculate theright-hand sides with a = h. The series solution ow rate then predicts a dimensionless owrate of Q =0.25-0.035144=0.214856, with the corresponding PDEase value of 0.214847m3=sor 0.004% low.

Next, we compared a numerical solution for a typical groove geometry (Table 1) toexact Stokes rectangular groove solutions that approximate this geometry. The numericalcalculations for the groove are shown in Fig. 6. This groove is a partial arc groove withradius � = 4:318 mm with an arc of 61.63o. The area is then 1:832 mm2. The base is 2:21mm and maximum depth is 0:635 mm as shown in Fig. 6. The ow rates are computed tobe Qs=W = 0:892 mm2 and Qp�L=�p = 0:0377 mm4.

We approximated this geometry two ways. The �rst was a rectangle enclosing the ac-tual groove geometry (hence h = 0:635 mm and a = 2:21 mm). The second approximationpreserved the aspect ratio and the total area of the partial arc groove geometry. The �rstapproximation results in a 102% greater pressure ow and 40% greater ow rate due toshaft velocity. The approximation with the same cross-sectional area has 6.2% less for thepressure ow and 4.7% less for the velocity ow. The approximation with the same areaand aspect ratio can also be extended to other geometries such as the trapezoidal shapeused in the experimental veri�cation.

6 Experimental Veri�cation

The test rig is represented by the schematic shown in Fig. 7 where one end of the bearingis pressure-fed with oil through a shaft attached to a non-rotating sleeve, while the otherend is at atmospheric pressure. The journal is attached to a second shaft which can rotatein either direction. Comparison with groove angle de�nition in Fig. 1 allows interpretationof clockwise (+ sign) and counterclockwise (� sign) journal rotation as \in the direction"and \against the direction" of the groove, respectively. A gauge measures supply pressure,a positive displacement ow meter placed in the sleeve passage measures the inlet ow anda thermocouple measures the inlet oil temperatures. The operating viscosity is calculatedusing Walther's ASTM temperature-viscosity relationships for ATF Type H oil based onthe inlet temperature.

Before proceeding to an angled groove bearing, we �rst ran experiments with an axially-grooved (� = 0) bearing. For a variety of pressure drops across the bearing, the shaftrotational speed had negligible e�ect on the ow rate through the bearing. This gives somecon�dence that the Reynolds number depending only on the shaft speed in the direction of

11

0.0

0.2

0.6

0.4

0.9

0.8

0.8

0.6

0.4

0.2

0.0

1.0

-0.5 0.0 0.5

x/a

y/h

Figure 5: Velocity contour lines for a square groove. The isovelocity contours have aninterval of 0.05. The minimum velocity at the bottom corners is -0.0023, with the velocityequal to 1 at the top wall. The volumetric ow rate Q is calculated as 0.214847 by PDEase,while the exact Stokes value is 0.004% higher.

12

-.2-.4 -.6 -.69

1.0.8

.6 .4.2 0

w wp s

Figure 6: Velocity contour lines for non-rectangular groove cross sectional geometry

Table 2: Common Parameters for Experimental Apparatus

bearing width d =13 mmgap �lm thickness hg =0.50 mmjournal radius R = 18.75 mm

number of grooves = 2groove angle � = 45 deg

radial clearance c = 42.5 �mviscosity � = 13.1 mPa-s (ATF Type H oil at 71.1 C)

the groove was chosen correctly.Fig. 8 shows a �nite element mesh for one-half of the trapezoidal groove cross-section

for one of the two diametrically-opposed angled grooves in the experimental application.Only one-half of the groove is modeled due to symmetry. The bearing radial clearance isadded to the groove height, assuming concentric journal and sleeve and no-slip boundaryconditions along the extended side walls of the groove. Table 2 gives the other commonparameters for the experimental bearing (bushing). The additional ow Qp through theungrooved portion (over the pad) is given approximately by Keller (1985) as

Qp =�Rc3�p

6�d: (6.1)

Evaluation of nine case studies shown in Table 3 indicates that the predicted total owscalculated from the �nite element computations added to that of (6.1) are in good agreementwith measured ows over a wide range of operating journal speeds and supply pressures.The shaft rotation direction signi�cantly increases oil ow when the journal rotates in thegroove direction. On average, the predictions are three percent too high, with the averagedeviation of 14 percent error. The relative errors increase for high journal speeds, possiblyindicating the increased importance of nonisothermal or non-Newtonian e�ects.

