Issue 
Mechanics & Industry
Volume 24, 2023



Article Number  2  
Number of page(s)  27  
DOI  https://doi.org/10.1051/meca/2022027  
Published online  16 January 2023 
Regular Article
Elastohydrodynamic lubrication analysis of a porous misaligned crankshaft bearing operating with nanolubricants
^{1}
Université 8 mai 1945 Guelma, Mechanical Engineering Department, Laboratory of Mechanics and Structures (LMS), BP 401, Guelma 24000, Algeria
^{2}
Université de Lyon, CNRS INSALyon, LaMCoS, UMR5259, 69621, France
^{*} email: benyebka.bouSaid@insalyon.fr
Received:
22
July
2022
Accepted:
2
October
2022
In this paper, the combined effects of the characteristic size and concentration of inorganic fullerenelike tungsten disulphide nanoparticles (IFWS_{2} NPs) or molybdenum disulphide nanoparticles (IFMoS_{2} NPs) on the nonlinear dynamic behaviour of a gasoline engine crankshaft bearing subject to an arbitrary force torsor (effective applied force and moment vector) are theoretically and numerically investigated using the V. K. Stokes microcontinuum theory. These NPs are the most common additives for lubrication purposes due to their excellent tribological characteristics along with their effect on reducing friction and wear. It is assumed that the journal (crankshaft) currently made of a forged steel is rigid and the main bearing consists of a thin poroelastic liner made of low elastic modulus materials like Babbitt metals fixed in a stiff housing as defined by ASTM B2300. The KriegerDougherty law is included in the proposed EHD model to account for the viscosity variation with respect to the volume fraction of nanoparticles dispersed in the base lubricant. On the other hand, the characteristic size of nanomaterials is introduced by a new material entity, denoted l, which is responsible for a couplestress property. The Reynolds equation is derived in transient conditions and modified to account for the size of nanoparticles and the bearingliner permeability property. For an arbitrary force torsor, the hydrodynamic pressure distribution, the squeeze film velocities, and the misalignment angular velocities are determined simultaneously by solving the discretized Reynolds equation and the equilibrium equations with the damped NewtonRaphson iterative method at each crank angle step. The crankshaft center trajectories in three sections of the main journal axis as well as the misalignment angles are deduced from the squeeze film velocities and the misalignment angular velocities by means of a RungeKutta scheme. According to the obtained results, the combined effects of the size and concentration of fullerenelike nanoparticles on the dynamic behavior of a compliant dynamically loaded crankshaft bearing operating with dynamic misalignment are significant and cannot be overlooked.
Key words: Layered crankshaft bearing / dynamic misalignment / couplestresses / nanolubricant / modified Reynolds' equation / relaxed NewtonRaphson method / thin poroelastic liner
© R. Hamel et al., Published by EDP Sciences 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Automotive industry requirements are currently of increasing importance. Each new generation of thermal engines requires a formulation of more efficient lubricants. The current trend in the design of modern engines is to optimize their operating characteristics in terms of power, fuel consumption, and compliance with increasingly stringent environmental standards. A consequence of these requirements is the severity of the stresses to which the various engine components are subjected, which results in an increase of undesirable friction. It is well known that friction is one of the main causes of loss of efficiency in an internal combustion engine. Indeed, from 10 to 20% of the power contained in the engine fuel is absorbed by the mechanical power which is dissipated by friction [1]. In this regard, petroleum refiners and engine lubricant manufacturers must be able to formulate lubricating oils that meet the increasingly stringent specifications such as environmental standards, but also efficient in terms of reducing friction and wear of moving parts.
To reduce friction in an engine, two technological solutions can be considered. The first solution consists of treating the active contact surfaces by depositing coatings such as thin layers of amorphous carbon (DLC: DiamondLike Carbon), lead or tinbased white metals (Babbitts), and polyetheretherketone (PEEK composites). The properties of these materials to reduce friction and wear have turned out to be very interesting [2]. The major drawback of such a solution lies in the difficulty, or even the impossibility, of depositing such coatings in certain geometrically complex areas of the engine.
The second solution concerns the choice of a more efficient lubricant with physicochemical properties other than its viscosity. It is then a question of formulating lubricants by adding additives to the base lubricant which can act when the lubricating film is no longer thick enough to avoid the destructive risks of direct metaltometal contact between the rough surfaces. To this end, the inclusion of antiwear additives to the base lubricant constitutes a technological solution to enable a reduction of friction and wear rates, especially in the mixed and boundary lubrication regimes.
It should be noted that the additives with tribological action such as antiwear additives currently used in the formulation of traditional motor oils are organic compounds based on sulfur and phosphorus. These two compounds are the cause of the deterioration of vehicle gas treatment systems and are called into question by antipollution standards. The substitution of these molecules by lubricating nanoparticles will allow the formulation of lubricating nanofluids (i.e., nanolubricants) that are more efficient and more respectful of the increasingly strict environmental standards.
The harmfulness of sulfur and phosphorus emissions of which the two molecules MoDTC and ZnDTP are at the origin, as well as the ineffectiveness of these two molecules at cold temperatures, have recently mobilized significant research on the development of new alternatives to these molecules. Such new materials should respect the environmental standards and have tribological properties at least equivalent to those of the additives being used currently. This research has led to a completely revolutionary new nanotechnology approach in the field of lubricating additives through the development of a new family of nanoparticle (NP) additives. It has been shown that some of these nanomaterials, when used in dispersion as lubricating additives, have, sometimes, clearly superior lubrication performance compared to those of traditional additives [3–5]. In addition, these nanoparticles are found to be more chemically inert than traditional molecular additives and therefore potentially less harmful to the environment. Accordingly, the use of nanotechnologies in the context of tribological applications (lubrication) is currently seeing increasing interest. This interest has resulted in a substantial development of nanomaterial synthesis techniques allowing the development of nanoobjects of different structures (fullerenes, nanotubes, etc.) and of different chemical nature (metallic, oxides, carbonaceous, etc.). Numerous studies on the tribological properties (friction and wear) of these nanoobjects dispersed in a lubricating base have shown substantial reductions of wear as nanoparticles were added to oil bases. According to Chen et al. [6], the multilayered carbon nanotubes dispersed in oil reduce friction by 10% and wear by 30% compared to the lubricating base.
On the other hand, the inorganic fullerenes IFMoS_{2} and IFWS_{2}, synthetized for the first time in 1992 by Tenne et al. [7], can reduce friction and wear by up to 70% compared to the lubricating base [8,9].
