Effect of lubricant inertia in externally pressurized conical bearings with temperature dependent viscosity

Transcription

1 Acta Mechanica 106, (1994) ACTA MECHANICA 9 Springer-Verlag 1994 Effect of lubricant inertia in externally pressurized conical bearings with temperature dependent viscosity P. Chandra, P. Sinha, and S. Saxena, Kanpur, India (Received September 8, 1992; revised March 23, 1993) Smnmary, The influence of convective inertia on the characteristics of a parallel gap conical bearing which rotates with a uniform angular velocity is investigated. The lubricant is assumed to be incompressible and the viscosity varies exponentially with temperature. Inertia terms in the governing equations have been averaged across the film-thickness and the resultant system of equations is solved numerically to determine various bearing characteristics. It has been observed that the inclusion of inertia terms causes a decrease in the dimensionless load capacity for a lubricant whose viscosity is weakly dependent on temperature, irrespective of the temperature setting of the slider. However, for a lubricant with strong temperature dependence of viscosity, there is a slight decrease in the load capacity when the slider temperature is lower than that of the pad. As the slider temperature increases, the situation reverses. The dimensionless torque on the slider increases with the inclusion of inertia. Notation c C D E h k P Pin er Q r, ~, z R Re R~* Rew T T,,T~ hi, V~ W U,V,W x, y, ~o Xim XO q qo Vo (D specific heat of the fluid convection parameter dissipation parameter a type of Eckert number (defined in Eq. (4)) lubricant film thickness thermal conductivity gauge pressure inlet gauge pressure Prandtl number (defined in Eq. (4)) flow rate cylindrical coordinate system rotational parameter (defined in Eq. (4)) Reynolds number (defined in Eq. (4)) modified Reynolds number (defined in Eq. (4)) rotational Reynolds number (defined in Eq. (4)) temperature of the fluid temperature of the pad and the slider, respectively velocity components in the x, y and ~b directions, respectively reference quantities for the velocity components conical coordinate system inlet, outlet positions angle between the stationary pad surface and the plane perpendicular to the bearing axis viscosity-temperature coefficient viscosity of the lubricant viscosity of the fluid at the temperature of the pad ~/o/~, kinematic viscosity of the lubricant angular velocity of the slider Variables with superscript + denote the corresponding dimensional quantity.

2 158 R Chandra et al. 1 Introduction With the current trend towards the design procedure for bearings, it is becoming increasingly important that the bearing operating characteristics be predicted accurately. Because of strong dependence of viscosity on temperature, it becomes necessary to predict the temperature rise in bearings. This calls for the consideration of the energy equation. One of the difficulties encountered in dealing with the energy relation is that there exists no exact relationship between viscosity and temperature. Various empirical expressions of t/+ = f(t have been developed and the one which is used frequently has the form tl + = floe -p~(r+- r') (1) where t/0 is the viscosity at temperature Tt and fl+ is a constant known as viscosity temperature coefficient. Consequently, several investigations have been made to analyse thermal effects in various bearings. A methodical development of these studies can be found in reviews by Khonsari ([1], [2]). Another important aspect which has generated much interest is that of fluid inertia. This may be due to bearings operating with high-speeds and/or with an unconventional low-kinematic viscosity lubricant. Kahlert [3] was perhaps the first one to deduce that the importance of the inertia terms depends upon the magnitude of the effective film Reynolds number Re* defined as Uh h Re* - v L' where U is the slider velocity, h and L are a typical film-thickness and length of the bearing, respectively, and v is the kinematic viscosity. In the lubrication theory with inertia terms, the pressure in a lubricant film can be expressed by the modified Reynolds equation under the following conditions: (i) h/l ~ 1 (ii) 1~Re,~ 1 (iii) Re* (= Re(h/L) 2) "~ 1 where Re - For many designing procedures the condition (iii) may not be valid whereas the conditions (i) and (ii) may still be valid. This can happen due to the following reasons: (a) the high value of the Reynolds number either as a consequence of the bearing speed higher than in usual applications or as a consequence of the lower values of v when unusual lubricants such as water, air or liquid metals are used. (b) the value of the parameter h/l, although it may be very small when compared to unity, is greater than its values in large size bearings with conventional lubricants. In view of these aspects several theoretical researches have been made to extend the classical lubrication theory and a brief review of them has been given by Szeri [4]. Since the inclusion of inertia terms introduces complexities in the determination of the solution, researchers have resorted to certain approximations. For instance, Kahlert [3] considered the inertia terms as perturbing terms inducing small perturbations in the lubricant flow. Slezin and Targ [5] averaged inertia terms across the film-thickness. This allowed the resulting equations to be readily integrated. This technique has also been used by Osterle and Saibel [6] and Osterle et al. [7] to obtain solutions for various bearing configurations. Snyder [8] obtained the solution by taking into account the variation of inertia terms across as well as in the direction of flow. In spite of the abundance of literature available on inertia effects in various bearing geometries, scant attention has been given to their effects in conical bearings. As far as thermal effects in conical bearings are concerned, Rodkiewicz and Mioduchowski [9] pioneered the study in this area and obtained the UL v

