Performance of two-lobe hole-entry hybrid journal bearing system under the combined influence of textured surface and couple stress lubricant

Textured surface in journal bearings is becoming an important area of investigation during the last few years. Surface textures have the shapes of micro-dimple with a small diameter and depth having order of magnitude of bearing clearance. This paper presents the influence of couple stress lubricant on the circular and non-circular hole-entry hybrid journal bearing system and reports the comparative study between the textured and non-textured circular/non-circular hybrid journal bearing system. The governing Reynolds equation has been modified for the couple stress lubricant flow in the clearance of bearing and journal. The FEM technique has been applied to solve the modified Reynolds equation together with restrictor flow equation. The numerically simulated results indicate that the influence of couple stress lubricant is significantly more in textured journal bearing than that of non-textured journal bearing. Further, it has been observed that the textured two-lobe (δ  = 1.1) hybrid journal bearing lubricated with couple stress lubricant provides larger values of fluid film stiffness coefficients and stability threshold speed against other bearings studied in the present paper.


Introduction
Circular fluid film journal bearings usually suffer from fluid induced instability in terms of oil-whirl or oil-whip phenomena at high speed applications. It produces large self-excited vibrations and may even damage the bearing system [1]. In order to overcome these problems, noncircular journal bearings were developed by researchers and bearing designers. The simplest configurations of the non-circular fluid film journal bearing available are a twolobe, three-lobe and four-lobe journal bearing configurations. They are characterized by their number of lobes, bearing aspect ratio and preload factor or offset factor. In the recent years most of the research activities both experimental as well as theoretical have been carried out related to performance of non-circular bearings in their research works. Pinkus et al. [2,3] studied the influence of various geometric and operating parameters on the noncircular journal bearing. They computed the load carrying capacity and power losses of elliptical journal bearing with the help of FDM numerical technique. Goenka and Booker [4] reported the effect of surface ellipticity on dynamically loaded circular bearing and optimized the bearing shape on the basis of maximizing the minimum fluid film thickness. Malik [5] computed the theoretical analysis of bearing characteristics for different configurations of two-lobe bearings. It was observed that the two-lobe bearing provides consistently better dynamic performance as compared to elliptical and offset-halves bearings. Further, Crosby [6] determined the thermodynamic solution of a finite two-lobe journal bearing and obtained the pressure and temperature distributions by solving the governing Reynolds equation and energy equation simultaneously. The theoretical and experimental comparison of dynamic coefficients for three-lobe bearing was presented by Kostrzewsky et al. [7]. They found a good agreement between theoretical and experimental dynamic coefficients with in lower range of Sommerfeld numbers. Sharana Basavaraja et al. [8] studied the influence of roughness on the performance of a misaligned two-lobe hole-entry hybrid journal bearing. They concluded that the roughened journal bearing gives the partial compensation to the effect of bearing misalignment on the value of minimum fluid film thickness especially with longitudinally surface roughness pattern. Sharma and Kushare [9] numerically analyzed the effect of surface roughness on a non-circular (two-lobe) hybrid journal bearing performance. They showed that the performance of journal bearing is significantly affected by orientation of roughness. They also concluded that the longitudinal roughness pattern improves the fluid film stiffness and damping coefficients of journal bearings.
Surface roughness is of great importance in lubrication studies as it influences friction and wear to a great extent [10]. In hydrodynamic lubrication, since the surfaces are fully separated, the surface topography at the nanometer scale is usually considered to have no influence when films of few micrometers are developed in the contact zone. The surface topography needs to be at the scale of the fluid film in order to have a noticeable influence on quantities such as pressure, friction and flow of the lubricant. Texturing is a way to control the surface topography with great accuracy. Further, the surface texture can be understood as the repetition of identical shapes along the surfaces. The presence of certain textured features is known to may benefit some lubricant systems by acting as microreservoirs, storing and providing the fluid directly to the contact zone. The use of a textured bearing system has been found to be more demanding due to its advantageous features such as low value of friction, higher load carrying capacity, decreasing surface damages and increasing service life of bearing [11,12]. In the recent year, many bearing designers focused on the influence of surface texture in the fluid film journal bearing system. Etsion et al. [13] investigated experimentally the effects of surface texture on the performance of parallel thrust bearing. They reported the influence of dimple geometry on the performance of bearing by providing a series of dimples on surface of parallel thrust bearing. Further, it has been noticed that the optimized texturing parameters may help to enhance the hydrodynamic action. Siripuram and Stephens [14] studied the influence of different types of microasperities for both protruding and recessed asperities with constant height. They showed that the value of friction coefficient is independent to the types of asperity profiles and orientation, but very sensitive to the size of aperities and side leakage greatly depends on asperity profile, orientation and size. Cupillard et al. [15,16] established the mechanism for the pressure build up in converging gap due to introducing the texture profiles on the bearing surfaces. The authors analyzed the effect of dimples on the bearing performance for different depths of dimples and convergence ratios. They reported that dimples produce significant changes in the load carrying capacity for low convergence ratios k = 0.1 and k = 0.2, where the gain goes up to 74% and 32%, respectively.
