**1. Introduction**

Many experimental investigations on the frictional behaviour of textured surfaces in sliding contacts have been done, and recently the theoretical research in this field gets more attention. In machine components like gears and bearings, the influence of a mixed lubrication regime could be significant where asperities and the fluid contribute to carrying the load [1]. Moreover, in these applications, the existence of starvation can influence the effect of lubricant film in contact, and increase the friction in contact [2]. Investigations of high-speed bearings have shown in many cases that they operate under the starved lubrication regime [3–5]. In many cases, the lubricant cannot ensure a full separation of surfaces, which can cause higher friction and wear. Therefore, in starved lubricated conditions it is important to identify the influential surface parameters in order to understand the frictional behaviour of these lubricated contacts.

The effect of starvation on lubrication performance is analyzed experimentally in studies by Wedeven et al. [6], Pemberton et al. [7] and Kingsbury [8], and theoretically by Chiu [9], Damiens et al. [10] and Chevalier et al. [11]. It is worth to mention that in these theoretical studies, the concept of "fractional film", introduced by Jakobsson and Floberg [12] and Olsson [13], has an important role.

In the theoretical study of Brewe and Hamrock [14] on the effect of starvation in hydrodynamicallylubricated contacts, they chose the start of the pressure build-up at the inlet meniscus boundary, and

by employing a systematic reduction of the fluid inlet level, they observed an increase in the contact pressure for a specified film thickness. By considering a wide range of geometry parameters, they solved the Reynolds equation to simulate the film thickness in the contact area; moreover, in their study the film thickness formula in the hydrodynamic lubrication regime is modified to incorporate the starvation effect into it. In the work of Boness [15] on the cage and roller slip, it was shown experimentally that the oil supply can have a significant effect upon the cage and roller motion, and that limiting the oil supply decreases the amount of slip. Chevalier et al. [16] employed an iso-viscous hydrodynamic model to analyse a non-deformable body; in this study the flow continuity equation is based upon the Elrod [17,18] theory. They conclude that the inlet film shape could affect the film thickness. In Cann and Lubrecht [19], a study of the relationship between the film thickness, velocity, load and viscosity, was the focus of investigation.

Investigations on starved lubrication show that the employment of surface modification methods could be a practical and efficient method for the reduction of friction. One of the first rational methods of decreasing the friction can be by a reduction of roughness, making the surfaces smooth; however, producing an extra smooth surface is expensive [20]. In this case, surface texturing proved to be a reliable method to influence the frictional behaviour in the contacts. A well-designed use of this technique can modify the hydrodynamic component of mixed lubrication, which results in the enhancement of several tribological parameters, such as load carrying capacity and friction coefficient [21–29]. In general, when the temperature increases, the shear stress of the boundary layer decreases except when the temperature is passing from the melting point of boundary layer, moreover an increase in temperature of the parts due to interaction between asperities can change the situation from effective lubrication to high wear [30]. In work of Kango et al. [31], based on the Reynolds equation and the JFO (Jakobsson and Floberg and Olsson) boundary conditions for non-Newtonian fluids, temperature effects on textured surfaces are theoretically studied. They show that in presence of surface texturing, the average temperature of the lubricating film reduces. In work of Guzek et al. [32], upon an optimization of the surface texturing parameters in parallel bearings, they numerically solved the Reynolds equation, considering mass-conserving cavitation and viscosity changes due to temperature change. They showed that the decrease in viscosity due to the temperature rise can reduce the load carrying capacity. Therefore, cavity height ratios should be higher in order to have a similar load carrying capacity to textured surfaces with a constant temperature assumption. In work of Gu et al. [33] on the performance of surface texturing under starved and mixed lubrication, they employed a thermal mixed lubrication model considering the oil supply. They found that the start-up conditions can affect the friction coefficient. Moreover, they showed that it is easier for the textured surface to form the hydrodynamic lubrication than it is for the smooth surface, which is helpful to separate the mixed lubricated contact surfaces, and thus less friction heat is generated at the start-up phase. In the work of Bijani et al. [34] the influence of surface texturing on mixed lubricated contacts, different texturing patterns and cavity shapes are studied, and a numerical model to predict the friction is proposed.

Although the starved lubrication influence on film thickness in different applications is studied extensively in more recent times, not much work has been done on mixed lubrication under starved lubrication conditions, and in the case of friction in starved lubricated textured surfaces, even fewer studies have been done. When the lubricant in contact is limited to a specific amount, a correction of the film thickness formula is necessary, so in this study, a corrected film thickness is presented for textured surfaces under starved lubrication conditions.