3 Externally pressurized conical bearings 159 mean temperature by assuming a constant viscosity. Of significant importance is the work done by Kennedy et al. [10] who studied the thermal effects with dissipation in externally pressurized conical bearings taking lubricant of variable viscosity. They examined the influence of varying slider temperature on the characteristics of a conical bearing with constant film-thickness and rotating with a uniform angular velocity. The work of Kennedy et al. [10] was extended by Sinha and Rodkiewicz [11] who considered thermal effects with both convection and dissipation in the same type of bearing. They observed that the inclusion of convection alone leads to an overestimation of load and torque whereas the inclusion of dissipation alone leads to an underestimation of them. On account of a strong dependence of viscosity on temperature, it is interesting to study both the thermal as well as inertia effects in conical bearings. Salem and Khalil [12] investigated thermal and rotational inertia effects on the performance of an externally pressurized conical bearing. But they neglected inertia effects due to fluid flow. In order to fully comprehend the effect of inertia, this paper presents a theoretical analysis of the influence of inertia as well as thermal effects with convection and dissipation in an externally pressurized conical bearing with a lubricant whose viscosity varies according to relation (1). 2 Governing equations The problem considers the analysis of an axially symmetric steady flow of an incompressible lubricant in a conical bearing (Fig. 1) externally pressurized through the central recess. The upper part of the bearing (slider) rotates with constant angular velocity o~ and both parts of the bearing are maintained at constant but different temperatures, T, and Tt for the upper and lower (pad) parts respectively. It is assumed that the slider and pad surfaces are parallel. Using the conical coordinate system, the transformed Navier-Stokes equations have been obtained by Kennedy et al. [10]. These equations were obtained under the assumptions (i) to (iii). For the present analysis, the assumption (iii) is relaxed which leads to the following equations in z t stider c~ pod I Fig. 1. Geometry of conical bearing