From the tribological point of view, fullerenes are then the most interesting systems and they can be used as potential substitutes to currently used molecular lubricating additives since they are environmental pollutants and less efficient in service under certain specific conditions. Indeed, the MoS_{2} and the WS_{2} act like a third body in lubricated rough contacts and essentially behave like tiny microscopic ball bearings thus reducing the undesirable friction between the two metal surfaces in contact as illustrated in Figure 1.
Many key questions were addressed such as the involved lubrication mechanisms as well as the influence of the concentration, size, structure, morphology, and aggregation of nanoparticles on the nanolubricant rheological behavior [10]. Experimental rheological studies demonstrated that nanolubricants (base oil + nanoadditives) have a nonNewtonian rheological behavior and their flow cannot be described by the classical continuum mechanics theory that neglects the size of particles [11]. This requires the use of alternative continuum theories that accurately describe the hydrodynamic flow of nanolubricants in the bearing clearance space, considering the effects of both concentration and size of NPs. Among them, the Vijay Kumar Stokes microcontinuum theory [12–14] was successfully applied to this kind of fluids and the theoretical analyses revealed an increase in load carrying capacity and an enhancement of dynamic stability of journal bearings [15–20].
As far as the authors know, very few research investigations have been carried out to study the journal bearing dynamic behavior operating with NP based lubricants. Nair et al. [21] showed, for a given eccentricity ratio, that the addition of NPs on commercially available lubricant may enhance the viscosity of lubricant and, in turn, change the steadystate and dynamic performance characteristics such as the load capacity, attitude angle, end leakage, the stiffness and damping coefficients, and the whirl frequency.
Nair et al. [22] studied the static performance characteristics in terms of load capacity, friction force, end leakage, and attitude angle using isoviscous and thermoviscous approaches for different values of eccentricity ratios and lubricants containing different nanoparticles (CuO, CeO_{2}, and Al_{2}O_{3}). They found that for the thermoviscous case at high values of eccentricity, addition of nanoparticles produces a significant effect on the performance characteristics. The load capacity and friction force of the bearing increase with the increase of concentration of nanoparticles for both isoviscous and thermoviscous cases. They also found that the end leakage and attitude angle decrease with increase of concentration of nanoparticles in both isoviscous lubricants and lubricants that contain nanoparticles.
Shenoy et al. [23] analyzed the effects of CuO, TiO_{2} and Nanodiamond nanoparticles additives in SAE 30 engine oil on static characteristics of an externally adjustable fluidfilm bearing. The obtained results showed, in the case of a bearing having negative radial and negative tilt adjustments, and operating with SAE engine oil added with TiO_{2} nanoparticles showed that the carrying load capacity was higher by about 23% and 35% than that obtained for SAE engine oil without nanoadditives and base oil, respectively. Furthermore, the friction force of an externally adjustable fluidfilm bearing increased by about 25% for the SAE engine oil without nanoparticles compared to the same oil blended with TiO_{2} nanoparticle additives.
Kalakada et al. [24] investigated the static performance characteristics of thermohydrodynamic journal bearings operating under nanolubricants (lubricants containing per cent weight concentration of nanoparticles). The lubricant and the nanoparticles used for the present work are SAE 15W40, copper oxide (CuO), cerium oxide (CeO_{2}), and aluminum oxide (Al_{2}O_{3}). Their study was focused on the distributions of pressure and temperature throughout the lubricant film as well as the static performance characteristics. They found that addition of nanoparticles increases the load capacity of journal bearings at any eccentricity ratio, especially in the thermoviscous case and at high values of the eccentricity ratio. They also found that the friction force increases with an increase in concentration of nanoparticles for both isothermal and thermoviscous cases. However, at any eccentricity ratio, both end leakage flow and attitude angle decrease with the increase in concentration of nanoparticles in both cases especially at higher eccentricity ratios.
Binu et al. [25] presented a theoretical study on static characteristics of a journal bearing operating with TiO_{2} based nanolubricant considering a variable viscosity model. The predicted shear viscosities of TiO_{2} based nanolubricant at different volume fractions and aggregate particle sizes were obtained using modified Krieger–Dougherty model and were found to be in good agreement with experimental shear viscosities. The modified Reynolds equation considered the Krieger–Dougherty viscosities and couplestress effects of TiO_{2} nanoparticle additives at different volume fractions and particle sizes. The obtained results revealed that an increase in load carrying capacity of 45%, in comparison with plain engine oils, for journal bearings operating with nanolubricants containing TiO_{2} nanoparticle additives at a concentration of 0.01 volume fraction and couple stress parameter, , of 0.03108, which corresponds to an average TiO_{2} nanoparticle aggregate size of 777 nm. The analysis also revealed a 40% increase in friction force for journal bearings using TiO_{2} based nanolubricants as compared to plain engine oil.
Sadabadi et al. [26] employed the CFD simulation approach to model the effect of nanofluid lubricant on load carrying capacity of highly loaded industrial journal bearings. Inorganic fullerenelike WS_{2} nanoparticles were used as an additive to the base synthetic industrial oil with different weight fraction percentages. The obtained results showed that the load carrying capacity of the journal bearing can be improved up to about 20% using a practical 5.0 wt.% IFWS_{2} NP additive. In addition, the SEM images showed that the high aggregation tendency of WS_{2} NPs limits the application of nanolubricant at weight fractions above 6.0 wt.%.
More recently, Kadhim et al. [27] have analyzed the TEHD performance of journal bearings with pure and Al_{2}O_{3} nanolubricant with different volume fractions by using fluidstructure interaction with CFD technique. The relevant results obtained show that the use of Al_{2}O_{3} nanolubricant with 5.0% of NPs increases the oil film thickness by about 18%, and the presence of the NPs in the base oil lightly decreases the maximum oil film temperature.
Dang et al [28] studied the static thermal performances of a pure elliptical bearing lubricated with NPs based mineral oils at different eccentricity ratios and journal speeds. Two types of NPs CuO and TiO_{2} with 0.5, 1.0, and 2.0 wt.% concentrations have been separately added in three different viscosity grades of oils. They found that the peak pressure and load capacity increase with addition of NPs and this increase was more pronounced at higher concentrations of NPs and at higher viscosity grade oils.
Gundarneeya and Vakharia [29] conducted an excellent experimental work by using a journal bearing test rig to investigate the performance analysis of a journal bearing operating on nanolubricants formulated by adding TiO_{2}, CuO, and Al_{2}O_{3} NPs as lubricant additives in ISO VG 46 oil. The experimental results revealed an increase in maximum pressure and load carrying capacity of the journal bearing by adding NPs to the base oil especially for the higher volume fraction of NPs.