In order to develop the starved lubrication model, the modified film thickness relation for starved contacts is solved, by taking the limited input film thickness into account, then the corrected film thickness is combined with the deterministic contact model. In this article the consequences of the existence of starvation in lubricated contacts on friction is discussed.

During the past decades, several efforts were devoted to study this mixed lubrication frictional behaviour [35–38]. Based on the contact model, mixed lubrication models can be divided in two types: Statistical and deterministic contact models. In the statistic models, the parameters represent the random characteristics of surface roughness. A major shortcoming of this model is its inability to provide detailed information on local roughness, which has an influence on the mechanisms of lubrication and friction. Another approach to simulate the frictional behaviour of contacting asperities results in a deterministic model, which employs the deterministic information of surface roughness. In these models, for a given separation, by summing up the local components of load and contact area, it is possible to deterministically calculate the real contact area and the total force carried by the contact.

In 1972 Johnson, Greenwood and Poon [39] developed a model in which the load carried by a contact in the mixed lubrication regime is shared between the asperity contact and the fluid film. In their model, they combined the well-known Greenwood and Williamson [40] theory of random rough surfaces in contact with the Elasto-hydrodynamic lubrication theory. This model was extended in 1999 by Gelinck and Schipper [41] to calculate the Stribeck curve for line contacts. Shi and Salant [42] introduced a mixed lubrication model, considering the inter-asperity cavitation and surface shear deformation for soft materials, and showed the occurrence of local cavitation. For moderately-loaded lubricated systems, the Jakobsson-Floberg-Olsson [12,13] cavitation theory is used. In 1970, Greenwood and Tripp introduced a deterministic contact model between two identical rough surfaces.

The flow factor method was introduced by Patir and Cheng [43,44]. They solved the Reynolds equation on a small area of the rough lubricating gap. The calculated micro flow is related to the flow of a perfectly smooth lubricating gap with similar mean height, resulting in flow factors. Fluid flow assumed to have two sources, shear driven flow and pressure. The flow factors are calculated independently by solving the local deterministic flow problem for a specified roughness topography. The main drawback of this method is due to nature of the roughness asperities that are not identical to the coordinate axes; this method is not effective in modelling the cross-flow of anisotropic roughness. Hu et al. [45] present a numerical solution for the contact of elastic bodies with three-dimensional roughness. The elastic contact has been modelled as a linear complementarity problem, and was solved by the Conjugate Gradient Method. Yu et al. [46,47] developed a full numerical solution to mixed lubrication in point contacts. They viewed the asperity contact as a result of a continuous decrease in the film thickness. By employing this assumption, the transition from contact to non-contact is continuous, and as a result, the same mathematical model should work for both regions. To calculate the asperity contact problem a multi-level integration method is used. In the work of Faraon et al. [48], by developing a numerical model for a real distribution of the asperities, the Stribeck curves were calculated; based on this model they compared the Stribeck curves between the deterministic and statistic model. They showed that the Stribeck curve results obtained with the statistic and the deterministic contact models are significantly different when the distribution of the surface heights deviates from the Gaussian height distribution; then by performing experimental measurements they showed that the deterministic mixed lubrication model is in good agreement with the measurements.

Recent developments in texturing techniques made it possible to employ different geometrical micro- and meso-scale patterns on the surface. These surface modification techniques include machining, photoetching, etching techniques, ion beam texturing and laser texturing [49]. Laser surface texturing proves to be more efficient, accurate, convenient and controllable for many materials [50], and is used to study the effect of micro-scale cavities on the frictional behaviour of contacts [25,27,29,37,51–56]. Kovalchenko et al. [27] show the influence of texturing on the transitions between the different lubricating regimes. They show that LST is able to enhance the hydrodynamic lubrication regime and thus increase the load carrying capacity of the contact; moreover, they found that the lapping after laser texturing that is carried out to remove the bulges at the edges of dimples is essential for increasing the positive effect of LST. In another study, Ryk et al. [29] theoretically and experimentally investigated on the beneficial effects of applying LST on piston rings.

They observed that the benefits of LST in both full and starved lubrication conditions results in fuel consumption reduction in combustion.

In the work of Bijani et al. [34] on the influence of surface texturing on mixed lubricated sliding contacts, the deterministic asperity contact model is applied. By employing this contact model and solving the Reynolds equation, Stribeck-like curves for several cavity patterns with different geometry were plotted, and the behaviour of the coefficient of friction based on these parameters was investigated.
