## Abstract

The governing equations for the flow of a granular material within the context of the lubrication theory are derived. The resulting analysis gives a generalized Reynolds equation that predicts the pressure generation capacity in a bearing with consideration of side flow. A series of simulations are presented that characterize the three-dimensional flow behaviour of powder in a slider bearing.

## 1. Introduction

Over a century has passed since Osborne Reynolds first introduced his celebrated theory of hydrodynamic lubrication. Reynolds was inspired by Tower's report (Tower 1883) which experimentally demonstrated that a thin film of lubricating oil is capable of generating an extraordinarily high pressure. Reynolds went on to lay the foundation of the thin-film lubrication theory and thus arrived at an equation that bears his name (Reynolds 1886). In its complete form, the Reynolds equation provides a realistic prediction of the pressure distribution as well as a complete description of the flow behaviour in a three-dimensional domain. With ample experimental verification testifying to its validity, the Reynolds equation has been widely used in the design and analysis of tribological components.

The derivation of the Reynolds equation was based on the assertion that the fluid is linearly viscous (Newtonian), which applies to most conventional liquid lubricants. Yet, many practical situations arise that require revisiting its derivation.

In this paper, we seek to examine the performance of a class of lubricants that is expected to have much greater flexibility in terms of its operating temperatures. It is well known that most conventional liquid lubricants are incapable of operating at temperatures exceeding 200 °C. Hence, the need for high-temperature engines call for development of alternative lubricants and methods by which their performance can be predicted. Suitable granular powders have been identified as promising lubricants for this purpose.

In a noteworthy undertaking, Heshmat (1992) reported a series of tests with an alternative lubricant where, instead of oil, a suitable type of granular powder was injected into the clearance space of a thrust bearing. The granular powder generated a hydrodynamic-type lift and with the aid of a new transducer, Heshmat recorded the ensuing pressure profile along the direction of motion. The shape of the pressure profile was remarkably similar to what is normally seen in liquid-lubricated bearings; hence he referred to it as the quasi-hydrodynamic lubrication. In quasi-hydrodynamic lubrication, the principal mechanism responsible for generation of pressure is the momentum and energy transportation from the collisions between powder particles. Therefore, in order to proceed with the analysis, an appropriate constitutive relation is needed.

Turning our attention to the pertinent literature on the granular materials, we begin by referring to the work of Savage & Jeffrey (1981), who developed a theory for determining the stress tensor for granular material undergoing a rapid shear flow. They assumed that momentum transport takes place based on the binary collisions between smooth perfect spheres. Jenkins & Savage (1983) and Lun *et al*. (1984) extended Savage and Jeffrey's theory to incorporate energy dissipation into the formulation for a more complete description of the problem. Another fundamental contribution is due to the work of Haff (1983), who offered a phenomenological approach for treating granular flow by applying the theory of dense gases to describe the granular collisions. Haff's formulation approach is simple, straightforward to derive, and quite attractive for describing the general features of granular flows. However, evaluation of a series of unknown constants in the granular constitutive equations is necessary to arrive at realistic quantitative predictions.

In a noteworthy contribution, Johnson & Jackson (1987) applied a more realistic constitutive equation—put forward by Lun *et al*. (1984)—in treating the granular flow between two infinitely wide, parallel plates. According to Johnson & Jackson (1987), the total stress tensor and the total energy flux vector are approximated as the sum of frictional and collisional–translational contributions, each calculated as if it acted alone. The frictional component of stress translates directly into thermal internal energy, i.e. the true heat flux (thermodynamic temperature), and the collisional–translational component is translated into pseudo-thermal energy.

In this paper, we set out to derive the appropriate governing equation for a granular flow within the context of the lubrication theory. We derive a generalized Reynolds-type equation capable of predicting the granular flow characteristics and pressure in a bearing with an arbitrarily shaped gap. The flow is three-dimensional and side leakage is taken into consideration, as required in the treatment of a bearing with a finite aspect ratio. Derivation of the equation, comparison with experimental results, and presentation of extensive parametric simulations comprise the content of this paper.