4 160 R Chandra et al. dimensionless form: c~u ~v u cqx + ~yy + x 0 0 ( O~yy) dp Rw 2 ( Ou O~yy) -- Oy ~/ -dxx +-=Re* x u ~xx +v (2.2) Oy ~1 Re* U ~x + V ~y + 7) (2.3) (2.1) --T 8y - \ 8x + ay ] + R~-;- r/\~y] = 0. (2.4) Equations (2.1)-(2.4) have five unknowns, namely u, v, w, T and p. The associated boundary conditions are u(x, o) = v(x, o) = w(x, o) = o u(x, 1) = v(x, 1) = 0 w(x, 1) = x (3) p(1) = 0 T(x, 0)=l, T(x, 1)= where Tu= Tu T~ The dimensionless quantities used in these equations are defined as x + y+ u + v+ w + Q x=--, y=-~-, u=-~-, v-, w=-- U= Xo V W ' 2rcxoh cos c~ h T + r/+ V= -- U, W= COXo cos ~, T= -- /1 = -- = e -~(T-1), fl = fl+t t Xo Tl ' r/o P +h2 ~10 r U2 (Rew) 2 = Vo= --, Pr- E= ~ R- P r/oxou ' ~ k ' ' R e Re m Uxo Re* = Re (h) 2, R~, Wh ~o \Xo/ Vo (4) 3 Solution and numerical method The coupled system of equations (2.1)- (2.4) together with the boundary conditions (3) cannot be solved analytically. So convective inertia terms in Eqs. (2.2) and (2.3) are first averaged across the film-thickness and v is eliminated from these equations with the help of the continuity equation (2.1) and the boundary conditions on v. Thus Eqs. (2.1) and (2.3) reduce to the following forms: 1 x ~ u(x, y) dy = 1 (5.1) 0

5 9 kay) 9 Externally pressurized conical bearings 161 OY za2u l(; ;) ] au u 2 Rw 2 dp fl OTay ayou - ep(r-l) R~* 2 U-~x dy + --x dy - --x + dx 0 0 (5.2) Ii ;;] aw au uw OZWay 2 fl atay away = Re*eP(r-1) u -~x dy + w ~x dy x dy. 0 e (5.3) at at In the energy equation (2.4), the contribution of the term v ~ is small compared to u t,y a-x at as the cross-convection term ~ has a negligible influence (Cheng and Sternlicht [13]) and v < u. Hence, Eq. (2.4) reduces to a2t - Cu at De_P(r_,) (awy (5.4) ay 2 ~x - where C = PrRe* and D = PrER/Re* represent convection and dissipation parameters. This resultant system of equations (5.1)-(5.4) is solved numerically using the finite difference method. To solve these equations, apart from boundary conditions on u, w, T and p given in Eq. (3), certain additional conditions are also needed because of the presence of partial derivatives of u, w and T with respect to x. It is assumed that u(xi,, y)= 1/x~,. (6) yl "7 '7 u(i,my)= 0,w(i,my) = Xin +(i-1) 6x,T(i,my)=T u j:rny n # "-i,j-1 3_ "5 (0,0) X=Xin ~ x=l X i =1 A X I i=mx u (i,1) = 0,w(i,1) =0,T (i,1) =1 Fig. 2. Grid details with associated inlet and boundary conditions

6 162 R Chandra et al. This choice of constant fluid velocity at the inlet satisfies the matching condition Eq. (5.1), for the flow rate at the inlet. The inlet condition for T is assumed to be linear as given by Rodkiewicz et al. [14]: T(x,., y) = 1 + (T, - 1) y. (7) Such a profile for T(x, y) at the inlet is generally used to overcome the discontinuities at the surfaces. Further, it is assumed that w(x, y) is linear at the inlet, i.e. w(xi., y) = xl.y. (8) This assumption has been made on th e basis of the inertia-less solution of w(x, y). Writing the finite difference scheme for Eqs. (5.2)- (5.4), with respect to the grid details shown in Fig. 2, one obtains u(i, j) = u(i, j + 1) + u(i, j -- 1) 2 (dy) 2 e p(r(id)-o [dp 2 Ux (i) R(w(i, j))2 x(i) + Re*(2Fl(i) + F2(i))] - -- [T(i, j + 1) -- T(i, j -- 1)1 [u(i, j + t) - u(i, j - 1)1 (9.1) 8 w(i, j) = w(i, j + 1) + w(i, j - 1) Re*(Ay) 2 ep(r(ia)-o 2 2 [F31(i) + F32(i) + 2F4(i)] fl - - [T(i, j + 1) - T(i, j - 1)] [w(i, j + 1) - w(i, j - 1)] (9.2) 8 T(i, j + 1) + T(i, j - 1) C(Ay) z u(i, j) T(i, j) = 2 2Ax [T(i, j) - T(i - 1, j)] D + -- e-p(r(id)-l)[w(i, j + 1) - w(i, j -- 1)] 2 (9.3) 8 where Ax and Ay are the mesh width in the x- and y-direction, respectively. Fl(i), F2(i), F31(i), F32(i) and F4(i) denote values of the following integrals at the i-th grid: 1 1 Iu2 FI(x) = u ~x dy, F2(x) = -- x dy (10) 0 0 I 1 i F31(x) = f u -~x Ow dy, F32(x) = f w ~x #u dy, f4(x) = f uw X dy, (11) which have been evaluated numerically using Simpson's rule. dp It is to be noted that in Eq. (9.1) the difference scheme for dxx (i) has not been used as the pressure condition at the inlet is not prescribed explicitly. To solve the coupled system of Eqs. (9.1)- (9.3) under the prescribed boundary conditions, the parameters R, C, D, Tu, fl and Re* are prescribed. After having prescribed the values of these

