MathCad J Lubrication Flow Kostic

8/2/2019 MathCad J Lubrication Flow Kostic

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Appl ied MathCAD Journal, Volume 2, Number 3, 1993

Lubrication Flow in a Narrow Gapby M. Kostic, Assoc. Professor of Mechanical EngineeringNorthern Illinois University Page 1 of 13

Fluid flow in a narrow gap is a very characteristic problem in Fluid Mechanics, or morespecifically in lubrication theory, developed by Reynolds in 1886. When two bodies moverelative to each other, the friction is reduced if a viscous fluid (lubricant) with viscosity Jl isallowed to move through a narrow but variable gap between them. An idealization of lubricationproblem is the slipper-pad bearing shown in Fig. 1. The bottom wall moves relative to theupper block at a velocity U and creates a Couette flow in the gap. It is important that the gapis very narrow (h L) and non-uniform (h(x) *- const). The pressure distribution p(x) alongthe gap (x =OL) is described by the following governing, second-order differential equation(see for example [1]) :

d ( 3 d )_ d- hex) ' - p (x ) -6 oJloUo-h(x)dx dx dx

Page 2 of 13Figure 1: Slipper-pad bearingo AplxJ J"UL

p pref l ref 2e will calculate the hexcess pressure Yl- inIdistribution, L\p. ~

Moving wall

u[y.x)I - - - - - ~ ' "

hout I - - - - - - - - - - - ; ~ h(x}

Fixed block


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Page 3 of 13Problem Statement:For a given (arbitrary) gap profile, Le. some equation h(x), solve for the pressure distributionprofile p(x) using the above differential equation. To see the effect of different parameters onthe result, calculate the excess pressure distribution L\p(x)=p(x)-Pref (where the referencepressure Pref is a known constant), for a decreasing, increasing and unifonn gap, and fordifferent gap types, with various viscosities and velocities. The physics and/or trends of theproblem will be better understood if the user "plays" interactively with different input values.

Input variables (arbitrary and in any system of units):The bottom wall The fluid (lubricant) 11 :=O.5PasccU : = 2 0 ~velocity: viscosity:secGap dimensions and profile:Height of the gap at Height of the gap at thethe inlet (h in < 0.01 L outlet (hOUI < h inrecommended): recommended):h in :=O.lin h out := 0.05in

Page 4 of 13Gap profile polynomial order:(e.g. n =0,1,2,3,... impliesrespectively a unifonn, linear, Length of the gapquadratic, cubic profile, etc.) n :=2 in flow direction: L :=5inThe gap profile equation (may be arbitrary; we chose this type for convenience):

hex) :=h out+ (h in - hout)'( 1 - ~ ) nSolution:The governing second-order differential equation may be expressed in more universal,dimensionless form if its dimensional variables are nonnalized using the above inputvariables.

First, the dimensionless gap length in the flow direction should be defined as a rangevariable for future calculations.x 1 =- so that : = 0 , - .. 1 or x ( ~ ) : = ~ L x L 20


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Page 5 of 13The dimensionless gap height is then given by

And the dimensionless pressure is2~ p ( x ) h .

1 ( ~ ) = III~ U L

where the function for dimensional gap excess pressure is ~ p ( x ) = p ( x ) - p re f

The reference pressure, Pref' is the environmental pressure (outside the gap), and is usually 0psi (gauge) atmospheric pressure. The boundary conditions at the inlet and outlet aretherefore

~ p O ) = ~ p ( L ) = O or 1( 0)=1(1 )=0

Page 6 of 13Finally, with the above substitutions, we obtain the governing (Reynolds') equation indimensionless form:

d ( 3d ) dT \ ( ~ ) ' - 1 ( ~ ) = 6 ' - T \ ( ~ ) d d d

Integrating both sides of this equation with respect t o ~ , dividing by T J ( ~ ) 3 , and integratingone more time with respect to gives the final solution for the dimensionless pressure 1 t ( ~ ) .

o


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Page 7 of 13The interior integral can easily be solved analytically without losing any generality. We mayexpress the dimensionless pressure, 1t, as a function of dimensionless length, and anintegration constant, C.

Note that

This is a common form of a boundary-value problem, which requires an iterative numericalsolution. In order to satisfy the known boundary condition, 1t(1,C) =0, the constant ofintegration can be calculated using the root finder.First, make an initial guess for C: C :=1Then, calculate the approximate value: C :=root(1t(l,C),C)

C =2.459

Page 8 of 13Now that the integration constant C is determined, the universal, dimensionless pressure 1tand the gap profiles" may be easily plotted as a function of dimensionless gap l e n g t h , ~ .

