2. Model of incompressible fluid flow in the thrust bearing

Generally, the fluid flow in the bearing is described by the Navier-Stokes equations for viscous incompressible fluid that has the following form in cylindrical coordinates (r, ', z):

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ELASTOHYDRODYNAMIC CONTACT MODEL FOR CALCULATION OF AXIAL AND ANGULAR. . .455

Fig. 1. 6- lobe thrust bearing structure: a – lobes locations; b, c – bearing cross-sections

8 @t + (V r) vrr = fr@r +vr r2r2 @!

@vr v2 1 @p vr 2 @v

< !

+ (V r) v + = f + v + 2 r ;(1)

@t r r @ r2 r2 @

> @v vr v 1 @p v @v

@v 1 @p vz

> @t + (V r) vz = fz@z +

:

where (vr , v', vz) is the fluid flow velocity vector, ( fr , f', fz) is a mass forces vector, is a fluid density, t is a time, p is a fluid pressure, is the fluid dynamic viscosity, operators (V r) and defined as:

(V r) f = vr @r + r @ + vz @z ;f = r @rr @r ! + r2 @ 2 + @z2 :

@ f v @ f @ f 1 @ @ f 1 @2 f @2 f

Laminar flow of the incompressible liquid without slippage, which is defined by the shaft revolution speeds and relative bearing clearance di-mensions, is considered in the mathematical model of the fluid flow in the bearing. Taking into account that the order of magnitude of terms in equations (1) is unequal for flow in thin layer, one can draw the following conclusions [5]: mass and inertia forces of the liquid particles are negligible

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456 MIKHAIL TEMIS, ALEXANDER LAZAREV

in comparison with the viscosity forces for the concerned bearing structure and operating conditions; velocity component perpendicular to the sliding surfaces is negligible in comparison with the components directed along the sliding surfaces (vz << vr ; v ); the derivatives of r and ' coordinates of any velocity component are smaller by the factor of h l in comparison with z coordinate derivative of the same velocity component (h and l are character-istic dimensions of gap and sliding surface length for the thrust bearing). The simplification of Navier-Stokes equations (1) and the table showing relative order of derivatives magnitudes are presented in detail in [6]. Taking into account the above-listed assumptions, we may represented the Navier-Stokes equations describing the fluid flow in thrust bearing in the following form:

8 @p @2vr

= ;

@r @z22

> 1 @p @ v ; (2)

=

r @ @z 2

@p

<

> @z = 0:

:

Expressions for the velocity vector components vr and v' are obtained by integration of equation (2) and substituting it into the fluid continuity equa-tion:

h(r; ) h(r; ) 1 @v h(r; )

Z0 1 @ (rv ) Z0 Z0 @v

r dz + dz + z dz = 0; (3)

r @r r @ @z

and after taking into account the boundary conditions for the thrust bearing are reformated into the two-dimensional Reynolds equation describing the

fluid flow in the small sized channel: (r;) ! = 6 !r ; (4)

@rrh3 (r;) @r ! + r @h3 @ @

@ @p(r;) 1 @ @p(r;) @h(r;)

where the coordinate axes r and ' are chosen as it is shown in Fig. 1a; p = p(r; ') – pressure function over the lobe surface; h = h(r; ')

– fluid film thickness determined, in general case, by the initial form and the changes in sliding surfaces mutual position (runner displacement and rotations) – hgeom(r; ') and also the sliding surfaces deformations under the action of hydrodynamic pressure hde f (r; ') [2-4]:

h(r;) = hgeom(r;) + hde f (r;): (5)

The sliding surfaces deformations hde f (r; '), caused by the action of fluid pressure p(r; ') in the gap are considered in the calculation of h(r; ')