The discrepancies between the experiments and predictions are obviously a�ected bymeasurement uncertainties in uid viscosity and supply pressure in addition to the groovecross-sectional geometry, as well as the neglected inertial and entrance e�ects in the model.In our test setup, supply pressures over the range given in Table 3 have been measured to

13

Figure 7: Experimental Flow Rig Schematic

14

y (m

m)

x (mm)

sym

met

ry

plan

e

shaft surfacebearing surface

Figure 8: Computational Mesh for Groove Cross-Section for Experimental Apparatus

Table 3: Case Studies { Diagonally Grooved Bearing

case journal supply calculated measured relativespeed pressure Q+Qp ow error(rpm) �p (kPa) (cm3/min) (cm3/min) (%)

1 0 40.74 266 + 11 = 277 305 { 9.22 0 110.48 722 + 29 = 751 735 +2.23 0 179.52 1174 + 48 = 1222 1056 +15.74 +4000 111.16 1016 + 30 = 1046 937 +11.65 { 4000 110.47 433+ 29 = 462 560 { 17.56 +9000 179.52 1825 + 48 = 1873 1465 +27.87 { 9000 179.52 523+ 48 = 571 750 { 23.98 +9000 41.43 922 + 11 = 933 809 +15.39 { 4000 180.21 889+ 48 = 937 904 +3.7

15

Table 4: Diagonally Grooved Bearing { Simpli�ed Analysis

case journal supply simple relative Muijderman relativespeed pressure prediction error prediction error(rpm) �p (kPa) (cm3/min) (%) (cm3/min) (%)

1 0 40.74 304 +3.5 195 {32.22 0 110.48 826 +16.4 529 {24.03 0 179.52 1342 +31.7 861 {14.04 +4000 111.16 1087 +16.4 853 {5.75 { 4000 110.47 570 +7.0 209 {57.56 +9000 179.52 1920 +34.3 1582 +11.17 { 9000 179.52 765 +8.5 540 {21.68 +9000 41.43 886 +10.9 919 +15.19 { 4000 180.21 1091 +26.0 543 {34.6

an accuracy of 2 kPa, resulting in an estimated one to �ve percent uncertainty in the total ow. The oil supply temperature of 71 C has been measured within 1 C, leading to anestimated four percent total ow uncertainty.

Discrepancies may also be attributed to deviations from the assumed journal-sleeveconcentricity. The test rig has an estimated maximum journal-sleeve eccentricity of 8 �mat maximum shaft speed. Under some simple assumptions, this leads to an estimate of 6percent uncertainty in the ow through the pad regions. However, as this bearing ow isonly a small fraction of the ow through the groove, a more likely e�ect of eccentricity isto alter the e�ective groove cross-sectional geometry. This may have a much stronger e�ecton the groove ow.

In contrast to the numerical solution of the Stokes equations, the use of the hydraulicradius solution and a multiplier of 0.4 for the shaft velocity typically over-predicts the owrate by 17 percent. The Muijderman results both over- and under-predict the ow rate witha typical discrepancy of 24% for the cases considered as shown in Table 4. For these errorcomparisons, the same cases are considered as in Table 3 and the simple prediction andMuijderman predictions have the same computed pad ow as in Table 3 before comparingto experiments.

7 Concluding Remarks

Muijderman's results do not adequately account for the groove aspect ratio and do not havethe proper groove angle dependence. The solutions we obtain are exact when inertial andend e�ects are neglected. These same assumptions are required in Muijderman's analysis,but the lubrication scaling he uses is not required here. The ow rate can be broken downinto a component due to the shaft velocity and one due to a pressure gradient. Rectangulargroove approximations with the same area and aspect ratio give adequate predictions of

16

typical groove geometries. Even simpler approximations using hydraulic diameters for thepressure contribution and using an average velocity close to 0.4 times the shaft velocitycomponent along the groove give accurate predictions for modest aspect ratio grooves.