## 2. Theory

### (a) Conservation laws

We begin by invoking the conservation laws for mass, momentum and pseudo-energy. When dealing with conventional lubricants without the consideration of temperature-dependent properties—as was the case in the classical Reynolds equation—only the mass and momentum conversations are needed. In granular flows, the equation governing the conservation of pseudo-energy is naturally coupled to the momentum equation, akin to a class of problems known as thermohydrodynamic theory (Khonsari *et al*. 1996). A schematic model of the slider bearing lubricated with powder is shown in figure 1.

Following the notation of Johnson & Jackson (1987), the conservation of mass is given by(2.1)where *ρ*=*ρ*_{p}*υ* is the bulk density of the granular material, *υ* is the solid volume fraction, *ρ*_{p} is the density of the individual granule and * U* is the bulk velocity.

The governing equation for the conservation of momentum is(2.2)where * g* is the gravity acceleration, and is the stress tensor caused by the collision and the kinetic motion.

The governing equation for the conservation of pseudo-energy is(2.3)where is the pseudo-thermal temperature defined in terms of the mean fluctuation velocity *v*_{m} of the granule, **q**_{PT} is the flux of the pseudo-thermal energy, is the rate of work by the collisional–translational component of the stress, and *γ* is the rate of energy dissipation due to the inelastic collision between granules. Following Johnson & Jackson (1987), it is assumed that the energy by the enduring contact contributes only to the true thermal energy and, therefore, it does not appear in the conservation of the pseudo-energy.

### (b) Constitutive equations

The appropriate constitutive equations for granular material are developed by Lun *et al*. (1984). The stress tensor due to collision and kinetic motion is(2.4)where is the identity tensor and is the deviatoric part of the rate of deformation tensor defined as(2.5)The parameter *g*_{0} represents the radial distribution function proposed by Carnahan & Starling (1969). It is defined in terms of solid volume fraction as(2.6)where *υ*_{max} is the value of *υ* at the closest random packing. The maximum volume fraction is assumed to be *υ*_{max}=0.65 in this analysis. Parameter *η* is a constant characterizing the inelastic collision between granules and is defined as *η*=(1+*e*_{p})/2. Parameter *e*_{p} represents the coefficient of restitution of granules, i.e. *e*_{p}=1 for perfectly elastic granules and *e*_{p}=0 for perfectly inelastic granules. The bulk viscosity *μ*_{b} and the shear viscosity *μ* for perfectly elastic particles and the thermal conductivity *λ* of the granules are defined as(2.7)where *M*=*πρ*_{p}*D*^{3}/6 represents the mass of each spherical granule.

The effect of enduring contact between granules may be considered following the work of Johnson & Jackson (1987). Based on Coulomb failure criterion, the stress tensor by enduring contact is(2.8)where *κ*=1 when and *κ*=−1 when . *Φ* is the internal angle of friction and *N*_{f} is the normal stress defined as(2.9)where *F*_{r} and *n* are constants. Therefore, the total stress is .

The total flux of the pseudo-energy due to the collision and the kinetic motion is (Lun *et al*. 1984)(2.10)The energy dissipation due to the particle collision is (Lun *et al*. 1984)(2.11)*γ*→0 as *e*_{p}→1 for a perfect elastic granules.