Bangotra and Sharma [30] investigated the impact of surface waviness on the static performance parameters of hydrodynamic journal bearings operating with lubricants containing copper oxide (Cu0) and cerium oxide (CeO_{2}) NPs. The static performance parameters were calculated for different waviness numbers and various wave amplitudes with variable viscosities of nanolubricants using a relationship between the relative viscosity, temperature, and weight fraction of NPs developed from experimental results. They found that the addition of such NPs to the lubricant enhanced its viscosity which further improved the steadystate parameters of the wave bearing, namely the carrying load capacity and the friction coefficient.
Byotra and Sharma [31] developed a finite element model to analyze the performance improvement of a journal bearing by applying the arcshaped textures on various regions of bearing operating with and without nanolubricants. They observed that the addition of Al_{2}O_{3} and CuO NPs increased the carryingload capacity.
In the existing technical literature, the studies of the dynamic behavior of the poroelastic internal combustion engine crankshaft journal bearings subjected to an arbitrary force torsor and operating with nonNewtonian nanolubricants are scarce. Therefore, it is felt that there is a need to carry out the dynamic performance characteristics of such journal bearings using the microcontinuum theory of Vijay Kumar Stokes [12,13] which considers the characteristic size of nanoparticles or agglomerates of nanoparticles to the extent that the latter is neglected when using classical or traditional continuum theory. The use of this theory, also known in literature as the theory of nonNewtonian polar fluids, involves stress couples and volume couples in addition to surface and volume forces in the transport equations. The nonNewtonian character of such fluids is particularly due to the skewsymmetry of the stress tensor.
The assumed incompressible and nonNewtonian nanolubricant is then characterized using three physical constants, namely: the density ρ_{nf}, the dynamic or absolute viscosity μ_{nf}, and a constant responsible for the presence of stress couples in the lubricant denoted η whose the dimensional equation is [η] = MLT^{−1} similar to that of momentum. The first two constants naturally depend on the concentration or the volume fraction ɸ of the nanoparticles in the basic lubricant. It is then necessary to use laws of variation of these constants with concentration such as the law of KriegerDougherty [32,33] currently used in literature for the calculation of the effective dynamic viscosity of the nanolubricant.
In the parametric study, we introduce a novel parameter denoted which has a dimension of length, i.e. [l] = L . This parameter accounts for the couple stress property and can be regarded as the characteristic diameter of the fullerene NPs or the agglomerates of fullerene NPs.
Basically, the present research is concerned with the study of the combined effects of the characteristic size and concentration of inorganic fullerenelike tungsten disulphide nanoparticles (IFWS_{2} NPs) or molybdenum disulphide nanoparticles (IFMoS_{2} NPs) on the nonlinear dynamic behaviour of a gasoline engine crankshaft bearing subject to an arbitrary force torsor. It is assumed that the journal (crankshaft) currently made of a forged steel is infinitely rigid and the main bearing consists of a thin poroelastic liner made of low elasticity modulus materials like Babbitt metals fixed in a stiff housing as defined by ASTM B2300. The simplified thin elastic liner model (TELM), derived from a semianalytical approach based on the use of complex Kolosov and Muskhelishvili potentials [17] considering asymptotic cases for plane strain hypothesis, is retaken to determine the pressure induced deformation at the bearing linerfluid film interface. Thus, this elasticity model avoids us the use of the 3D finite element analysis of the bushing structure.
Reynolds equation is derived in transient conditions and modified to account for the size of NPs or the aggregates of NPs dispersed in the base lubricant, and the bearingliner permeability property. This material property is introduced by means of the MorganCameron approximation [34].
In the proposed PEHD model, the normalized Reynolds equation is first written in residual form and discretized by the finite difference method using two different computational molecules (stencils).
For an arbitrary force torsor, the hydrodynamic pressure and film thickness distributions, the squeeze film velocities, and the misalignment angular velocities are determined simultaneously by solving the discretized Reynolds equation and the equilibrium equations with the damped NewtonRaphson iterative method at each crank angle step. The crankshaft center trajectories in three sections of the main journal axis, as well as the misalignment angles, are deduced from the squeeze film velocities and the misalignment angular velocities by means of a RungeKutta scheme.
Note that the dynamic misalignment considered in the present investigation is due to the noncentral application of the dynamic load W which therefore causes a misalignment torque T over the entire cycle, i.e. where T_{X} (t) andT_{Y} (t) arethetorque components, ℓ_{w} is the lever arm, and W_{X} and W_{Y} are the dynamic load components expressed in the frame related to the engine cylinder.
Fig. 1 Representation of a rough contact lubricated by a fluid with fullerenelike NPs. 
2 Governing equations
Figure 2 shows the details of the geometric configuration of a layered engine crankshaft bearing subject to an arbitrary force torsor. A coordinate system (X, Y, Z) is fixed to the nonrotating bushing with its origin located at the bearing center. The cylindrical main journal of radius R, made of a hard steel, is assumed to be rigid and rotates at a constant angular velocity about the Zaxis.
Fig. 2 Geometry of a five degree of freedom main journal bearing, and the system of coordinates. 
2.1 Modified Reynolds equation and pressure boundary conditions
Using the usual assumptions of thin film lubrication theory, and considering the nanolubricant as an incompressible nonNewtonian couplestress fluid [12–14], the transient modified Reynolds equation for laminar flow in the connectingrod bigend bearing with a poroelastic layer may be derived in its more general form as [18]
For the main crankshaft bearing, the modified Reynolds equation (1a) reduces to
where
and l is the characteristic size of nanoparticle added to the base lubricant.
The last term appearing in equation (2) is incorporated by using the MorganCameron approximation [24] to consider the bearingliner permeability property k.
In equations (1) and (2), h is the film thickness expressed by equation (3a) considering the initial geometrical clearance, misalignment angles of the main journal, and the elastic deformation [35]:
Its rate of change is given by
ℒ being the compliance operator and which is the total temporal derivative.
In equation (3a),e_{X} and e_{Y} are the eccentricity components of the main journal axis in the midplane section. The terms , , appearing in equation (3b) stand for squeeze film velocities in X and Ydirection, and misalignment angular velocities about X and Y axis, respectively.
Pressure within the lubricating film must satisfy the modified Reynolds equation (1) and the following boundary conditions:
Equation (4) represents conditions at the bearing ends, while (5) expresses the periodicity of pressure in the circumferential direction.
Additionally, the SwiftStieber conditions or simply the Reynolds conditions are needed to take into account the cavitation phenomenon (fluidfilm rupture) occurring in the diverging space of the bearing clearance filled with lubricant, i.e.,
where θ_{cav} is the cavitation angle at which the film rupture occurs.
If no cavitation condition is prescribed, the pressure obtained from the Reynolds equation corresponds to the Sommerfeld solution. This solution is physically unrealistic because the pressure becomes negative in the diverging part of the clearance.