7 Externally pressurized conical bearings 163 parameters, the boundary and inlet (i = 1) conditions are fixed for u, w, and T. A value of the pressure gradient is subsequently assumed at i = 2. Fictitious values of u, w and T are prescribed at all points of the grid. Various integrals involved in these equations are calculated using Simpson's rule. Then new values of u(2, j), w(2, j) and T(2, j) are obtained for alll using old values of u(2, j), w(2, j) and T(2, j) and inlet conditions for u(1, j), w(1, j) and T(1, j) by the Gauss-Seidel iteration method. New values thus obtained are compared with the values obtained in the previous iteration, till a desired accuracy is reached. Continuity of flow, Eq. (5.1), is then used to check the validity of the assumed pressure gradient. In case the assumed pressure gradient is correct, u(2, j) finally obtained must satisfy the matching condition (5.1). If these values of u(2, j) so obtained fail to satisfy (5.1), the assumed pressure gradient is modified and the process is repeated. In this manner we obtain u(i, j), w(i, j), T(i, j) and the pressure gradient at i = 2. Then we move on to the next point of the grid and the procedure is repeated. This procedure is continued for each mesh point until (5.1) is satisfied for all points of the mesh. Proceeding in this manner, the values of u(i, j), w(i, j), T(i, j) and ~ (i) are obtained for all i and j except for i = 1. dp. dxx (1) is calculated by extrapolation. Then the pressure is calculated by using the finite-difference formula for dp/dx, dp p(i) = p(i + 1) --/Ix dxx (i), for i _>- 1, (12) and assuming a zero gauge pressure at the outlet. In this manner the pressure distribution at each point of the mesh is found. 4 Load capacity and torque The dimensionless load capacity and torque for the present geometry as defined by Kennedy et al. [10] is L= M= L+h 2 ~X03~O U COS 2 O~ M+h 2n~/oXo4CO COS =pinxi2n-b2 f pxdx=-2 f (x2 d~xx) dx xin Join 1 = f x2e_~(r(~,y)_t)~3w ~--~ dx (13) (14) where L + and M + represent dimensional load capacity and torque, respectively. Integrals (13) and (14) have been evaluated numerically using Simpson's rule. 5 Results and discussions The bearing characteristics in the present study appear as functions of the inlet point xln, the temperature of the slider Tu, the convection and dissipation parameters C and D and the rotational parameter R. For all numerical computations in the present study, the value of the