### (c) Boundary conditions

Boundary conditions in granular flows are considerably more involved than those involving the flow of liquids. The principal mechanism for the transport of properties such as momentum and energy are through the interaction between granules and the bounding surfaces as well as between the granules themselves. The granules possess both a flow velocity and a fluctuation velocity. The latter is commonly referred to as pseudo-temperature (and is not a measure of thermodynamic temperature). Campbell's (1993) extensive computer simulations show that behaviour of granular materials is strongly influenced by the collisions between the grains and the boundaries. Slip at the boundaries and momentum transfer as the particles collide with the walls are two important features of the granular flows. Jenkins & Richman (1986) derived the appropriate boundary conditions for the flow of nearly elastic particles by adopting the statistical averaging method. Later Jenkins (1992) applied the continuum granular theory with a proper treatment of boundary conditions for a flat plate with the friction force. Following the work by Hui *et al*. (1984) and Jenkins & Richman (1986), McKeague & Khonsari (1995) focused their attention to the investigation of boundary conditions in Couette flow. They derived the boundary conditions for both the flow velocity and the flow temperature by considering the granular slip using the Haff's theory of granular materials. Later, McKeague & Khonsari (1996) extended their analysis to predict the behaviour of a powder lubricant in an infinitely wide slider bearing.

The condition for the slip velocity between the granules and a boundary surface can be obtained equating the tangential force acting on the boundary and the rate of momentum transfer to the wall by granule impact (Jenkins & Richman 1986). The boundary conditions for slip in the *x*- and *z*-directions are, respectively,(2.12)where *u*_{s} is the slip velocity component in the *x*-direction and *w*_{s} is the slip velocity component in the *z*-direction. * n* is the unit normal vector to the boundary pointing into the powder flow. The parameter

*ϕ*

_{w}is the specularity coefficient whose value depends on the large-scale roughness of the surface. The magnitude of the tangential component of enduring contact is

*N*

_{f}tan

*δ*, where

*δ*is the angle of friction between the surface and the particle.

The boundary condition for the pseudo-temperature is obtained by equating the rate of heat generation due to slip at the boundary and the rate of dissipation of pseudo-thermal energy due to inelastic collisions of granules with unit area of the boundary (Jenkins & Richman 1986; Zhou & Khonsari 2000). It can be shown that the boundary condition for pseudo-temperature is(2.13)where *e*_{w} is the wall coefficient of restitution and varies from zero to unity.

### (d) Velocity distributions

The following set of dimensionless parameters are used to dimensionalize the governing equations and the boundary conditions:(2.14)where *s* and *B*_{o} are characteristic lengths and *U*_{o} represents the sliding velocity. Inserting stresses (2.4) and (2.8) into (2.2), one arrives at the following equations for the conservation of momentum in dimensionless form:(2.15)where *α*=*s*/*D*, *β*=*s*/*B*_{o} and *g* is the gravitational acceleration. The parameters used in equation (2.15) are listed in table 1 and functions from *f*_{1}(*υ*) to *f*_{9}(*υ*) are listed in table 2. In table 1, terms containing parameters *B*_{o}/*s* and *B*_{o}/*D* are the dominant terms, since typically *B*_{o}/*s*≈400 and *B*_{o}/*D*≈5000. Terms containing *ρ*_{p}*f*_{1}(*υ*)*T* can not be neglected due to the fact that *f*_{1}(*υ*) is large. In *y*-momentum equation, the pressure term is and the gravitational term is . Typically which shows that gravitational term is small compared to the pressure term (*B*_{o}/*D*≈5000). The inertia terms are also neglected since they are lower order terms than the viscous term. Neglecting lower order terms reduces (2.2) to(2.16)Note that *P*=*ρ*_{p}*f*_{1}(*υ*)*T*+*N*_{f} represents the pressure in powder lubrication. The second equation in (2.16) reveals that pressure remains constant across the gap.

The flow velocity profiles are determined by integrating the first and the third equations in (2.16), and the results are(2.17)where(2.18)The parameter *θ* is analogous to viscosity in fluid-film lubrication. In equation (2.17), *u*_{H} and *u*_{L} are the flow velocities of granules on the upper surface and lower surface in the direction of sliding, respectively. *w*_{H} and *w*_{L} are the flow velocities perpendicular to the direction of motion in the lateral direction. The flow velocities on the surfaces can be determined from the slip boundary conditions (2.12) through the relationships of *u*_{sH}=*u*_{H}−*U*_{H} and *u*_{sL}=*u*_{L}−*U*_{L}, where *U*_{H} and *U*_{L} are the velocities of the upper and lower surfaces in the *x*-direction, respectively. Note that the slip velocities in the *z*-direction *w*_{sH}=*w*_{H} and *w*_{sL}=*w*_{L} since the velocity of the upper and lower plates in the lateral direction is nil, i.e. *W*_{H}=*W*_{L}=0. Using *m*_{xH}=*u*_{sH}/|*u*_{sH}| and *m*_{xL}=*u*_{sL}/|*u*_{sL}|, the slip boundary conditions (2.12) reduce to the following equations:(2.19)