Recently, Kango et al. [17] studied the influence of different types of dimples on the performance of hydrodynamic journal bearing. They concluded that the use of surface texturing on the surface of journal bearing enhances the load carrying capacity as well as reduces the friction coefficient. Influence of lubricant inertia in the textured bearing was analyzed by Syed and Sarangi [18]. They also studied the effect of dimple orientation on the performance of unit dimple slider thrust bearing. Their results showed that texture shape and size plays a vital role in the improvement of lubrication performance.
Due to the rapid technological developments in the recent decade, the bearings are required to be operated with improved lubricants. To improve the properties of lubricant, i.e., viscosity index, resistance to oxidation and volatility, etc., some polymer additives are added by tribologists and lubricant engineers. Due to the addition of these additives, lubricant behaves as a non-Newtonian lubricant [19]. The couple stress lubricant is one case of non-Newtonian lubricant, which contains the long-chain polymer molecules. The special feature of couple stress lubricant is that the stress tensor is anti-symmetric and their flow behaviour cannot be anticipated by the classical Newtonian hypothesis. In the year 1966, Stokes [20] developed the theory of couple stress with consideration of the effects of particle sizes. This theory is based on the generalization of classical theory of lubricants by accounting for the effects of anti-symmetrical stresses, couple stresses and body couples. It is proposed that a linearized constitutive equation for the force and couple stresses. Later, Stokes [21] derived the expressions for velocity and pressure distributions for creeping flow past a sphere. It was indicated that the value of drag increases with couple stresses and the effect of couple stresses is equivalent to an apparent increase in the viscosity.
Further, many investigators used the couple stress fluid theory to analyze the effect of squeeze film behaviour [22][23][24][25][26]. It was concluded that the squeeze film characteristics are significantly affected in the presence of couple stresses. Ramanaiah and Sarkar [22] presented the performance of slider bearings lubricated by fluids with couple stress. They derived analytical expressions for the load carrying capacity, frictional force and the centre of pressure. They observed that the load carrying capacity increases and frictional coefficient decreases with increasing the couple stress parameter. A study by Lin [23,24] deals with the influence of couple stress lubricant on the dynamic characteristics of journal bearing system. It was found that the couple stress lubricant gives the largest values of fluid film stiffness and damping characteristics as compared to Newtonian lubricants. It was also observed that for the applied external loads corresponding to the eccentricity ratio e 0.7, the rotor bearing operated with couple stress fluids under a small disturbance gives more stability than Newtonian lubricant. Ma et al. [25] investigated the behaviour of dynamically loaded journal bearings lubricated with couple stress fluids. They revealed that the couple stress fluids lubrication enhances the bearing performance under dynamic loads. Further, it was also reported that the lubricants with couple stresses increase the value of fluid film pressure and the attitude angle as well as reduce the friction coefficient as compared to Newtonian lubricant. Crosby and Chetti [26] studied the effect of couple stress lubricant behaviour on the performance of two-lobe journal bearing. They concluded that the use of couple stress lubricant improved the stability of the bearing against a bearing operated with Newtonian lubricant.
A thorough review of the published literature pertaining to the journal/thrust bearing system reveals that the textured surface significantly enhances the performance of bearing. To the best of authors knowledge, a very few researchers focused their studies on the influence of surface texture on the performance of hybrid journal bearing system. Thus, it was felt more work is required that can emphasize the influence of surface texture in the area of hybrid journal bearings to provide the much needful information for the design process. However, there is no study in the literature reported so far to deal with the influence of couple stress lubricant on the performance of textured two-lobe hole-entry hybrid journal bearing system. Thus, the present work is planned to bridge this gap in the literature and to study the performance analysis of textured two-lobe hole-entry journal bearing lubricating with couple stress lubricant. The textured surfaces have been provided circumferentially on the front of each hole of bearing land area. In this study, the circular and non-circular bearing profiles are differentiating by using the non-dimensional offset factor (d). Further, the influence of couple stress lubricant has been accounted for by defining a couple stress parameter ðl c Þ. The geometry of a two-lobe textured hole-entry hybrid journal bearing is presented in Figures 1-3.

