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Slip status in lubricated point-contact based on layered oil slip lubrication model
Tribology International 144 (2020) 106104
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Slip status in lubricated point-contact based on layered oil slip lubrication model Yaoguang Zhang, Wenzhong Wang *, He Liang, Ziqiang Zhao School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Slip extent Limiting shear stress Rheological properties Layered slip model
Obvious decrease of film thickness is found in recent experimental tests of high-speed elastohydrodynamic lubrication (EHL) point contact, which shows a close correlation with interface slip. This work investigates the effects of operation conditions and rheological properties of lubricant on slip behaviours based on a layered oil slip model. The results show that slip extent generally increases with the aggravation of sliding, the increase of applied load and the decrease of limiting shear stress; furthermore, the slip extent will also be affected by the contacting surface properties. Essentially, as internal factors, the rheological properties of lubricant may determine the slip behaviour under various operation conditions.
1. Introduction Oil film under elastohydrodynamic lubrication (EHL) usually suffers from high contact pressure and high shear stress, especially in high speeds conditions while sliding occurs. Many widely used machine el ements such as rolling bearing and gears usually work at such conditions that the lubricant will not behave as Newtonian fluid. Along with the increase of shear rate, the shear stress in the lubricant will increase at first, and usually approach limiting shear stress eventually. Therefore, the limiting shear stress of lubricant would inevitably have an effect on EHL film behaviour when the components are operated under high speeds condition. Limiting shear stress was found to be a property of lubricant in traction measuring experiments. The earlier studies on lubricant rheology at high pressure were conducted in two-disc or four-disc ma chines for EHL contact [1–3]. Results under sliding condition show the phenomenon that the traction coefficient no longer increases with the increase of shear rate after having reached a limit value, which is similar to plastic yield of solid material. Thus, limiting shear stress was intro duced to characterize this property of lubricant. Like viscosity, the limiting shear stress decreases with the increase of temperature, while increases with the applied pressure. Subsequently, Houpert et al. [4] proposed Eq. (1) to approximate limiting shear stress considering pres sure and temperature effects through fitting the traction curves obtained from two-disc machine. Other models that considering both the effects of pressure and temperature were also established based on impacting
ball apparatus [5], high pressure viscometer [6], or falling viscometer [7]. � � �� 1 1 τL ¼ ðτ0 þ γPÞexp β (1) Ti T0 When limiting shear stress is reached, usually at high shear rate condition, lubricant flow behaviour may change. Velocity variations across the lubricant film and near the solid surface are deemed to be related to limiting shear stress, such as shear banding, wall slip and plug slip. Shear banding was observed at the shear rate around 10 1 by Bair et al. [8] in the oil with large viscosity confined between parallel plates with separation distance of 150 μm. In their observed results, shear banding was found in an inclination angle to solid surface, but does not extend to the surface. By contrast, wall slip occurs at the interface be tween lubricant and surface when the velocity of the lubricant near the surface is unequal to the velocity of the surface or there is very large velocity gradient in lubricant near the surface, as captured in experi ments [9–11] and found in molecular dynamic simulations [12–16]. It can be found that the significant change of speed occurs in the fluid layer next to the surface as the density and viscosity of liquids adjacent to solid surfaces are distinct from the bulk ones as manifested in references such as Ref. [12], and low viscosity can be found in this boundary layer depending on the surface properties and interaction between surface and fluid. Slip occurs at the interface between the solid phase and fluid phase, and takes place in the lubricant layer below or on the order of molecular size as shown in the flow profiles in Refs. [14,17,18]. Slip is closely related to the limiting shear stress of lubricant, and the operation
* Corresponding author. E-mail address: [email protected] (W. Wang). https://doi.org/10.1016/j.triboint.2019.106104 Received 4 September 2019; Received in revised form 10 November 2019; Accepted 4 December 2019 Available online 6 December 2019 0301-679X/© 2019 Elsevier Ltd. All rights reserved.