### (e) Reynolds equation for granular powder lubrication

Integrating the conservative of mass across the channel gap yields(2.20)where, in general, the channel gap *h*=*h*(*x*, *z*, *t*). The wedge shape ∂*h*/∂*x*<0 is necessary to generate a positive pressure. The time variation can exist if the gap thickness is changed with time, by forcing the surfaces to approach each other, causing squeezing action.

Inserting the velocity distributions (2.17) into equation (2.20), performing the integration using the Leibnitz rule, and simplifying the resulting equation yields(2.21)where(2.22)Equation (2.21) has the same form as the classical Reynolds equation for fluid-film lubrication. However, the major difference is that equation (2.22) is coupled with the pseudo-temperature energy (2.3) through . This coupling is discussed in the §2*f*. The first two terms in (2.21) are Poiseuille terms and describe the net flow rate due to pressure gradient. The third term describes the net flow rate due to local expansion. The other terms are a combination of Couette terms and squeeze terms. Couette terms describe the net entraining flow rates due to sliding velocities.

### (f) Pseudo-energy equation

Similar to the conservation of momentum, the pseudo-energy equation (2.3) reduces to(2.23)The terms in the first bracket represent the divergent of the flux of the pseudo-thermal energy, the terms in the second bracket corresponds to the rate of work by the collisional–translational component of the stress and the last term accounts for the rate of energy dissipation due to the inelastic collision between granules. The boundary conditions (2.13) for pseudo-temperature reduce to(2.24)where *T*_{H} and *T*_{L} are the pseudo-temperatures and *v*_{H} and *v*_{L} are the volume fractions on the upper and lower boundaries.

### (g) Dimensionless lubrication equations

The dimensionless form of the Reynolds equation for the powder lubrication becomes(2.25)where *Λ*_{o}=*L*/*B*_{o}. The dimensionless pressure is defined as . Additional functions in (2.25) are defined as follows(2.26)where . The dimensionless velocity distributions are(2.27)The dimensionless slip boundary conditions are(2.28)The dimensionless form of the pseudo-energy equation for the slider bearing and the related boundary conditions are(2.29)(2.30)The solution to equations (2.25) and (2.29) gives the pressure and pseudo-temperature field in a slider bearing. Once determined, one can proceed to determine the so-called bearing performance parameters.

### (h) Special case: slider bearings

In this section, we apply the theory to study the behaviour of a powder-lubricated thrust bearing, where two surfaces—one stationary and the other rotating at a constant speed—form a hydrodynamic wedge. A constant normal load is applied to the bearing surfaces. This load must be supported by the pressure generated within the gap. Our attention is on the steadily loaded bearing, henceforth Reynolds equation (2.31) is used in the simulations. For a steadily loaded slider bearing, , , , *m*_{xH}=1, *m*_{xL}=−1 and *κ*=1. The channel gap in which the lubricant flows is *h*=*h*(*x*). The references for the slider bearing are: *s*=*h*_{2} (minimum film thickness), *B*_{o}=*B* (bearing length) and *U*_{o}=*U*_{L} (bearing speed) and, therefore, *α*=*h*_{2}/*D* and *β*=*h*_{2}/*B*. Note that for the journal bearing, *s*=*C* (clearance), *B*_{o}=*R* (journal radius) and *U*_{o}=*U*_{H} (rotational speed). The dimensionless form of the Reynolds equation for the powder lubrication in slider bearings becomes(2.31)where *Λ*_{s}=*L*/*B* represents the bearing aspect ratio.