Analysis
The momentum and continuity equations for the lubricant with couple stresses proposed by Stokes are given as [20,27] For the steady state flow of incompressible fluids, ∂r ∂t ¼ 0; ∇⋅ṽ ¼ 0, equations (1) and (2) reduce to By neglecting the body forces and moments, the momentum equation (3) is further reduced to The velocity and pressure component for twodimensional flow of lubricant are expressed as [27,28] V ¼ ðV 1 ðx; zÞ; V 2 ðx; zÞ; 0Þ ð 6Þ By incorporating the appropriate boundary conditions in equations (6) and (7) and with the help of continuity equation of flow, the modified Reynolds equation is obtained as follows [29,30]: where f c (l c , h) = couple stress function By introducing the following non-dimensional parameters:

Fluid film thickness
The dimensionless nominal fluid film thickness for two-lobe non-recessed hybrid journal bearing is expressed as [8,9,31] where X J and Z J are the journal center co-ordinates, X i L and Z i L are the coordinates for lobe center. The dimensionless fluid film thickness for micro spherical dimple is expressed as [32,33] The dimensionless fluid film thickness for two-lobe textured non-recessed hybrid journal bearing is expressed as

Finite element formulation
The fluid flow domain of couple stress lubricant has been discretized using four-noded quadrilateral elements for FEM calculation. The unknown fluid film pressure is considered to be distributed linearly over an element and is defined by [31,32,34] P ¼ where N j is the Lagrangian polynomial function and n e l is the number of nodes per element.
By applying the Galerkin's and FEM techniques, the global system equation is expressed as [8,32,34] where n x and n y are direction cosines.

Restrictor flow equation
The flow rate of couple stress lubricant through the capillary compensating device is expressed by following non-dimensional equation [9,34,35]: where C s2 is the non-dimensional parameter for capillary discharge and is obtained by

Performance characteristics
The two-lobe hybrid journal bearing performance characteristics, i.e., minimum fluid film thickness, frictional torque, external load carrying capacity, lubricant, etc., fall under the category of steady state performance characteristic. Whereas in the category of dynamic performance characteristic, the stiffness and damping coefficients, critical mass and stability threshold speed parameters are included.

Static performance characteristics
The static performance characteristics are simulated after journal centre occupies its equilibrium position ði:e:; _ X J ¼ _ Z J ¼ 0Þ under a given vertical load ðW 0 Þ.

Load carrying capacity
The fluid film pressure obtained from the Reynolds equation is integrated to obtain the fluid film reaction components along X and Z direction as follows [9,30]: p cos a da db ð17:1Þ The resultant fluid film reaction is obtained as The attitude angle is defined as the angle between the line of action of external load, W o and line of center. The line of action of load (i.e., vertical load line) is taken as a reference for computing attitude angle. The attitude angle for different equilibrium positions of journal center are expressed as follows:

Dynamic performance characteristics
The dynamic performance characteristics are evaluated under the disturbance of the journal centre from its equilibrium position (i.e., _ X J ≠ 0; _ Z J ≠ 0Þ.

Fluid-film stiffness coefficients
The pressure derivative with respect to journal center displacement ðq j ¼ X J ; Z J Þ gives the expression for fluid film stiffness coefficients and is written by [9,31,34] where "i" indicates the force direction and "q j " indicates the direction for journal center displacement ðX J or Z J Þ.
Stiffness coefficients in matrix form may be expressed as:

Fluid-film damping coefficients
The pressure derivative with respect to journal center velocity ð_ q j ¼ _ X J ; _ Z J Þ gives the expression for fluid film damping coefficients and is written by [9,31,34] where _ q j indicates the journal center velocity ð _ X J or _ Z J Þ components.
Damping coefficients in matrix may be expressed as:

Linearized equations of motion and stability margin
The linearized equation of disturbed motion of the journal about its equilibrium position is framed by equating the inertia force components to the out of balance fluid film force and moment components. In compact form the linearized equation of motion is described as [32,36] ½M J f€ q g þ ½Cf_ q g þ ½Sfqg ¼ 0 2   6  6  6  6  6  6  6  4   3   7  7  7  7  7  7  7  5 ð19:3Þ ð20:3Þ The linearized system representing the lateral motion of the journal can be reduced in matrix form as [32,36] The stability threshold speed margin ðv th Þ is calculated by using the relation given below [32,36]: where F 0 is the resultant fluid-film force ∂h ðM c Þ is the non-dimensional critical mass and is expressed as [32,36] The stability threshold speed margin of the journal bearing system is defined in terms of critical mass ðM c Þ. The system becomes stable, when the journal mass is smaller than the critical mass ði:e:; M J < M c Þ.

Boundary conditions
The following boundary conditions are used to obtain a unique solution from partial differential equation for twolobe hybrid journal bearing [8,9,31,35]: -All nodes at the ends of the bearing boundary were assigned ambient pressure; The nodes located on the boundary of the holes have the same pressure. -Restrictor input flow is equal to bearing lubricant flow rate. p ¼ ∂p ∂a ¼ 0:00 is at trailing edge of the positive region for symmetric pressure distribution in the a direction. It is known as Reynolds boundary condition (RBC). This RBC suggested that pressure will smoothly approach zero at the position where cavitation occurs.

Solution strategy
The complete details of the solution algorithm to study the combined influence of textured surfaces and behaviour of couple stress lubricant on the performance of tow-lobe holeentry hybrid journal bearing is illustrated in Figure 4. As the governing Reynolds equation is non-linear in pressure and cannot be solved by using analytical methods, therefore, a MATLAB program has been developed with the help of FEM formulation as described in the above section. To obtain accurate solution, a uniform fine mesh of 49 Â 20 is used in analysis. The solution strategy is implemented in the following steps: 1. The lubricant domain is discretized by using four-node iso-parametric quadrilateral elements. 2. Operating and geometric parameters of the bearing are defined for two-dimensional lubricant grid. 3. Computation of the fluid film thickness for nontextured, textured, circular/non-circular journal bearing using equations (10) and (12) for steady state conditions. 4. Computation of the elements matrices using Gauss points and assemblage into global fluidity matrices. 5. Apply the boundary conditions to global fluidity matrices. 6. The modified Reynolds equation (9) for flow of couple stress lubricant is solved by Gauss-Siedel iterative method to evaluate the nodal fluid film pressure distribution. 7. Steps 3-6 are repeated until the following convergence criteria is satisfied for journal center equilibrium position:  of hydrodynamic journal bearing has been computed for different couple stress parameters at a wide range of eccentricity ratios and validated with published data of Mokhiamer et al. [30]. From Figure 5, the minor deviation has been noticed between present and published results because of the use of different solution technique. Figure 6 shows the comparison of fluid film pressure distribution with the study of Brizmer and Kligerman [33] for textured hydrodynamic journal bearing. The simulated results have reasonably good in trend and values. In this study, the variation of the static and dynamic bearing characteristics has been computed with external load ðW o Þ and couple stress parameter ðl c Þ for different types of bearing configurations. The representative operating and geometric parameters have been chosen as reported in Table 2. The numerical values of these parameters have been judiciously chosen from the available literature [8,9,26,29,30,36]. Further, many available studies [8,9,31,36] confirmed that multi-lobe hybrid journal bearings having offset factor greater than one (d > 1) exhibits better bearing performance characteristics than the bearing having offset factor (d 1). Thus, in the present analysis offset factor d > 1 has been taken into account. Furthermore, the simulated static and dynamic characteristics such as frictional torque, minimum fluid film thickness, fluid film pressure distribution, lubricant flow, attitude angle, fluid film dynamic coefficients and threshold speed are presented in Figures 7-19.