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Nomenclature A cf h kf P SRR T T0 u ue v W x y z
α
β γ δ
slip area ratio specific heat of fluid film thickness heat conductivity of fluid pressure slide-to-roll ratio temperature ambient temperature speed in x direction average speed (uF þ uS)/2 speed in y direction load coordinate along the motion direction coordinate perpendicular to the motion direction coordinate through the film thickness pressure-viscosity coefficient
ε η η0 θ λ
ρ τ0 τL ϕ F LF LS max S
conditions lubricant sustains. Echeverri [19] conducted the nonequi librium molecular dynamics (NEMD) study of n-alkanes, and the slip was proved to be related to the length of n-alkanes. The results of the NEMD work by Gattinoni et al. [20] show that the phase behaviour of lubricant in confined gap is depending on various operation conditions, and slip occurs under high pressure and shearing conditions. Consequently, slip may cause the formation of dimple [21,22], or large sliding under tractive rolling mode which was detected in ball-on-disc [23] and ball-on-ring [24] experimental investigations. Affected by the properties of contact surfaces, slip could occur at one or both surfaces under pure sliding condition as discussed in the experi mental work by Li et al. [25]. Theoretical models were established for lubrication analysis with slip. Ehret et al. [26] proposed a slip coefficient to express the degree of slip, which is proportional to the pressure, their model well reproduced the dimples observed in experiments; however, the detailed slip status in the contact area, such as slip area and the degree of slip in different velocity conditions is not solely related to pressure. Therefore, slip should relate to the shearing condition and rheological properties of lubricant. Zhang et al. [27] developed a slip model with the assumption
limiting shear stress-temperature coefficient limiting shear stress-pressure coefficient thermal expansion coefficient the thickness coefficient of slip layer viscosity of lubricant viscosity of lubricant at ambient pressure and temperature viscosity coefficient of slip layer coefficient in modified Reynolds equation density ambient limiting shear stress limiting shear stress temperature-viscosity coefficient fast surface slip layer of lubricant near the fast surface slip layer of lubricant near the slow surface maximum value slow surface
that wall slip occurs when shear stress at the surface exceeds limiting shear stress. Load effects on lubricant flow, friction and so on are ana lysed in their isothermal line contact model. Their strategy they used to determine slippage is adopted in some researches [28–30], where the isothermal expression of limiting shear stress was used to investigate the wall slip in line contact. In the recent work by Zhang et al. [30], a layered oil slip model considering the slip and thermal effect is proposed to explore the mechanism of film thickness behaviours at high speed. The film thickness distribution obtained show good agreement with experimental results; moreover, local slip status and velocity profile through film thickness in contact area are obtained. However, in-depth theoretical investigations of slip effect are still required at present. In this work, an comprehensive analysis of slip status in ball-on-ring contact is conducted based on the layered oil slip model in Ref. [30]. The factors that may affect slip behaviour including sliding condition, load conditions and rheological properties of lubricant are parametrically discussed in detail by characterizing the extent of slip through slip amplitude and slip area.
Fig. 1. Schematic diagram of layered slip lubrication model, adapted from Ref. [30]. 2
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adjacent layers including solid surfaces.