#### (i) Bearing performance parameters

The bearing load-carrying capacity can be obtained by integrating the pressure over the appropriate surface area(2.32)The dimensionless load carrying capacity is(2.33)In a similar way, the friction force is(2.34)The friction coefficient is defined as(2.35)The mass flow rates in the direction of motion *x* and the axial direction *z*, i.e. the side leakage, are given below.(2.36)

## 3. Numerical simulation method

The governing equation presented in §2*h* was solved using the finite difference method. Basically, there are three loops in the computations. The inner loop is for solving the pressure and velocities and the outer loop is for solving the volume fraction. The loop between them is for solving the pseudo-temperature. The solution procedure is as follows: first, with an initial guess of the volume fraction and pseudo-temperature, functions *F* and *G* are estimated from the equation (2.26). The pressure is then computed by equation (2.31). Generalized Reynolds equation (2.31) is an elliptic partial differential equation and is solved by successive over relaxation method. Then, the velocities *u* and *w* are computed by equation (2.27) with the slip boundary conditions (2.28). This process is iterated until the difference in computed pressure between two successive iterations falls below a specified tolerance. The tolerance 0.0001 is used in the present analysis. Next, the pseudo-temperature is computed by equation (2.29) with the appropriate boundary condition (2.30). The finite differencing of the energy equation yields a tri-diagonal matrix which is conveniently solved using the Thomas method. Following the convergence of pseudo-temperature, the volume fraction is computed from the relation, i.e. . The volume fraction cannot be computed analytically and, therefore, the bisection method is used. The mesh points are 51 in the moving direction and 21 in the lateral direction. 21 mesh points are used in the film gap. Numerical simulations indicated that a finer mesh does not affect the results but requires significantly more computational time.

## 4. Results and discussion

In this section, the results of a series of simulations are presented that predict the performance of a powder-lubricated slider bearing of a finite aspect ratio. We shall first begin by presenting the prediction of the pressure distribution and compare our simulations with Heshmat's measurements. Heshmat (1992) presented the first experimental evidence of the formation of hydrodynamic pressure profiles for a granular powder in a pivoted-pad thrust bearing. The powder lubricant was titanium dioxide (TiO_{2}) granules of 5 μm in diameter, which Heshmat injected at the inlet zone of the slider pad by means of an airbrush. The dimensions of the square shaped slider pad were 26.4×26.4 mm. The test thrust runner was operated at a speed of 4.6 m s^{−1}. Additional parameters used in the simulations are summarized in table 3. It is difficult to quantify the coefficient of restitution and the surface roughness. In the simulations presented, the coefficient of restitution and the surface roughness are chosen based on the work of McKeague & Khonsari (1996) and Zhou & Khonsari (2000).

### (a) Enduring contact

The effect of the enduring contact between two parallel plates is investigated. Johnson & Jackson (1987) introduced two parameters *F*_{r} and *n* in the Coulomb friction (2.9). They assumed that *F*_{r}=3.65×10^{−32} kg m^{−1} s^{−2} with *n*=40 for glass beads and *F*_{r}=4×10^{−32} kg m^{−1} s^{−2} with *n*=40 for polystyrene beads. These parameters are determined by experimental measurements. However, there are no experimental data available for *F*_{r} and *n* for TiO_{2} powders. In order to proceed, *F*_{r} is fixed at *F*_{r}=4×10^{−32} kg m^{−1} s^{−2} and the following analysis is used to determine the value of parameter *n* (Zhou 1998).