Once the pressure distribution is obtained, resultant film force and moment components acting on the main journal surface are evaluated by the following relations
2.2 Viscosity model of nanolubricants
Nanolubricants are fluids containing nanoparticles such as fullerene NPs. These nontraditional fluids, used as efficient liquid lubricants in machinery, are suspensions of particles in a base lubricant.
The effective viscosity of the nanolubricant, μ_{nf}, is calculated using the KriegerDougherty viscosity model [23].
where μ_{0} is the viscosity of base fluid, φ is the volume fraction of spherical fullerene NPs in base fluid. The symbol φ_{m} is the maximum particle packing fraction, which varies from 0.495 to 0.54 under quiescent condition, and is approximately 0.605 at high shear rates [28], and [ is the intrinsic viscosity whose typical value for mono disperse suspensies of hard spheres is 2.5 [36].
Typical variation of the relative viscosity, , with the volume fraction φ, is presented in Figure 3 for φ_{m} = 0.605.
Fig. 3 Graphical representation of the KriegerDougherty viscosity model for a range of volume fraction and φ_{m} = 0.605. 
2.3 Force balance and journal center orbits
For a misaligned journal bearing, subject to an arbitrary force torsor (Fig. 2), the equations of motion are given as follows [37,38]:
where are the angular acceleration components about X and Yaxis We denote T_{X} = ℓ _{w}W_{Y}, T_{Y} = ℓ _{w}W_{X} as the applied couples,m_{j} is the journal mass, and I_{0X}, I_{0Y}, and I_{0Z} are components of the inertia tensor:
The four equations of motion (9) have eight unknowns, namely e_{X} (t) , e_{Y} (t) , e '_{X} (t) , e '_{Y} (t) , α_{X} (t) , α_{Y} (t) , α '_{X} (t), and α '_{Y} (t) . Therefore, four more equations are required and are given below [39]
Taking into account equations (10), the equations of motion (9) become
See equation (11) below.
If the journal bearing is aligned and the inertia forces are neglected, equations (11) reduce to
The coupling of equations (1) and (11) results in a nonlinear system whose unknowns are the fluid film pressure, the instantaneous translational journal velocities (e ' _{X}, e ' _{Y}) also called squeeze velocities, and the instantaneous misalignment angular velocities (α '_{X}, α '_{Y}).
The instantaneous journal center position at the midplane section (e_{X}(t), e_{Y}(t)) of bearing as well as the instantaneous misalignment angles (α_{X} (t) , α_{Y} (t)) are then calculated from equations (10) using the modified Euler scheme or the RungeKutta method of second order.
The journal eccentricity components (e_{XF, }e_{YF}) at the front bearing end (z = +L/2) and the journal eccentricity components(e_{XR}, e_{YR})at the rear bearing end (z = −L/2) (Fig. 2) are determined by the following relation:
3 Numerical treatment of the governing equations
There is no analytical solution of the transient nonlinear inverse problem governed by equations (1) and (11), thus numerical calculation using the finite difference method is used to determine the pressure distribution within the oil film. Detailed description of the procedure is given in Annex A.
4 Elasticity model: expression of the compliance factor ℒ
In the film thickness equation (3), the radial component of the elastic displacement field U = ℒ p at the fluid filmbearing liner interface is obtained from elastic analysis of the bearingliner considered as a cylindrical shell made of low elasticity modulus materials like white metals or PEEK (Poly Ether Ether Ketone, ASTM B2300) rigidly fixed in a stiff housing (Cast steel).
Hydrodynamic pressures developed in the fluid film to endure external forces act as normal stresses on the fluid filmbearing liner interface and produce elastic deformations of the bearing shell structure. These deformations occur in the three dimensions of space; i.e. radial, axial, and circumferential.
It has been demonstrated [17] that if the relative thickness of the bearingliner is very small; i.e. ≪1, the axial and circumferential components of the deformations are negligible compared to the radial. In a such case, the radial deformation at any point U (θ, z, t) belonging to the bearingliner interface can be expressed in terms of the pressure at that point by the relation (14):
This relationship is derived assuming plane strain hypothesis [40].
From equation (14), the compliance factor is then a scalar defined by the following expression
where E and σ are Young's elasticity modulus and Poisson's ratio of the homogeneous and isotropic bearingliner, respectively. The compliance matrix [L] only contains nonzero coefficients in the principal diagonal, i.e. L_{ij} = ℒ δ_{ij} where δ_{ij} is the Kronecker symbol or the unit second order tensor.
The scalar compliance factor ℒ is also defined as the deformation coefficient and can be regarded as a measure of flexibility and compressibility of the bearingliner since it involves Young's modulus and Poisson's ratio.
The simplified elasticity model derived for a thin cylindrical shell with finite dimensions and expressed by equation (14) is commonly known as the “Thin Elastic Liner Model” and has been successfully integrated by several authors [41–43] in EHD simulations of singlelayered cylindrical journal bearings.
The reason for using the TELM is the easy implementation and very low CPU time requirement since very little time is needed to find the values of the radial displacement U .
However, this model is only approximate and is expected to be less accurate when a journal bearing with large shell thickness particularly made of incompressible or almostincompressible materials σ > 0.4 is to be investigated. In such cases, the threedimensional FEM is more suitable to generate the compliance matrix [L] of the bearingliner structure. Hence, the radial displacement at node k should be calculated by equation (16).
where k = (j − 1) (N − 1) + i, and K = (N − 1) M. If L_{km} = ℒ δ_{km}, equation (39) becomes U_{k} = ℒ p_{k}.
5 Hydrodynamic characteristics
5.1 Side leakage flow
The side leakage flow at bearing edges is obtained from the following relation:
See equation (17) below.
where θ_{1}(t) and θ_{2}(t) are the angles delimiting the active zone of the bearing.
In the above equations, h is the film thickness and is the axial mean flow velocity:
The axial flow velocity w as well as the circumferential velocity u are determined from integration of equations governing the flow of the nano lubricant assumed as a nonNewtonian couplestress fluid in x and zdirections [17]
The general solutions have the following form:
Using the boundary conditions:
we obtain
See equation (22a),(22b) below.
5.2 Power loss
The total power loss is evaluated on the active zone of the bearing from
where
In equation (24), Φ_{1} and Φ_{2} are the dissipation functions due to the shear stress and the couple stress effects, respectively. These two functions can be defined in hydrodynamic lubrication theory as [44]
After integration with respect to y, we get
where
and
6 Solution procedure of the inverse and transient EHD lubrication problem for dynamically loaded journal bearings operating with dynamic misalignment
After selecting the input parameters of the problem, for each time step t or crankshaft rotation angle θ_{c} ranging from 0 to 1440 degrees (two engine load cycles), the procedure for obtaining a numerical solution to the inverse EHD problem governed by equations (1) to (6), as well as equation (9) is outlined as follows:
Step 1
Initialize the iteration counter n to zero, choose initial values of , and the initial dimensionless pressure at each interior grid point.
Step 2
Construct the initial solution such that for j = 1, K, and
Step 3
Calculate the dimensionless film thickness and its rate of change at each grid point, and effective viscosity of nanolubricant .
Step 4
Calculate the residual functions and f_{i} for i = 1, K + 4.
Step 5
Generate the Jacobian matrix and solve the linear algebraic system E_{ij} δX_{j} = − f_{i} for δX_{j} such that i, j = 1, K + 4.