An angled groove in the bearing design can act as an oil pumping mechanism. Ap-plication to a production-level bearing reveals that calculated oil ow agrees well withexperimentally determined ow taken from a test oil ow rig, thus supporting the use ofthis analysis as a bearing design capability.

Further planned experimental and analytical studies will investigate e�ects of shaft mis-alignment, journal loading, and heat generation. Analyzing wider bearings will test theentrance e�ects in the model. The relative errors increase for high journal speeds, possiblyindicating the increased importance of nonisothermal or non-Newtonian e�ects.

Acknowledgments

We acknowledge partial support of National Science Foundation ContractsDMC-8716766and ECS-90012263.

References

Bender, C.M. & Orszag, S.A, Advanced Mathematical Methods for Scientists and Engi-

neers, McGraw-Hill, New York, 1978.

Ettles, C. Solutions for ow in a bearing groove. Proc. Inst. Mech. Engrs., Vol. 182,1967, pp. 120-131.

Fox, R.W & McDonald, A.T. Introduction to Fluid Mechanics, John Wiley & Sons, NewYork, 1992.

Keller, G.R., Hydraulic System Analysis, 2nd ed., Penton/IPC, Cleveland, 1985.

Muijderman, E.A., Spiral Groove Bearings, Springer-Verlag, New York, 1966

White, F.M., Viscous Fluid Flow, McGraw-Hill, New York, 1974.

Appendices

A Rectangular Geometry

The problem is broken into two parts, wp, for the pressure-driven ow

52wp =1��pL (A1a)

wp(x = 0; y) = wp(x = a; y) = wp(x; y = 0) = wp(x; y = h) = 0; (A1b)

17

and ws is the solution to the Laplace equation:

52ws = 0 (A2a)

ws(x = 0; y) = ws(x = a; y) = ws(x; y = 0) = ws(x; y = h)�W = 0; (A2b)

where W is the component of shaft velocity along the z-direction. These problems can besolved by separation of variables and eigenfunction expansions such as found in Bender andOrszag (1978). In both cases, the Laplacian operator is in the plane perpendicular to the ow, i.e. 52 = @2=@x2 + @2=@y2. The solution of ws uses separation of variables withproduct solutions of the form

ws(x; y) = X(x)Y (y) (A3)

with boundary conditions

X(0) = X(a) = Y (0) = Y (h)�W = 0: (A4)

Substituting the product into the Laplace equation and separating variables yields:

1

X

@2X

@x2= � 1

Y

@2Y

@y2= ��: (A5)

We have chosen �� since X(x) must oscillate. Then Y (y) is composed of exponentials.These equations are now standard ordinary di�erential equations. The x-dependent problemis a boundary-value problem and can be used to determine the separation constant �. Theboundary conditions dictate that the eigenvalues are determined from the sine terms suchthat:

X(x) = sinn�x

a; � =

�n�

a

�2n = 1; 2; 3; ::: (A6)

We can now solve the other ODE:

@2Y

@y2=

�n�

a

�2Y: (A7)

The most convenient solution to this equation is a linear combination of hyperbolic func-tions. Again, from the Y (0) = 0 boundary condition we infer:

Y (y) = A sinhn�Y

a: (A8)

We now have the product solution:

ws(x; y) = A sinn�x

asinh

n�y

a: (A9)

To satisfy the inhomogeneous boundary condition at ws(x; h) we apply superposition toobtain:

ws(x; y) =1Xn=1

An sinn�x

asinh

n�y

a: (A10)

18

To determine the coe�cients An we associate the hyperbolic term with the coe�cients,and use orthogonality to obtain:

An sinhn�h

a=

2W

a

Z a

0

sinn�x

adx =

2W

n�[1 + (�1)n+1]: (A11)

Equation (A10) and (A11) form the �rst half of the superposition solution where (A10)is the velocity �eld in the groove and (A11) yields the coe�cients An. We also note thatthis is the complete solution when �p = 0.