The normal stress contributed by the enduring contact *N*_{f} must be less than or equal to the normal stress *P*_{L} applied on the plate. Therefore, the parameter *n* can be determined by the following equation(4.1)The results are shown in table 4. It is shown that the effect of *P*_{L} on *n* is very small while the difference of volume fraction (*υ*_{max}−*υ*) plays an important role on *n*. At *P*_{L}=50 KPa, the value of (*υ*_{max}−*υ*) is roughly 0.01, and table 4 shows that the maximum value of the parameter *n* is 18. The effects of the enduring contact on velocity, pseudo-temperature and volume fraction are discussed in §4*f*. In these simulations, angle of friction *Φ*=25° and *δ*=22.9°. The enduring contact is a strong function of constant *n*. When *n*≤16, the effect of enduring contact is negligibly small. As *n* increases, the profile of the volume fraction in the *y*-direction is flattened, indicating that the enduring contact force between the granules begins to play a role in supporting the applied load. However, in the simulations presented, *n*≤18 and, therefore, the effect of the enduring contact is not pronounced. Research shows that if the operating speed is below 4 m s^{−1}, the effect of the enduring contact becomes large and cannot be neglected.

There are two obvious time scales in granular shear flows. One is the time between particle-to-particle collisions and the other is the contact time during a collision. If the contact time is much shorter than the time between collisions, constitutive laws derived from the kinetic theory are valid. The particle collision must depend on quantities like the solid volume fraction, particle size and velocity. The time between collisions derived by Gidaspow (1994) for monosized particles is(4.2)Using the volume fraction and pseudo-temperature on the boundary, where the fluctuation of particles is more active, the time between collisions is *t*_{c}=2.1×10^{−8} s. The contact time during a collision is defined as(4.3)where *K* is the stiffness constant, *M* is the mass of the particle and the damping coefficient is defined as . According to Hertzian model, the stiffness constant (Johnson 1985) is equal to , where *E* represents Young's modulus, *ν* is Poisson's ratio and *F* is the load acting on the particle. The load *F* can be determined from the conservation of momentum. For TiO_{2}, *K*=1.3×10^{4} N m^{−1} since *E*=270 GPa and *ν*=0.27. From equation (4.3), the contact time is *t*_{f}=7.2×10^{−9} s. Since *t*_{f}/*t*_{c}=0.34<1, the assumption based on the binary collision is still valid. For the slider bearing, it is expected that the ratio *t*_{f}/*t*_{c} remains small since the volume fraction is flattened to a lower value when the enduring contact is involved. Note that a small increment in the volume fraction reduces both *t*_{f} and *t*_{c}.

### (b) Pressure distribution

Figure 2 shows the results of the simulation of the pressure according to the present theory along with the results reported by Heshmat (1992) as measured at the bearing midplane. The pressure distribution of the powder lubrication is analogous to that of hydrodynamic lubrication. The film thickness ratio in this simulation is *a*=2.116. The maximum pressure is *P*_{max}=770 kPa and it occurs at , i.e. 19.8 mm away from the leading edge on the bearing midplane. These findings are in good agreement with the experimental measurements. The boundary pressure *N*=50 kPa is computed iteratively which corresponds to the supplied mass flow rate at the inlet. In Heshmat's experiment, an air brush was used to supply powder into the gap of the slider bearing. This gap was originally present and exposed to the ambient pressure. The centre of load, , must and does coincide with the position of the pivot. In the experiment, the pivot position was at and the present theory yielded a close prediction of . The shape of the pressure distribution is akin to that of a liquid-lubricated bearing. The predictions show that the maximum pressure remains relatively flat over a major portion of the bearing width and then rapidly drops near the bearing's edge along the width boundary. In contrast, fluid-film thrust bearings generally show a parabolic-type pressure distribution in that direction. The number of particles at the inlet is 18 and the number of particles at the outlet is eight in this simulation. Note that according to Johnson & Jackson (1987) if the number of particles becomes less than five or so, then the continuum assumption would not be valid. Figure 2 shows that the trend of the shear stress on the lower surface is similar to that of the pressure. The maximum shear stress occurs at the same location of the maximum pressure. The prediction for total friction force in this simulation is: *F*=87 N.