Step 6
Update the solution according to the NewtonRaphson recurrence formula
where ω_{NR} is an underrelaxation factor which ranges from 0 to 1 (i.e., 0 < ω_{NR} ≤ 1), and n = 0, 1, 2, 3 … , n_{max}.
Step 7
Extract the nodal film pressures from the solution ⟨X^{n+1}⟩, set all negative values of pressure to zero and calculate the corresponding radial deformations .
Step 8
Check convergence of the Newton Raphson algorithm: <ϵ_{tol} where ϵ_{tol} = 10^{−6}. If the convergence test is not satisfied, increment the iteration counter n (n ← n + 1), and return to step 3 to perform another iteration while n < n_{max}.
Step 9
Extract the velocities of the journal center and the angular velocities of misalignment ε^{′}_{X}, ε^{′}_{Y}, α^{′}_{X}, and α^{′}_{Y} from ⟨X^{n+1}⟩ and calculate the new values of the eccentricity components and misalignment angles using the modified Euler's scheme or the RungeKutta method of second order, i.e.,
and similar relations for ϵ_{Y} (t + Δt) , α_{X} (t + Δt) , andα_{Y} (t + Δt) where , Δθ_{c} being the crank angle step [degrees], and n_{j} is the engine rotational speed [rpm].
Step 10
Calculate the instantaneous hydrodynamic performance characteristics of the journal bearing and store the obtained numerical results for postprocessing.
7 Results and discussion
7.1 Validation of the thin elastic liner model (TELM)
To confirm the validity of the TELM, a separate 3DFE computer code using eightnode hexahedral elements has been developed to generate the compliance matrix [L] for a cylindrical shell structure. The FE formulation is obtained using the principle of virtual work which corresponds to the Galerkin weak formulation of the threedimensional elasticity problem [45].
The bearingliner is modeled as a cylindrical shell structure made of a homogenous and isotropic material having Young's elasticity modulus of the leadbase white metal (E = 29 GPa) [46].
Two values of Poisson's ratio have been selected to study the compressibility effect on the accuracy of the TELM.
The comparison between the more accurate 3DFE results and results obtained from the analytical TELM (Eq. (14)) for a given pressure field (Fig. 4) confirms the accuracy of the TELM, when the shell thickness t_{l} is much smaller than the bearing radius R and the shell material is compressible (σ < 0.40) as depicted in Figure 5.
Fig. 4 Circumferential distribution of the hydrodynamic pressure field exerted on the Bearing liner—fluid film interface. 
Fig. 5 Comparison of the radial displacement field calculated by the simplified thin elastic liner model at the bearingliner—fluid film interface with that obtained by the 3D finite element analysis. 
7.2 Validation of the computer code
There are no benchmark results available for crankshaft bearing operating with nonNewtonian couplestress nanolubricant. Therefore, the proposed PEHD model is first compared with the published results of Paranjpe et al. [47], and Hirani et al. [48] for a gasoline engine crankshaft main journal bearing using a Newtonian fluid as lubricant. The bearing and lubricant data are reported in Table 1. Figure 6 shows the load components acting on the crankshaft main bearing at 5000 rpm obtained by interpolation with cubic splines for each onedegree crank rotation angle from data given in Table 2 to ensure convergence of the Euler's method and to faithfully represent the peak loads. The two peaks in the Ycomponent of the load at crank angles, θ_{c}, at approximately 135 degrees and 255 degrees crank angle are caused by firing of the neighboring cylinder. In Table 3, we compare the predicted values of both the greatest peak pressure and the smallest film thickness with those published by Paranjpe et al. [47] and Hirani et al. [48]. It may be deduced that the computed results lie within the scatter of the results obtained by using different approaches such that the powerful technique of FEM (FEJOB) which uses the complementarity principle for implementing SwiftStieber or Reynolds condition at both film rupture and reformation boundaries. However, there are slight discrepancies between the results which may be attributed to the feeding conditions of the bearing that were not considered in the present calculations. The journal bearing system is supplied with lubricant through a halfcircumferential groove which is machined in the midplane section of the bearing with a supply pressure p_{s} = 2.758 × 10^{5} Pa.
To further validate the correctness and accuracy of the algorithm and the computer code, the Diesel Ruston and Hornsby 6 VEBX Mk III connecting rod bigend bearing with circumferential groove was analyzed because the data are widely available in technical literature [49,50] related to dynamically loaded bearings. The engine and bearing data are given below: crank length ℓ_{2} = 0.184 m, connecting rod length ℓ_{3} = 0.782 m, angular velocity of the crankshaft ω_{c} = −20π rad/s, length of the halfbearing L = 0.057 m, bearing diameter = 0.2032 m, bearing radial clearance C = 82.55 × 10^{−6} m, and dynamic viscosity of the lubricant μ = 0.015 Pa s.The values of the greatest peak pressure obtained by the proposed methodology and those obtained by various other techniques are reported in Table 4. The results show that the value of the greatest peak pressure is comparable to that obtained by the FEA.
Results for the Paranjpe's gasoline engine crankshaft main bearing.
Results for the Ruston Hornsby Diesel engine bigend bearing.
7.3 Parametric study
After the successful validation, the code has been used to study the combined effects of dynamic misalignment, size and concentration of NPs; and poroelasticity on the crankshaft axis stationary orbits, the minimum film thickness calculated in rear, middle, and front sections of the main bearing. The side leakage flow, and the power loss for an entire engine cycle are also computed. All calculations have been carried out for the Paranjpe's main bearing [47] consisting of an elastic layer made of lead base white metal for which the geometric and elastic characteristics are t_{l} = 1 mm, E = 29 GPa, σ = 0.33. The obtained results correspond to the same dynamic loading as depicted in Figure 6.
7.3.1 Effects of dynamic misalignment
Various numerical simulations have been performed considering a misaligned case. The dynamic misalignment is due to the noncentral application of the dynamic load that therefore causes a misalignment torque over the entire cycle. The graphical results are presented in Figures 7–14 for Newtonian and nonNewtonian fluids, i. e. , and two values of the relative lever arm . The aligned case correponds to L_{W} = 0.
Figures 7 and 8 show the comparison of the stationary main journal bearing axis trajectories for both Newtonian and nonNewtonian couple stress cases. At the ends of the layered bearing (rear and front sections), the influence of the misalignment is clearly significant. However, at the midplane section of the bearing, the same shape of trajectories is obtained in both aligned and misaligned cases. On the other hand, the EHD simulations performed in Newtonian case (Fig. 7) for L_{W} = 10% showed that the maximum eccentricity occurs at the front section plane (Fig. 7c) which tends perilously towards the bearing radial clearance. In contrast, the EHD calculations made for the same bearing using a nanofluid as lubricant predicted that the presence of NPs leads to a contraction of the orbits at the three sections for both aligned and misaligned cases as depicted in Figure 8. Therefore, the presence of NPs or aggregates of NPs moves the crankshaft center orbit towards the bushing center. This allows the system to operate safely.
In both the Newtonian and nonNewtonian couple stress cases, the minimum film thickness occurs at one of the bearing edges, i.e. the front section. Figures 9 and 10 show the variation of the minimum film thickness as a function of crank angle for both Newtonian and nonNewtonian cases. In almost the entire cycle, the values of the minimum film thickness calcultated in the Newtonian case are smaller than those obtained in the nonNewtonian couple stress case.