Influence on maximum fluid film pressure ðp max Þ
The three-dimensional contour plots of fluid film pressure distribution for circular/non-circular textured/non-textured hole-entry hybrid journal bearing are illustrated in Figures 7-9. Figures 7-9 show the value of maximum fluid film pressure ðp max Þ increase with an increase in the value of couple stress parameter ðl c Þ for all types of bearing configurations. It can be concluded that the additives    rees deg  with larger chain length molecules, enhances the load carrying capacity of the bearing. Further, it has been observed that the two-lobe journal bearing increases the value of p max with respect to circular journal bearing. The reason is that the two-lobe journal bearing gives a more convergent gap at the bottom of the bearing. From Figures 7-9, it can also be seen that the textured journal bearing reduces the value of p max for Newtonian and couple stress lubricant ðl c ¼ 0:1Þ, respectively. Whereas the couple stress lubricant ðl c ¼ 0:2Þ lubricated journal bearing enhanced the value of p max for both textured and nontextured journal bearing. For the given value of W o ¼ 1:0, when the two-lobe (d = 1.1) non-recessed hole-entry hybrid journal bearing lubricates with couple stress lubricant ðl c ¼ 0:2Þ, the value of p max gets enhanced by 21.7% and 15.4% for textured and non-textured journal bearing, respectively, over a circular journal bearing lubricating with Newtonian lubricant. Figure 10 shows the influence of couple stress lubricant behaviour on the value of minimum fluid film thickness ðh min Þ of textured/non-textured circular/two-lobe hybrid journal bearing. As seen from Figure 10, it may be noticed that by applying the surface texturing on the bearing surface reduces the minimum fluid film thickness ðh min Þ. Whereas the minimum fluid film thickness increases for non-circular (d = 1.1) profile journal bearing. This is due to the fact that the offset factor, d = 1.1, provides a maximum clearance at the bottom of the bearing. As a result of this, the minimum fluid film thickness increased by adding extra lubricant in the clearance gap. It may also be observed that the value of h min increases with an increase in the value of couple stress parameter ðl c Þ. Therefore, a bearing with couple stress lubricant can sustain higher value of external load and permits the larger value of h min . At a chosen value of external load, W o ¼ 1:3, the circular textured journal bearing reduces the value of h min by order of 2.7%, 2.4% and 2.0% for Newtonian, couple stress ðl c ¼ 0:1Þ and couple stress ðl c ¼ 0:2Þ lubricant, respectively, as compared to the non-textured journal bearing. While the two-lobe (d = 1.1) textured journal bearing reduces the value of h min in order of 2.6%, 2.4% and 2.1% for Newtonian, couple stress ðl c ¼ 0:1Þ and couple stress ðl c ¼ 0:2Þ lubricant, respectively, with respect to non-textured tow-lobe journal bearing. In order to maintain the safe limiting value of h min , the following trend may be useful: h min j d¼1:1 > h min j d¼1:0 ; h min j Non-textured > h min j Textured and h min j l c ¼0:2 > h min j l c ¼0:1 > h min j Newtonian

Influence on lubricant flow ðQÞ
The variation of dimensionless lubricant flow ðQÞ with increasing the load carrying capacity ðW o Þ is depicted in Figure 11. The lubricant flow ðQÞ decreases as couple stress parameter ðl c Þ increases at chosen value of W o . This is because of the larger chain length additives presence in lubricant provides anti-symmetrical stresses and increases the resistance to the flow as well as the wall shear stress. It may be observed that the requirement of lubricant flow rate gets significantly increased for textured journal bearing over a non-textured journal bearing for either of Newtonian ðl c ¼ 0:0Þ and couple stress lubricated ðl c ¼ 0:1; l c ¼ 0:2Þ hole-entry hybrid journal bearing configuration. Whereas the two-lobe (d = 1.1) journal bearing reduces the requirement of lubricant flow rate as compared to circular journal bearing. As seen in Figure 11, the circular textured journal bearing lubricating with Newtonian lubricant gives the Table 2. Non-dimensional operating and geometric input parameters for two-lobe non-recessed hybrid journal bearing [8,9,26,29,30,36].