Table 1 Lubricant properties and operation conditions. Parameters
2.1. Governing equations
Values
Viscosity at 25 C, η0 Viscosity-pressure coefficient, α Viscosity-temperature coefficient, ϕ Ambient limiting shear stress (LSS), τ0 LSS-pressure coefficient, γ LSS-temperature coefficient, β Load, w Hertzian contact pressure, PH Entrainment speed, u Slide-to-roll ratio, SRR �
0.048 Pa⋅s 15.0E-9 Pa 1 0.0342 K-1 2.0 to 8.0 MPa 0.01 to 0.15 1500 K 30 to 500 N 492.8 to 1258 MPa 0 to 20 m/s 0 to 2.0
As shown in Fig. 1, the shear stress equilibrium equations in each layer across the film can be obtained based on Navier-Stokes equation as follows, 8 ∂p η ∂2 u > > > ¼ z 2 ½ð1 εLF Þh; h � > > ∂x θ ∂x2 > > > < 2 ∂p ∂u (2) z 2 ½εLS h; ð1 εLF Þh � ¼η 2 > ∂ x ∂x > > > > 2 > > > ∂p η ∂ u : z 2 ½0; εLS h� ¼ ∂x θ ∂x2
2. Brief review of layered oil slip model
Then, based on continuity conditions of the shear stress and velocity at each interface of lubricant layers, the velocity distributions are ob tained:
A lubrication model with the consideration of slip effect was pro
� 8 ∂p 2 2 2 > > ðθ 1ÞðεLF þ εLS 1ÞuF þ uS θ θ ∂x h ðθ 1Þ εLF εLS þ εLS εLF > > ¼ u LF > > 2η½ðθ 1ÞðεLF þ εLS Þ þ 1 � ðθ 1ÞðεLF þ εLS Þ þ 1 > > > > > > > � � � ∂p ∂p > > θ h ðθ 1Þ ε2LS ε2LF þ 2εLF þ 1 θ z2 > > θðuS uF Þ > ∂ x ∂x > z z þ > > 2η½ðθ 1ÞðεLF þ εLS Þ þ 1 � h½ðθ 1ÞðεLF þ εLS Þ þ 1 � 2η > > > > > > > � � � ∂p 2 < 2 ðθ 1ÞðεLF uS þ εLS uF Þ þ uS εLS ∂xh ðθ 1Þ ðθ 1Þ εLF εLS εLF 2εLF þ εLS 1 uLM ¼ > > 2η½ðθ 1ÞðεLF þ εLS Þ þ 1 � ðθ 1ÞðεLF þ εLS Þ þ 1 > > > > > > > � � ∂p � ∂p 2 > > > h ðθ 1Þ ε2LS ε2LF þ 2εLF þ 1 z > ðuS uF Þ > ∂ x > z z þ ∂x > > > h½ðθ 1Þð 2 ε þ ε Þ þ 1 � η ½ðθ 1Þð ε þ ε Þ þ 1 � 2 η LF LS LF LS > > > > > > � � ∂p � ∂p > > > θ h ðθ 1Þ ε2LS ε2LF þ 2εLF þ 1 θ z2 > θðuS uF Þ : ∂ x uLS ¼ uS z z þ ∂x 2η½ðθ 1ÞðεLF þ εLS Þ þ 1 � h½ðθ 1ÞðεLF þ εLS Þ þ 1 � 2η
posed by authors in recent work [30]. As shown in Fig. 1, slip may occur at either or both contacting surfaces. Slip status of each position in contact area is determined by the local rheological and operating con ditions. Limiting shear stress is considered as the criteria to determine whether slip occurs or not. Once the shear stress of lubricant at one surface exceeds limiting shear stress, there will be a presumed thin lubricant slip layer with much lower viscosity between the middle lubricant and surface. The viscosity of slip layer is 1/θ of middle layer, and the ratio between the thicknesses of slip layer and whole film thickness is ε. Velocity is continuous at the interfaces between any
(3)
Therefore, the modified Reynolds equation with consideration of slip can be obtained by substituting Eq. (3) into the mass flow conservation equation, � � � � ∂ ρh3 ∂p ∂ ρh3 ∂p ∂½ρhðλS uS þ λF uF Þ � λP þ λP ¼ ∂x ∂y 12η ∂x 12η ∂y ∂x þ
3
∂½ρhðλS vS þ λF vF Þ � ∂y
(4)
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b
where
i
h
�2 ðεLS þ εLF Þ5 ε2LS þ εLS εLF λP ¼ þ ðεLS þ εLF Þ2 ½ðθ 1ÞðεLS þ εLF Þ þ 1 � 3 ε2LF
ðεLS þ εLF
h 1Þ ðεLS þ εLF Þ4
12εLS εLF ðεLS þ εLF ÞðεLS þ εLF
2ðεLS þ εLF Þ3 þ 4ε2LS þ 4ε2LF
1Þ θ
ðεLS þ εLF Þ2 i 4εLS εLF
ðεLS þ εLF Þ2 λS ¼ λF ¼
(5)
ε2LS
ε2LF þ 2εLF ε2LF ε2LS þ εLS εLF þ 2ðεLS þ εLF Þ 2ðεLS þ εLF Þ½ðθ 1ÞðεLS þ εLF Þ þ 1 �
ε2LF
ε2LS þ 2εLS 2ðεLS þ εLF Þ
ε2LF
ε2LS þ εLS εLF 1ÞðεLS þ εLF Þ þ 1 �
2ðεLS þ εLF Þ½ðθ
etween oil and solid. The smaller one among the limiting shear stress of lubricant and the critical shear stress at oil-solid interface should be regarded as the limiting shear stress actually adopted in the determi nation of slip. Thus, the coefficients in Eq. (1) of lubricant at one contact surface may be different from that at another surface in some occasions.