8 164 R Chandra et al. inlet point is 0.1. The slider temperature varies from 0.8 to 1.2 which signifies three different settings (A) Cool slider hot pad (T, = 0.8) (B) Equal slider and pad temperature (T, = 1.0) and (C) Hot slider-cool pad (Tu = 1.2). It has been observed by Sinha and Rodkiewicz [11] that for the representative values of fluid properties and operational parameters for a conical bearing, 0 < C -< 0.1 and 0 < D < 2.0. In the present study C and D are chosen to be 0.1 and 1.5 respectively. The rotational parameter R is taken as 60. The effect of variation of slider temperature on bearing characteristics was discussed by Kennedy et al. [10] and the qualitative behaviour in the present study is observed to be similar. Another parameter which has been taken into account is fl, it signifies the viscosity-temperature dependence. A small value of //characterizes a lubricant with a weak viscosity-temperature dependence whereas a large value of it indicates a strong viscosity-temperature dependence. Computational values of/? are taken as 0, 1 and 5./~ = 0 signifies no viscosity-temperature dependence. The main purpose of this paper is to study the influence of convective inertia and variable viscosity on the performance of the conical bearing via Re* and/~ under different settings of the slider temperature. For the present study =< Re* _-< Re* signifies the inertia-less situation. In view of the numerical example 2 (Kennedy et al. [10]), the upper limit of Re* is chosen to be It must be mentioned here that the influence of the cone angle on the bearing's load capacity has been studied in the past (one may refer to the references given in Kennedy et al. [10]) and hence no emphasis has been placed on this aspect. 5.1 Velocity distribution Figures 3 a (/~ = 1) and 3 b (fl = 5) show the effect of Re* on u(x, y) at the outlet (x = 1.0) for three different settings of the slider temperature. From Fig. 3 a it is seen that in all the three cases (A), (B) and (C), u(x, y) decreases with an increase in Re* in the film-region 0.6 < y < 1 whereas it increases in the region 0 < y < 0.6. For fl = 5 (Fig. 3 b) the effect of Re* on u(x, y) is predominant at higher values of T, (case (C)) and becomes less significant as the slider is cooled (cases (A) and (B)). Moreover, the variation of u(x, y) with respect to Re* is indistinguishable near the slider, specially in case (A), whereas in the other two cases ((B) and (C)) u(x, y) decreases in the upper half while it increases in the lower region as R,* increases. It is also observed in case (C) that at Re* =.001 values of u(x, y) closer to the pad surface become negative, indicating a reverse flow which in turn suggests a separation. This separation occurs due to the combined effect of ever-increasing cross-sectional area of the bearing and rotation of the slider. However, at Re* =.19 no reverse flow is observed. Thus it can be concluded that for a fixed flow rate and rotational velocity flow separation can be avoided at higher values of Re*. Figures 4a and 4b illustrate the effect of Re* on w(x, y) at the outlet for/~ -- 1 and/~ = 5 respectively. It is seen that for fl = 1, w(x, y) is approximately linear. It is observed that values of w(x, y) at Re* =.19 are always less than those at Re* =.001 (inertia-less situation). The velocity gradient in the y-direction increases with Re* in the film-region which is in proximity to the slider, whereas it decreases with Re* in the film-region which is near to the pad surface.

9 Externally pressurized conical bearings R~= R~ = Tu = Tu= ~ -,- Tu= ') 0./-,,0.2 O.C i.o a ~ 0.2 o.o u(x. y) > b Fig. 3 a, b. Velocity distribution u(x, y) versus y at the outlet for different slider temperatures

10 166 E Chandra et al [ IR~ = 0.001, R~= Tu = Tu=l ~- Tu= 1.2 y a 1.0 ~ ~. ~ 0.6,...,.OX. y 0.4 /, 0.2 * / ~ nn~,,,,, b w(x,y) > Fig. 4a, b. Velocity distribution w(x, y) versus y at the outlet for different slider temperatures 5.2 Temperature distribution Figures 5 a, 5 b and 5 c show the effect of Re* on the variation of the temperature T(x, y) with respect to y at the outlet for different temperature boundary conditions. From the curves, it is seen that T(x, y) increases with an increase in Re* in the region which is nearer to the slider whereas it decreases in the region which is away from the slider. However, these variations in T(x, y) with Re* appear in different ranges of y at different temperature boundary conditions. 5.3 Pressure, load and torque Figures 6 and 7 show the effect of Re* on the pressure distribution versus x for different values of T. and at fl = 1 and 5, respectively. At Re* =.001, the pressure distribution is qualitatively similar to that of Kennedy et al. [10]. For all the three settings of T,, the inlet pressure falls with an increase in Re* but for values of x >.35, a slight increase in the pressure is observed (Fig. 7).