Influence on frictional torque ðT f Þ
The influence of surface texturing along with behaviour of couple stress lubricant on the value of frictional torque ðT f Þ is presented in Figure 12. It can be clearly seen that the non-circular two-lobe (d = 1.1) journal bearing increases the value of T f for both textured and non-textured journal bearing, consequently increases the power loss than that of circular journal bearing. Therefore, to design an efficient bearing the value of T f should be minimize. Further, the presence of the textured surfaces on the bearing surface reduces the value of T f for both circular and two-lobe holeentry hybrid journal bearing. It is because of the reason that, textured surfaces acts as a lubricant micro-reservoir and providing the lubricant directly to the contact zone. Further, Figure 12 shows that the value of frictional torque ðT f Þ decreases with increasing the value of couple stress parameter ðl c Þ for all types of bearing configurations. For two-lobe (d = 1.1) textured journal bearing, the percentage reduction in the value of T f is of order of 5.6-6.0%, 5.3-5.9% and 5.1-5.4% for Newtonian, couple stress ðl c ¼ 0:1Þ and couple stress ðl c ¼ 0:2Þ lubricant has been observed, respectively, as compared to non-textured journal bearing. From the numerically computed results, the bearing designer may use the following definite trend for the value The similar pattern has also been observed in the works reported by Guha [29] and Crosby and Chetti [26] for the value of fluid film stiffness coefficients in the case of hydrodynamic and two-lobe journal bearings. The non-circular (d = 1.1) profile of the bearing enhanced the values of S 11 and S 22 against circular journal bearing. The noticeable observation is that the textured journal bearing gives larger values of S 11 and S 22 as compared to nontextured journal bearing. It may be observed that the influence of couple stress lubricant in the values of fluid film stiffness coefficients ðS 11 and S 22 Þ is significantly more for textured journal bearing against non-textured journal bearing. Further, maximum influence of offset factor (d) in the values of S 11 and S 22 is observed for textured journal bearing as compared to non-textured journal bearing. As seen from Figure 14, when the two-lobe (d = 1.1) journal bearing operates with couple stress ðl c ¼ 0:2Þ lubricant, the percentage increment in the value of S 11 is 29.9-31.2% and 3.4-4.0% for textured and non-textured journal bearing, respectively, as compared to corresponding bearing lubricating with Newtonian lubricant. Further, it may be noticed that from Figure 15, the two-lobe (d = 1.1) textured journal bearing improves the value of S 22 in order of 5.1-5.8%, 5.3-6.0% and 5.6-6.1% for Newtonian, couple stress ðl c ¼ 0:1Þ and couple stress ðl c ¼ 0:2Þ lubricant, respectively, against circular textured journal bearing. While the two-lobe (d = 1.1) non-textured journal bearing enhance the value of S 22 by an amount of 3.6-3.75%, 3.4-3.6% and 3.2-3.3% for Newtonian, couple stress ðl c ¼ 0:1Þ and couple stress ðl c ¼ 0:2Þ lubricant, respectively, with respect to circular non-textured journal bearing. From the computed results, the following general pattern has been obtained for the values of S 11 and S 22 : S 11=22 j Textured > S 11=22 j Non-textured and S 11=22 j l c ¼0:2 > S 11=22 j l c ¼0:1 > S 11=22 j Newtonian The effect of couple stress lubricant behaviour on the value of fluid film damping coefficients ðC 11 and C 22 Þ is shown in Figures 16 and 17. From Figures 16 and 17, it may be seen that for the textured journal bearing, the values of C 11 and C 22 get reduced as compared to the non-textured journal bearing. The trend for the values of C 11 and C 22 is consistent with observation of already published study [34]. From the vibration point of view, a proper value of fluid film damping coefficients is essential to damp out the shaft vibration under dynamic conditions. Further, the use of couple stress lubricant and non-circular (d = 1.1) profile bearing partially compensated the reduction in the values of C 11 and C 22 because of the textured surfaces on the bearing surface. As seen in Figure 16, for the given value of external load W o ¼ 1:3, the textured circular journal bearing reduces the value of C 11 by amount of 22.86% for Newtonian lubricant as compared to non-textured circular journal bearing. While the two-lobe (d = 1.1) textured journal bearing lubricates with couple stress ðl c ¼ 0:2Þ lubricant, the value of C 11 is enhanced in order of 6.4% with respect to non-textured circular journal bearing lubricating with Newtonian lubricant. Further, from Figure 17, it may be observed that the two-lobe (d = 1.1) textured journal bearing improves the value of C 22 in order of 4.9-5.4%, 5.1-5.6% and 5.6-6.1% for Newtonian, couple stress ðl c ¼ 0:1Þ and couple stress ðl c ¼ 0:2Þ lubricant, respectively, over a textured circular journal bearing. From the simulated   results, following