1.5 r - - - - . . . . - - - - - , r - - - - - r - - - , . - - ~ - - . . . , The pressureprofile peaksjust past thell(I;,C) center of the--*" gap, where theTI(I;)-0 - 0.5 1 - - - ~ ~ - - - 1 - - - - + ~ ~ " " " ' I ' ~ ( 9 _ 4 . . . . .- -_ ; gap profilenarrows to awaist.

0"----'----"----"-----'------""-----'o 0.2 0.4 0.6 0.8 1.2I;Corresponding dimensional variables, like gap excess pressure .1.p or gap height h as functionof gap length x may be easily calculated and/or plotted.


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A (V U) '= 7 t ( ~ , C ) l ! U L Page 9 of 13LlP ';,l!, , 2h inThe following graph compares pressure profiles for several combinations of input variables, showingthe results of increasing viscosity or decreasing velocity from the original conditions.

5

-----.. ."

~ - ~ - - - -t( \)( , 'J(. ,- ... - ... - _ : : x ~... - ... - ... - ..... .... ~ 80

" p ( ~ , 2'1L,U) 60psi // / J ( ' )C.

,l(

- ... ... - ....L6-.)('. ) ( ' " '.... - ... V

" P ( ~ , I L , U)psi 40- j (

- . ) ( ."p ( ~ , I L , ~ 20psi - ........o o

x ( ~ ) in

Page 10 of 13It is interesting to notice that the solution for the dimensionless (normalized) pressure profile isuniversal, Le. independent on fluid type or velocity. The fluid viscosity and relative velocity areonly used for, and do influence, the dimensional pressure as seen above, but the basic shape ofthe pressure distribution will depend on the gap profile alone.

Verifying the Solution:Any solution looks good on paper, but now we should check that it makes sense. First, verifythe boundary conditions used to solve the problem.

t.p( O,l!,U) =0"psi andt.p(l ,l!,U) = 4.284"10- l l "psi

which is nearly zero. The discrepancy is the result of the numerical process of root finding andintegration.


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Page 11 of 13Next, note that the bearing's load per unit area has to be smaller than the average gap excesspressure, ~ P a v . . If this condition is not met, the lubricant will not prevent the bearing surfacesfrom rubbing against one another. This number can also be extracted using these calculations.

1t a v e ' ~ ' U ' L 1t ave = 0.689 ~ ave: - - - - ~ ave = 19.662 psi2h in

We may calculate the pressure extremes (maximum or minimum) within the gap byfinding its location ( ~ . x t or Xext ) where the first derivative of the pressure profile is zero.

Guess: ini := 0.5

~ e x t =0.575 1t ext = 1.11Page 12 of 13

The corresponding dimensional, extreme excess pressure, and its location, are

~ ext = 31.69 psix ext = 2.877 in

The above procedure will be applicable for any input gap profile equation, h(x). The governingdimensionless differential equation will always maintain the same closed form symbolically interms of 1 t ( ~ ) and T J ( ~ ) . You may conveniently investigate the influence of different inputvariables on the results. For example, to find the inverse solution (reversed flow), simply swapthe values for hin and hou t in the beginning of the document. You should find that thisproduces undesirable, negative excess pressure, Le. vacuum. For a uniform gap profile, n = 0,the excess pressure ~ will be zero everywhere (again undesirable). Therefore, for effectivelubrication flow, it is important to have a decreasing narrow gap in the direction of flow to obtainenough excess pressure differential to prevent two bearing surfaces to rub (touch) each otherunder their load, thereby reducing the friction. You are encouraged to experiment with otherconfigurations and see the influence on the results.

Page 13 of 13For an example of the type of calculations to be done when there is no lubrication betweenbearings, see the article in this issue on Hertzian stress analYsis.

Reference[1] F. M. White, Viscous Fluid Flow, 2nd Ed., McGraw-Hili Pub. Co., (New York, 1991).


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Math SoftL + \ x J -7 f,

July 14, 1993Dr. Milivoje KosticNorthern Illinois UniversityDepartment of Mechanical EngineeringDekalb, IL 60115-2854Dear Dr. Kostic:Thank you for submitting your article on Lubrication Flow throuth a Narrow Gap to Applied Mathcad.I am delighted to tell you that your article has been reviewed ~ m accepted for publication in our journal.It will appear in the Mathcad in Education section of the Fall issue: Volume 2, Number 3, 1993.It has been a pleasure to work with a professional both in the area of Mechanical Engineering, and in thearea of Engineering Education. I mn impressed with your command of the subject and your commitmentto your students.Good luck in endeavors, and please let me know if you have other submissions I may consider for futureissues.Sincerely,

Leslie MeicerEditor. Applied MathcadMathSoft, Inc.210 BroadwayCambridge, MA 02139Phone: 617-577-1017 ext. 817

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MathCad J Lubrication Flow Kostic