11 Externally pressurized conical bearings =t 13=5 I - - R~=O.O01... R~=0.19 l y 0.5 J Tu =0.8 a Tu(x,y) Tu=l.0 :) b C Tu =1.2 Tu(x,y) ~- Fig. 5a-e. Temperature distribution T(x, y) versus y at the outlet for different settings of the slider temperature A similar behaviour in the inlet zone was observed by Salem and Khalil [12] who considered only rotational inertia. The appearance of negative pressure at the outlet is not very unusual for such bearings. Similar negative pressure regions were observed by Kennedy et al. [10], Sinha and Rodkiewicz [11] and Kalita et al. [15]. The negative pressure region can possibly be eliminated by increasing the inlet pressure, i.e. by lowering the value of the parameter R. This aspect has been discussed in great detail by Kennedy et al. [10]. Figure 8 shows the effect of Re* on the load capacity versus T, for different values of ft. The effect of T, on the load has already been discussed by Kennedy et al. [10] and is found to be similar in the present analysis. At fl = 1, the decrease in the load capacity is linear with T,, even when inertia terms are included. For a given 2r,, the load capacity decreases as R~* increases. At fi = 5,

12 168 R Chandra et al. ~o R~ =o.oo~ R~=O. IO Rex-=O. 19 ILl W" o~ w [12 o_ 20 ~Tu =0.8 O ',, 0.3 O X > Fig. 6. Pressure distribution versus x for different slider temperature settings at fl = Re~= R~=0.~O -- Re~=0.19 Ld n-" 20 u") 33 U") W rv" Q._ Tu:o /./IT u =1.2 _it I I I I,,,I ~ X > Fig. 7. Pressure distribution versus x for different slider temperature settings at fl = 5

13 Externally pressurized conical bearings _ Rr -- R~= Re~=0.19 f F ',.x'k <,,\ cl < =-.-2.~ - - 7_ = 0 < 0 J 1.0 ~ 13=1 0.0 ' X- i [ '~'2B = Tu---~ Fig. & Load capacity versus slider temperature Re~= R~= 0.10 _._ Re~=O.19 \,\ l "\. B=O uj 0.30~". ~-~=~... o 0.20 ~ B = 5 0~) I I 1.0 I I 1.2 Tu --" Fig. 9. Torque of the bearing versus slider temperature