At the front section of the bearing (Figs. 9c and 10c), the value of minimum film thickness is calculated as 1.56 [µm] at θ_{c} = 231° which is about seven times smaller than the value 10.73 [µm] at θ_{c} = 238° obtained in the nonNewtonian couple stress case for the misaligned case, i. e. L_{W} = 10%.
It can be concluded that the presence of NPs which are responsible for the couple stress in the lubricating fluid can prevent the occurrence of metaltometal contact at the bearing edges.
Figures 11–14 show that there is no significant influence of the dynamic misalignment on side leakage flow and power loss for the entire engine cycle in comparison to the aligned case. Furthermore, the average side leakage flow calculated in the nonNewtonian case when there is a presence of NPs is smaller than that predicted in Newtonian case as depicted in Figures 11 and 13, i.e., when the bearing is lubricated with a pure base fluid. However, the opposite trend may be observed for the average power loss but only slightly Figures 12 and 14.
Fig. 7 Effects of the dynamic misalignment parameter L_{W} on the stationary crankshaft center orbits for the Newtonian case. 
Fig. 8 Effects of the dynamic misalignment parameter L_{W} on the stationary Journal center orbits for the nonNewtonian case. 
Fig. 9 Effects of the dynamic misalignment parameter L_{W} on the minimum film thickness over an engine cycle for the Newtonian case. 
Fig. 10 Effects of the dynamic misalignment parameter L_{W} on the minimum film thickness over an engine cycle for the nonNewtonian case. 
Fig. 11 Effects of dynamic misalignment on side leakage flow over an entire cycle. 
Fig. 12 Effects of dynamic misalignment on power loss over an entire cycle. 
Fig. 13 Effects of dynamic misalignment on side leakage flow over an entire cycle. 
Fig. 14 Effects of dynamic misalignment on power loss over an entire cycle. 
7.3.2 Effects of NPs size
Figure 15 shows the influence of the NPs size parameter, l/C, on the main journal center orbits for both Newtonian and nonNewtonian cases. We note that the size of the crankshaft axis trajectories decreases with the increasing of l/C leading to an increase of the minimum film thickness as shown in Figure 16. The orbits calculated for the various values of the size parameter have the same shape.
The effect of the size of NPs gives a slight reduction of the side leakage flow as shown in Figure 17. However, the power loss is significantly affected by the NP size over the entire engine cycle as depicted in Figure 18. For the same crank angle, the power loss calculated in Newtonian case () is more pronounced than that obtained in nonNewtonian case especially for the higher values of the NPs size parameter ().
Fig. 15 Effects of the particle size parameter, l/C, on the stationary Journal center orbits for the deformable case. 
Fig. 16 Effects of the particle size parameter, l/C, on the minimum film thickness over an engine cycle for the deformable case. 
Fig. 17 Effects of the particle size parameter on side leakage flow over an engine cycle. 
Fig. 18 Effects of the particle size parameter on power loss over an engine cycle. 
7.3.3 Effects of NPs concentration
Figure 19 depicts for the case of aligned and layered bearing, the influence of the NPs concentration parameter φ on the stationary main journal center orbits for both Newtonian and nonNewtonian cases. One can observe that the increase of this parameter decreases the size of the crankshaft axis trajectories and increases the minimum film thickness in both cases Figure 20. This moves the shaft center orbits towards the bushing center due to the increase of the nanolubricant viscosity. The orbits calculated for the various values of the NPs concentration have the same shapes.
As expected, the increase of the dynamic viscosity due to the presence of NPs leads to an increase of the power loss and a decrease of the side leakage flow as depicted in Figures 21 and 22.
Fig. 19 Effects of the NPs concentration parameter, φ, on the stationary journal center orbits for the deformable case. 
Fig. 20 Effects of the nanoparticles concentration parameter, φ, on the minimum film thickness over an engine cycle for the deformable case. 
Fig. 21 Effects of the nanoparticles concentration parameter on side leakage flow over an engine cycle. 
Fig. 22 Effects of the nanoparticles concentration parameter on power loss over an engine cycle. 
7.3.4 Effects of the poroelasticity of the bearing liner
Figure 23 depicts the effect of the liner permeability on the shaft trajectory in the case of an aligned bearing. When the shaft is far from the bearing, the effect of permeability is not important. The pressure is not sufficient enough to allow the fluid to flow into the porous layer. As the shaft approaches the bearing the porosity of the material begins to play its role as a flow regulator. The compressed fluid can penetrate the porous layer and flow inside it. The trajectory compared to the impermeable case is larger.
Figure 24 represents the minimum film thickness evolution over an engine cycle. It confirms that for the porous liner case the minimum film thickness is lower when compared to the impermeable case. The fluid is incompressible and offers greater resistance to squeeze when the material is not porous.
The effect of a permeable layer on the leakage rate is not significant as depicted in Figure 25. This is due to the low thickness of the thin layer (1 mm) and the moderate pressure level in this layer.
As the film thicknesses are lower in the case of a permeable thin layer, the shear rates are higher, and a higher power dissipation is thus observed (Fig. 26).
Fig. 23 Effects of bearingliner permeability parameter on the stationary Journal axis orbits. 
Fig. 24 Effects of bearingliner permeability parameter on the minimum film thickness over an engine cycle. 
Fig. 25 Effects of bearingliner permeability parameter on side leakage flow over entire cycle. 
Fig. 26 Effects of bearingliner permeability parameter on power loss over entire cycle. 
8 Conclusions
On the basis of the V. K. Stokes microcontinuum theory for describing the flow of nanolubricants, the combined effects of size and concentration of NPs or aggregates of NPs on the dynamic behavior of a layered main crankshaft bearing subject to an arbitrary force torsor were investigated.
The analytical thin elastic liner model (TELM) was successfully used to determine the induced pressure and the radial deformation at the bearing linerfluid film interface.
The KriegerDougherty law was included in the proposed PEHD model to account for the viscosity variation with respect to the fraction volume of NPs dispersed in the base lubricant.
A transient modified Reynolds equation was derived to take into consideration the effects of couple stresses resulting from the presence of NPs in the base oil as well as the porosity of elastic thin layer. This latter was incorporated in the pressure equation by means of the MorganCameron approximation.
The damped NewtonRaphson iterative method was applied to predict the hydrodynamic pressure distribution, the squeeze film velocities, and the misalignment angular velocities. The RungeKutta of second order scheme was used to calculate the crankshaft center trajectories and misalignment angles.
From this investigation, the following conclusions may be drawn:
The presence of the NPs in base oil produces higher oilfilm thickness, more contracted trajectories and decreases the average side leakage flow and power loss.
The dynamic misalignment due to the noncentral application of the dynamic load produces smaller oilfilm thickness at the ends of bearing (front section) and more divergent trajectories for both Newtonian and nonNewtonian cases but have no influence on the leakage flow rate and power loss.