### (c) Flow velocity

Figure 3 shows the velocity distributions at 18 different locations (6 locations in the *x*-direction and 3 locations in the *z*-direction). Location represents the mid-plane of the slider bearing and is at the bearing's edge along the width boundary. The bottom surface undergoes a sliding motion at the imposed velocity *U*_{L}, and the top surface represents the stationary pad. The simulations show that the magnitude of granular slip at the boundaries vary in both *x*- and *z*-directions. At the bearing midplane, the magnitude of slip velocity on the upper surface increases along the sliding direction, but the opposite is true for the lower surface. The reason is that slip velocities directly depend upon the gradient of the flow velocities at the surfaces. The velocity gradients, in turn, depend upon the pressure gradient. At , the velocity profile is roughly linear since the pressure gradient is . When , the velocity gradient on the upper surface is small and the velocity gradient on the lower surface is large. The reverse is true when . The velocity profiles are in good qualitative agreement with McKeague & Khonsari (1996), who treated an infinitely wide slider bearing based on governing equations proposed by Haff (1983).

Figure 4 shows the prediction of the velocity distributions in *z*-direction, i.e. the leakage at . The leakage velocity increases along the sliding direction except at the inlet and outlet. At the inlet and the outlet the leakage velocity is nil since the pressure gradient . The leakage velocity is roughly symmetric in the *y*-direction about . The mass leakage flow is also shown in figure 4. The maximum mass leakage occurs at .

### (d) Granular pseudo-temperature

Figure 5 shows the predicted pseudo-temperature distributions at the same locations as the velocity profiles presented in figure 3. Examining the behaviour of the granular particles at the boundaries, we note that at the bearing midplane the granular pseudo-temperature increases along the length of the upper plate in the direction of motion. For the sliding plate, the opposite is true: the pseudo-temperature decreases along the length of the sliding plate. Note that momentum generated by granular slip serves as a source of pseudo-energy emerging from the boundaries. Therefore, pseudo-temperature is large on the upper surface at the outlet and on the lower surface at the inlet where the magnitude of slip velocity is large (see figure 3). This energy is transferred to the granules ‘flowing’ within the channel. However, the inelastic granular collisions cause an irreversible loss of energy proportional to the coefficient of restitution and therefore granular pseudo energy is at a minimum in the interior of the channel. In the pseudo-energy equation, the viscous dissipation is governed by the coefficient of restitution which characterizes the inelastic collision. Therefore, the shape of the pseudo-temperature profile is determined by the slip velocity and the severity of the inelastic collision. The maximum pseudo-temperature occurs at the trailing edge of the upper surface where the velocity gradient is at its maximum leading to the large viscous dissipation.

The behaviour of the granules can also be explained in terms of their fluctuations. The momentum supplied due to the slip at the boundary yields a greater fluctuation near the source. In the middle of the gap, the granules are in much closer proximity compared to those granules at the boundaries. At the trailing edge in the slider bearing, as the grains are forced into the gap, the granules near the sliding body tend to gain a higher flow velocity. This tends to increase the granular slip at the top plate since it is stationary, bringing about an increase in the pseudo-temperature at that location.

At the bearing's edge , the pseudo-temperature on the upper plate has roughly the same value of that on the lower plate since the pressure gradient is zero and, therefore, the velocity distribution becomes approximately linear.

### (e) Solid volume fraction

The behaviour of the volume fraction and the pseudo-temperature are intimately related. Figure 6 shows the how volume friction changes along the slider's length and width. Away from the top and bottom boundaries and towards the middle of the gap, the volume fraction increases implying a greater concentration of granular solids there, with a concomitant reduction in fluctuation. In locations where the granules are more densely packed, i.e. in the middle of the gap, there will be less fluctuation, the amount of energy dissipation is lower, and the pseudo-temperature is reduced. Near the boundaries the granules are loosely packed and, as a result of greater fluctuation, there will be a rise in the pseudo-temperature.

The volume fraction profiles along the width of the slider in figure 6 reveal that the volume fraction at the mid-plane is generally higher than elsewhere within the bearing clearance, due to the higher pressure. Thus, the region where the pressure reaches its peak value also corresponds to the maximum volume fraction.