14 170 E Chandra et al. an unusual character of the load capacity is observed. For ~Pu < 0.9, there is a slight decrease in the load capacity with an increase in Re*, but as Tu increases further, the effect of R~* is reversed. Thus heating of the slider increases the load capacity of the bearing with the inclusion of inertia terms. It is observed that at Re* =.001, which indicates an inertia-less situation, curves for fi = 0, 1 and 5 (fl = 10 in Kennedy et al. [10]) are qualitatively similar to those of Kennedy et al. [10]. The negative load capacity is perhaps the consequence of negative pressure regions, seen in Figs. 6 and 7. As mentioned earlier, this may be due to either a higher angular velocity or a reduced flow rate. A reduced flow rate in turn indicates a lower supply pressure. Figure 9 shows the effect of Re* on the torque of the slider for different T~ and ft. As expected, for fl = 0, the torque on the slider remains constant irrespective of temperature settings of the slider. The curves are perfectly linear at fl = 1 for all values of T~. However, at fl = 5, the curves are linear upto T~ = 1.0 and as the slider temperature is increased further, they show slight departure from linearity. It is observed that the torque always increases with an increase in the values of Re* for all values of ft. It may be noted that this increase in the torque could be a direct consequence of an increased ~w/~y with an increase in Re*, as discussed earlier. 6 Conclusions The present work analyzes the influence of convective inertia on the characteristics of a rotating constant gap bearing with incompressible lubricant having temperature dependent viscosity. It is observed that the convective inertia accelerates the movement of the lubricant along the x-direction in the lower half of the gap, and retards it in the upper half. The temperature of the lubricant increases with an increase in the inertia parameter Re*, in the region which is nearer to the slider, whereas it decreases in the region which is nearer to the pad. Further, it is seen that the inclusion of inertia causes a decrease in the inlet pressure, which results in a reduced load capacity for a cool slider. However, for a hotter slider there is a slight increase in the load capacity. All of these results are further accentuated for a lubricant having strong viscosity temperature dependence. The torque on the slider increases as Re* increases from.001 to 0.19 for all values of ft. It goes down as the slider temperature increases. It may thus be concluded that the absence of convective inertia under-estimates the torque and over-estimates the load. From the design point of view it would be advisable to work with a high inlet supply pressure and a cool slider. References [1] Khonsari, M. M.: A review of thermal effects in hydrodynamic bearings. Part I: Slider thrust bearings. ASLE Trans. 30, (1987). [2] Khonsari, M. M.: A review of thermal effects in hydrodynamic bearings. Part II: Journal bearings. ASLE Trans. 30, (1987). [3] Kahlert, W.: The influence of inertia forces on hydrodynamic lubrication theory. Ing. Arch. 16, (1947). [4] Szeri, A. Z.: Some extensions of the lubrication theory of Osborne Reynolds. Trans. ASME J. Trib. 109, (1987).

15 Externally pressurized conical bearings 171 [5] Slezin, N. A., Targ, S. M.: The generalized equation of Reynolds. Comptes Rendus de L'Academic des Science USSR 54, (1946). [6] Osterle, J. E, Saibel, E.: On the effects of lubricant inertia in hydrodynamic lubrication. J. Appl. Math. Phys. (ZAMP) 6, (1955). [7] Osterle, J. E, Chou, Y. T., Saibel, E. A.: Effect of lubricant inertia in journal bearing lubrication. Trans. ASME J. Appl. Mech. 24, (1957). [8] Snyder, W. T.: The nonlinear hydrodynamic slider bearing. Trans. ASME J. Basic Eng. Series D 85, (1963). [9] Rodkiewicz, Cz. M., Mioduchowski, C. M.: The mean temperature of a conical bearing. Wear 31, (1975). [10] Kennedy, J. S., Sinha, E, Rodkiewicz, Cz. M.: Thermal effects in externally pressurized conical bearings with variable viscosity. Trans. ASME J. Trib. 110, (1988). [11] Sinha, P., Rodkiewicz, Cz. M.: Convection and dissipation effects in oil lubricated conical bearings with variable viscosity. Trans. ASME J. Trib. 113, (1991). [12] Salem, E., Khalil, F.: Thermal and inertia effects in externally pressurized conical oil bearings. Wear 56, (1979). [13] Cheng, H. S., Sternlicht, B.: A numerical solution for pressure, temperature and film-thickness between two infinitely long lubricated rolling and sliding cylinders, under heavy loads. J. Basic Eng. 87, (1965). [14] Rodkiewicz, Cz. M., Hinds, J. C., Dayson, C.: Inertia, convection and dissipation effects in the thermally boosted oil lubricated sliding thrust bearing. Trans. ASME J. Lub. Tech. 97, (1975). [15] Kalita, W., Rodkiewicz, Cz. M., Kennedy, J. S.: On the laminar flow characteristics of conical bearings. Part I: Analytical approach. Trans. ASME J. Trib. 108, (1986). Authors' address: P. Chandra, E Sinha and S. Saxena, Department of Mathematics, Indian Institute of Technology, Kanpur , India