The second half of our problem is a Poisson equation with homogeneous boundary con-ditions. The most direct approach to this problem to form a two dimensional eigenfunctionsolution as follows:

52� = ���; (A12)

with homogeneous (� = 0) boundary conditions.For a rectangle this implies a sine series in both x and y:

�mn = sinn�x

asin

m�y

h(A13)

where

�mn =

�n�

a

�2+

�m�

h

�2: (A14)

We now expand the solution wp as a series of eigenfunctions:

wp =1Xn=1

1Xm=1

�bmn�mn sinn�x

asin

m�y

h: (A15)

These eigenfunctions are also orthogonal, with a weight of 1. Thus substituting into(A12) and rearranging gives

bmn = � �p

�mn�L

R h0

R a0 sin

n�xa sin m�y

h dx dyR h0

R a0 sin

2 n�xa sin2 m�y

h dx dy: (A16)

Now the total velocity solution can be written as:

w =1Xn=1

1Xm=1

�bmn�mn sinn�x

asin

m�y

h+

1Xn=1

An sinn�x

asinh

n�y

a: (A17)

The ow rate in the groove is now a straightforward integration of the velocity over thegroove area.

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B Semicircular groove

52wp =1

r

@

@r

�r@wp

@r

�+

1

r2@2wp

@�2=

�p

�L(B18a)

52ws =1

r

@

@r

�r@ws

@r

�+

1

r2@2ws

@�2= 0: (B18b)

Again, the solution of ws is the solution to the Laplace equation inside a semi-circlewith one inhomogeneous boundary condition. Applying separation of variables we look forproduct solutions of the form:

ws(r; �) = �(�)R(r); (B19)

To have homogeneous boundary conditions from which to determine the separationconstant, we �x the coordinate frame to the shaft instead of the bearing. Then the boundaryconditions become:

�(0) = �(�) = R(0) = R(�)�W = 0: (B20)

Substituting the product solution into the Laplace equation and separating variables yields:

r

R(r)

@

@r

�r@R

@r

�= � 1

�

@�2

@�2= �: (B21)

We have chosen �� since �(�) must oscillate. The ��dependent problem is a boundaryvalue problem and can be used to determine the separation constant �. The boundaryconditions require that all of the eigenvalues are positive sine terms over the interval (0; �).This implies the solution:

�(�) = sinn� � = n n = 1; 2; 3; ::: (B22)

The second half of the separation problem for R(r) is a solution to

r2@2R

@r2+ r

@R

@r� �R = 0: (B23)

Since this ODE is equidimensional, solutions are in the form rp. Substituting this form into(B23) yields:

[p(p� 1) + p� �]rp = 0: (B24)

Solutions are p = �p�. The general solution becomes:

G(r) = Csrp� + Cpr

�p� � 6= 0: (B25)

When � = 0, the solution becomes trivial and without loss of generality we can exclude� < 0. Using superposition the solution becomes:

ws(r; �) =1Xn=1

Bnrn sinn�: (B26)

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To determine the coe�cients Bn, we apply orthogonality to obtain:

Bn =2W

��n

Z �

0

sinn�: (B27)

Equations (B26) and (B27) form the �rst half of the superposition solution where (B26)is the velocity �eld in the groove and (B27) yields the coe�cients Bn. We note again thatthis is the complete solution when �p = 0.

The second half of the problem is a Poisson equation with homogeneous boundaryconditions. We look for two dimensional eigenfunction solutions of the form:

52� = ��� (B28)

with homogeneous boundary conditions. The solution is then constructed from

wp =Xm

Xn

Amn�mn: (B29)

The coe�cients are obtained by orthogonality:

Amn = � �p

L�mn

R �0

R �0 �mnr dr d�R �

0

R �0 �

2mnr dr d�

: (B30)

While the denominator can be evaluated explicitly in closed form, the numerator must beevaluated numerically. We chose to do an exact integration of the Taylor series expansionof the Bessel equation for small argument. We now need the appropriate eigenfunctions�mn that satisfy the homogeneous boundary conditions:

�mn = Jm�p

�mnr�sinn�; (B31)

where Jm is the Bessel function of the �rst kind and order m and �mn are the roots ofJm(

p�mn�) = 0. Thus, wp becomes:

wp(r; �) =1X

m=1

1Xn=1

AmnJm�p

�mnr�sin n�: (B32)

Again, the velocity �eld is a superposition of (B32) and (B26)

w(r; �) =1Xm=1

1Xn=1

AmnJm�p

�mnr�sin n� +

1Xn=1

Bnrn sinn�: (B33)

21