The minimum film thickness increases with the increase of NPs size parameter.
In Newtonian cases, the porosity of the thin layer decreases the minimum film thickness and increases the trajectories of the main crankshaft center.
The current findings will be beneficial for automotive engine designers to improve the performance characteristics of crankshaft bearings.
Nomenclature
Symbols: Definitions, SI Units
C: Bearing radial clearance, m
E: Young's elasticity modulus of the bearingliner, Pa
e: Operating eccentricity, , m
F_{X} (t) F_{Y} (t): Applied load components, N
T_{X} (t) , T_{Y} (t): Torque components, N.m
m_{j}: Crankshaft mass per main bearing, kg
W_{X} (t) , W_{Y} (t): Carrying load components, N
n_{j}: Engine rotational speed, rpm
k: Permeability of the porous matrix, m^{2}
l: Size of NP (or agglomerates of NPs), m
Q_{z}: Side leakage flow, m^{3}/s
t_{l}: Thickness of bearingliner, m
ℓ_{2}: Crankshaftarm length, m
ℓ_{3}: Connectingrod length, m
e_{X}, e_{Y}: Eccentricity components of the main journal axis in the midplane section., m
e '_{X}, e '_{Y}: Instantaneous translational journal velocities, m/s
α '_{X}, α '_{Y}: Misalignment angular velocities, rad/s
e_{XF}, e_{YF}: Journal eccentricity components at the front bearing end, m
e_{XR}, e_{YR}: Journal eccentricity componentsat the rear bearing end, m
α '_{X}, α '_{Y}: Misalignment angular acceleration components, rad/s^{2}
M_{X}, M_{Y}: Misalignment moment components, N.m
I_{0X}, I_{0Y}, I_{0Z}: components of the inertia tensor, kg.m^{2}
U (θ, z, t): Radialdeformation of the oil filmbearing liner interface, m
φ: Volume fraction of spherical fullerenes NPs in base fluid, −
ψ: Nondimensional permeability, −
ω_{NR}: Underrelaxation factor for the Newton Raphson algorithm, −
μ_{0}: Dynamic viscosity of base fluid, Pa.s
: Relative viscosity of nanolubricant, −
ϵ: Eccentricity ratio, ϵ = e/C, −
η: Material constant responsible for couplestresses, N.s
σ: Poisson's ratio of the bearingliner, −
θ_{c}: Crank rotation angle, rad
ρ: Mass density of lubricant, kg/m^{3}
Abbreviations
ASTM: American Society for Testing and Materials
CFD: Computational Fluid Dynamics
TELM: Thin Elastic Liner Model
PEHD: PoroElastoHydroDynamic
TEHD: Thermal ElastoHydroDynamic
SAE: Society of automotive Engineers
SEM: Scanning Electron Microscopy
Frames
(O, X, Y, Z): Frame related to cylinder of engine
(x, y, z): Local frame for writing of Reynolds equation
Annex A: Numerical treatment of the governing equations
There is no analytical solution of the transient nonlinear inverse problem governed by equations (1) and (13), thus numerical calculation using the finite difference method is used to determine the pressure distribution within the oil film.
Figure A1 shows the (2D) finite difference grid of fluid medium showing both doubleindexednode numbers and renumbered singleindexed nodes. As illustrated in this figure, the unique number or the global index, k, assigned to each gridnode can actually be derived by using the node's i and j indices, as follows:
where k ∈ {1, K}.
Fig. A1 2D orthogonal Cartesian finite difference grid used for spatial discretization of modified Reynolds equation. 
It is to be noted that equation (A1) represents just one way of converting the two indices to a single index and is, by no means, the only way. Based on this numbering pattern for a periodic bearing, the five nodes in the stencil (computational molecule) now have the indices k, k − 1, k + 1, k − N + 1, and k + N − 1 instead of the central node (i, j), the western node (i − 1, j), the eastern node (i + 1, j), the southern node (i, j − 1), and the northern node (i, j + 1), respectively.
For a nonperiodic bearing, equation (A1) becomes:
where N stands for the number of nodes in the circumferential direction.
The advantage of using an automatic formula, such as equation (A1) in the case of periodic bearing, for generation of the unique node number is that it allows numerical generation of the Jacobian matrix more easily, as will be seen later.
For simplicity, we will assume that the nodes are uniformly spaced both in the θ and z direction such that and where N and M are the number of nodes in the θ and zdirection, respectively.
To solve the nonlinear problem, we employ two finite difference schemes in the approximation of the modified Reynolds equation. Equation (1b) written for the main crankshaft bearing is first normalized and expressed in the following residual form:
where
and ,
The other parameters such as β and ψ are defined in the Nomenclature.
When the correct pressure distribution is found, the residual function f is identically zero. Thus, to obtain an accurate numerical solution to the inverse isothermal PEHD problem, the residual function should be accurately discretized.
To this end, the standard secondorder central difference formulas are used to approximate the residual function. For ease of discretization, we first rewrite equation (A3) as:
Now applying the central difference approximation, for the interior nodes (i.e. 1≤i ≤ N − 1, 2 ≤ j ≤ M − 1) we get
See equation (A8) below.
where
Next, we rewrite the discrete equation (A8) in terms of the unique global index k to yield
where
In all, we have to solveK nonlinear equations where = (N − 1) M . To these K equations, we must add the following equations also written in residual form:
where
Recall that k = (j − 1) (N − 1) + i such that i = 1, N − 1andj = 2, M − 1.
The computational molecule shown in Figure A2 is used to approximate the residual function f, from which the Jacobian matrix is generated. Thus, the discrete form of equation (A3) reads as
where the midpoint values of are computed by the following relationships:
Fig. A2 Computational molecule (stencil) used for the generation of the Jacobian matrix coefficients by forward finite difference scheme. 
; ;
and the circumferential and axial pressure gradients are evaluated by the firstorder forward difference formulas:
; ;
Using the unique global index k, equation (A17) becomes
∀k ∈ interior and
for k = 1, K, and f_{k} for k = K + 1, K + 2, K + 3, and K + 4
The damped NewtonRaphson method is employed to iteratively solve equations (A13) and (A15). To accomplish this, let
denote the n^{th} trial solution of Equations (A14) and (A15). According to the NewtonRaphson algorithm with damping factor ω_{NR}, the (n+1)^{th} trial solution is:
{X^{ (n+1)}} = {X^{ (n)} } + ω_{NR} {δX^{ (n)}}
where 0 <ω_{NR} ≤ 1 and
⟨δX^{ (n)}⟩ = ⟨ δX_{1} ⋮ δX_{2} ⋮ … δX_{k} ⋮ … δX_{K} ⋮ δX_{K+1} ⋮ δX_{K+2} ⋮ δX_{K+3} ⋮ δX_{K+4} ⟩
The correction vector {δX^{ (n)}} is found by means of firstorder Taylor series expansions of the functions ; and given by the following expressions:
See equation (A20a) below.
∀k ∈ interior; i.e. i = 1, N − 1 andj = 2, M − 1.
See equation (A20b) below.
The matrix form of equations (A20a) and (A20b) is
where [E^{ (n)}] is the Jacobian matrix of size (K + 4) × (K + 4) having the following general structure:
See equation next page.
and
⟨F^{ (n)}⟩ = ⟨ f_{1} ⋮ f_{2} ⋮ … …… … ⋮ f_{K} ⋮ f_{K+1} ⋮ f_{K+2} ⋮ f_{K+3} ⋮ f_{K+4} ⟩
The firstorder forward difference formula is used to evaluate the Jacobian matrix coefficients, i. e.,
where δ = 10^{−8} for the calculations performed in double precision.
Note that for nodes belonging to the bearing edges(i . e . , forj = 1andj = M), we have according to equation (A18).
For a crank rotation angle θ_{c}, the algebraic system (A21) is solved for each iteration using the Gauss elimination method with partial pivoting after [L] [U] decomposition of the coefficient matrix [E] by means of the Crout's algorithm.
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Cite this article as: R. Hamel, M. Lahmar, B. BouSaïd, Elastohydrodynamic lubrication analysis of a porous misaligned crankshaft bearing operating with nanolubricants, Mechanics & Industry 24, 2 (2023)
All Tables
All Figures
Fig. 1 Representation of a rough contact lubricated by a fluid with fullerenelike NPs. 