useful trend has been obtained for values of C 11 and C 22 : and C 11=22 j l c ¼0:2 > C 11=22 j l c ¼0:1 > C 11=22 j Newtonian 4.8 Influence on stability threshold speed ðv th Þ Figure 18 depicts the variation of stability threshold speed margin ðv th Þ with respect to the load carrying capacity ðW o Þ. The computed result in Figure 18 indicates that the textured journal bearing provides the greater value of v th for both circular and two-lobe journal bearing. Similarly, the two-lobe (d = 1.1) journal bearing gives better stability threshold speed margin ðv th Þ for both textured and nontextured journal bearing than that of circular journal bearing. From Figure 18, it may also be noticed that the influence of couple stress lubricant behaviour and noncircular profile of bearing is significantly more in the value of v th of textured journal bearing than that of non-textured journal bearing. Further, it can be seen that the value of threshold speed margin ðv th Þ increases with increasing the value of couple stress parameter ðl c Þ. The value of threshold speed ðv th Þ depends on the value of bearing rotor dynamic coefficients. As the values of bearing dynamic coefficients get affected due to couple stress parameter ðl c Þ. Hence, the bearing stability speed margin is bound to change. For the non-recessed hybrid journal bearing, the textured two-lobe (d = 1.1) bearing lubricating with couple stress lubricant ðl c ¼ 0:2Þ gives the highest stability parameter over the other journal bearing configurations. When the two-lobe (d = 1.1) journal bearing operates with couple stress lubricant ðl c ¼ 0:2Þ, the percentage increment in the value of v th is found to be of the order of 12-12.7% and 2.2-2.5% for textured and non-textured journal bearing with respect   C.B. Khatri and S.C. Sharma: Mechanics & Industry 18, 603 (2017) to corresponding bearing operating with Newtonian lubricant. To design an efficient stable journal bearing, the following trend may follow: v th d¼1: and v th l c ¼0:2 > v th l c ¼0:1 > v th Newtonian j . Furthermore, Figure 19 shows the linear journal motion trajectories for circular/non-circular textured/non-textured journal bearings. From Figure 19, it may be revealed that the textured bearing surfaces traces a small orbit for come back from disturbed journal centre position to equilibrium journal centre position in comparison to nontextured journal bearing. Smaller whirl orbit size indicates that journal centre fluctuates around its equilibrium position with lower amplitude of vibration. It has been also noticed that the use of couple stress lubricant and noncircular profile on bearing gives small locus of journal centre and settles down quickly to equilibrium position of journal centre. Hence, based on simulated trajectories, we can say that the couple stress lubricant lubricated two-lobe (d = 1.1) textured journal bearing is highly beneficial from stability point of view.
The comparative performance of two-lobe textured hybrid journal bearing against non-textured circular journal bearing considering the influence of couple stress lubricant is presented in Table 3. It has been noticed that the two-lobe textured journal bearing lubricated with couple stress lubricant ðl c ¼ 0:2Þ provides improved performance with respect to circular non-textured journal bearing. However, at a chosen value of external load W o ¼ 1:3, the two-lobe textured journal bearing operating with couple stress lubricant ðl c ¼ 0:2Þ gives enhancement in the value of stability threshold speed margin ðv th Þ with 38.14% as compared to base journal bearing as seen in Table 3.

Conclusion
In the present study, the performance of two-lobe holeentry hybrid journal bearing has been analyzed by considering the combined effects of surface texturing on bearing surface and behaviour of couple stress lubricant. On the basis of computed results in present study, the following conclusions are summarized: -The use of couple stress lubricant, two-lobe (d =     textured journal bearing. Further, it may also be seen that the combination of two-lobe (d = 1.1) profile, couple stress lubricant and textured surface provides the small radius of whirl orbit to settle down the journal centre to equilibrium position after some disturbances owing to unbalanced fluid film forces. Hence, it may be point out that the couple stress lubricant lubricated two-lobe (d = 1.1) textured journal bearing is highly beneficial from stability point of view. -The textured hole-entry hybrid journal bearing system reduces the values of fluid film, damping coefficients (C 11 and C 22 ). However, the use of couple stress lubricant in a two-lobe (d = 1.1) journal bearing gives improvement in the values of C 11 and C 22 . Hence, to significantly damp out the unwanted oscillations of journal bearing, the combination of textured surface, non-circular profile and couple stress lubricant is beneficial than the Newtonian lubricant lubricated circular non-textured journal bearing.