At the bearing's edge , the volume fraction on the upper plate has roughly the same value of that on the lower plate since the pressure is an ambient pressure and roughly symmetric pseudo-temperature across the gap.

### (f) Couette-type flow

Figure 7 shows the variations of velocity, pseudo-temperature and volume fraction for the Couette-type flow where the top and bottom plates are parallel. In this Couette-type flow simulation, the load per unit area is *N*=50 kPa, which corresponds to the boundary pressure *P*_{s}=50 kPa of the slider bearing. The mass flow rate is used in this simulation to match with the mass flow rate of the slider bearing in the previous sections. To satisfy the load and the mass flow rate, the inlet film thickness *h*=59.9 μm is computed iteratively. It is shown that the velocity profile is roughly linear across the gap. In this simulation, the effect of the enduring contact on the velocity profile is small. The trends of the pseudo-temperature and the volume fraction for Couette-type flow are very similar to those for the slider bearing at the bearing's edge . As described previously, under the conditions simulated, the consideration of enduring contact tends to predict the flatter volume fraction profile, indicating the importance of the enduring contact.

### (g) Coefficient of restitution

Figure 8 shows the variation of the velocity, pseudo-temperature and the volume fraction at the mid-plane for different coefficient of restitutions, *e*_{p}. The film thickness ratio used in this simulation is *a*=1.6. Other input data are the same as the list in table 3. As the coefficient of restitution increases, the collision between the powders becomes more elastic. Therefore, the fluctuation of the powder in the middle of the gap becomes greater. Figure 8 also shows that as the coefficient of restitution increases, the slip velocity on the boundaries decreases and the velocity gradient increases in the middle of the gap. As a result, the viscous dissipation is greater. On the boundaries, increasing coefficient of restitution results a reduction in the slip velocity and consequently lowering the pseudo-temperature. However, near the trailing edge of the upper surface, the effect of the elastic collision is more dominant due to the very small change of the slip velocity and the velocity gradient. Therefore, pseudo-temperature for the larger coefficient of restitution is higher there. As mentioned above, the volume fraction strongly depends upon the pseudo-temperature. The volume fraction is high where the pseudo-temperature is low.

### (h) Surface roughness

Figure 9 shows the variation of the velocity, pseudo-temperature and the volume fraction at the mid-plane for different roughness of the both surfaces *ϕ*. The extreme limits are *ϕ*=0 representing a very smooth and *ϕ*=1 representing a very rough surface. As the surface roughness increases, the slip on the surfaces decreases and the energy transferred from the surfaces decreases. Less energy results in the lower pseudo-temperature and the higher volume fraction on the boundaries. In the middle of the gap, the pseudo-temperature increases with increasing surface roughness since the velocity gradient increases leading to the viscous dissipation. A rough surface tends to bring about an increase in the volume fraction on and near the boundary.

## 5. Concluding remarks

Granular powder lubrication offers an important alternative to conventional liquid lubricant with a unique capability of operating under higher temperature. This paper develops a three-dimensional general theory for analysing the behaviour of granular material within the context of the lubrication theory. Starting from the conservation laws, a generalized Reynolds equation is derived that enables one to study the characteristics of powder lubricated bearings with finite aspect ratio. The theory properly takes the side flow into account. A series of results are presented to study the behaviour of powder lubricated thrust bearing. Comparison with experimental pressure results reveals good agreement in general trend and magnitude. It is shown that the pressure profile is very similar to the convectional fluid-film hydrodynamic lubrication. The relationships between the pressure, pseudo-temperature and solid volume fraction are presented for three-dimensional flows.

## Acknowledgments

The authors wish to gratefully acknowledge the financial support of the National Science Foundation under the award number CMS-0096141. Dr Jorn Larsen-Basse was the Program Director. We also thank an anonymous referee for constructive reviews of the manuscript.

## Footnotes

As this paper exceeds the maximum length normally permitted, the authors have agreed to contribute to production costs.

- Received November 11, 2004.
- Accepted April 29, 2005.

- © 2005 The Royal Society