In the text 
Fig. 2 Geometry of a five degree of freedom main journal bearing, and the system of coordinates. 

In the text 
Fig. 3 Graphical representation of the KriegerDougherty viscosity model for a range of volume fraction and φ_{m} = 0.605. 

In the text 
Fig. 4 Circumferential distribution of the hydrodynamic pressure field exerted on the Bearing liner—fluid film interface. 

In the text 
Fig. 5 Comparison of the radial displacement field calculated by the simplified thin elastic liner model at the bearingliner—fluid film interface with that obtained by the 3D finite element analysis. 

In the text 
Fig. 6 Cartesian representation of dynamic loading on the Paranjpe's main bearing [47]. 

In the text 
Fig. 7 Effects of the dynamic misalignment parameter L_{W} on the stationary crankshaft center orbits for the Newtonian case. 

In the text 
Fig. 8 Effects of the dynamic misalignment parameter L_{W} on the stationary Journal center orbits for the nonNewtonian case. 

In the text 
Fig. 9 Effects of the dynamic misalignment parameter L_{W} on the minimum film thickness over an engine cycle for the Newtonian case. 

In the text 
Fig. 10 Effects of the dynamic misalignment parameter L_{W} on the minimum film thickness over an engine cycle for the nonNewtonian case. 

In the text 
Fig. 11 Effects of dynamic misalignment on side leakage flow over an entire cycle. 

In the text 
Fig. 12 Effects of dynamic misalignment on power loss over an entire cycle. 

In the text 
Fig. 13 Effects of dynamic misalignment on side leakage flow over an entire cycle. 

In the text 
Fig. 14 Effects of dynamic misalignment on power loss over an entire cycle. 

In the text 
Fig. 15 Effects of the particle size parameter, l/C, on the stationary Journal center orbits for the deformable case. 

In the text 
Fig. 16 Effects of the particle size parameter, l/C, on the minimum film thickness over an engine cycle for the deformable case. 

In the text 
Fig. 17 Effects of the particle size parameter on side leakage flow over an engine cycle. 

In the text 
Fig. 18 Effects of the particle size parameter on power loss over an engine cycle. 

In the text 
Fig. 19 Effects of the NPs concentration parameter, φ, on the stationary journal center orbits for the deformable case. 

In the text 
Fig. 20 Effects of the nanoparticles concentration parameter, φ, on the minimum film thickness over an engine cycle for the deformable case. 

In the text 
Fig. 21 Effects of the nanoparticles concentration parameter on side leakage flow over an engine cycle. 

In the text 
Fig. 22 Effects of the nanoparticles concentration parameter on power loss over an engine cycle. 

In the text 
Fig. 23 Effects of bearingliner permeability parameter on the stationary Journal axis orbits. 

In the text 
Fig. 24 Effects of bearingliner permeability parameter on the minimum film thickness over an engine cycle. 

In the text 
Fig. 25 Effects of bearingliner permeability parameter on side leakage flow over entire cycle. 

In the text 
Fig. 26 Effects of bearingliner permeability parameter on power loss over entire cycle. 

In the text 
Fig. A1 2D orthogonal Cartesian finite difference grid used for spatial discretization of modified Reynolds equation. 

In the text 
Fig. A2 Computational molecule (stencil) used for the generation of the Jacobian matrix coefficients by forward finite difference scheme. 

In the text 
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