λP, λS, and λF are only dependent on ε and θ and various with different slip cases. The modified Reynolds equation for the case that slip does not occur can be obtained by setting εLS as 0 and εLF as 0 successively. The modified Reynolds equation for the cases that slip occurs at slow or fast surfaces can be obtained by setting εLS as 0 or εLF as 0 respectively. The detailed derivations of the equation and parameters involved can be referred to Ref. [30]. Energy equation of oil film for considering the thermal effect is expressed as � � �� �2 � �2 � � � ∂T ∂T ∂2 T ∂p ∂p ∂u ∂v ρc f u þ v þ (6) ¼ kf 2 þ δT u þ v þη ∂x ∂y ∂z ∂x ∂y ∂z ∂z
3. Results and discussions In the following numerical analysis, the point contact is considered formed between a glass ring with inner radius of 51.4 mm and a steel ball of 25.4 mm in diameter. The elastic moduli of steel and glass are 206 GPa and 88 GPa respectively. The Poisson’s ratios of steel and glass are 0.3 and 0.215. The initial temperature is 25 � C. The viscosity coef ficient of slip layer θ is set as 10000. In order to study the slip effect in EHL point contact, numerical calculations based on layered slip model are conducted under various operating conditions with different rheological properties of lubricant. Related parameters are listed in Table 1. For the variable parameters that are concerned factors in following discussion, their values are given in a range. τ0 is equal to 2.0 MPa and γ is equal to 0.03 in Sessions 3.1 and 3.2. The change of lubricant viscosity with pressure and tempera ture follows Eq. (7). � � � α � η ¼ η0 exp ðlnη0 þ 9:67Þ 1 þ 5:1 � 10 9 P 5:1�10 9 ðlnη0 þ9:67Þ 1 ϕðT
The thermal effect of slip layers is neglected in the thermal calcula tion due to the negligible thickness of slip layers. u and v in Eq. (6) are the velocity distributions in the middle layer of oil. 2.2. Determination of slip Slip status at each location should be determined before solving the governing equations in each iteration. In layered slip lubrication model, on which surface slip occurs depends on whether shear stress at the surface exceeds limiting shear stress. The thickness of slip layer repre sents the amplitude of slip, which depends on the theoretical shear stress condition at each position. When the thickness of slip layer gets thicker, the slip layer with much lower viscosity will sustain more velocity variation so that the shearing of lubricant in the middle layer is weakened. The thickness of slip layer is determined by the extent to which the shear stress on the surface is greater than limiting shear stress. The detailed processes to determine the slip status can be inferred in Ref. [30].
� T0 Þ
(7)
3.1. Slip status under different sliding conditions A matter worth of attention to delve into is the occurring condition of slip. Some results of molecular dynamic simulations [13,31] have indicated that slip occurs at the conditions within an approximate range of pressure and shear rate. However, the contact models in molecular dynamic simulation usually consist of two parallel solid walls with confined lubricant in between, which have large discrepancy with practical point contact EHL. Thus, in point contact EHL, the occurrence and escalation of slip at each position should be different. In EHL contact, the shear rate is usually at very high level when sliding occurs. Therefore, sliding condition directly influences the shear stress at the interface of lubricant and contact surface and the status of slip. Fig. 2 shows the slip extents at the centreline along the motion direction at different slide-to-roll ratios with the entrainment speeds of 4 m/s, 10 m/s and 16 m/s. In these figures, the profile of εLF looks like trapezoid shape when entrainment speed is 4 m/s and slide-to-roll ratio is 1.0, while all the other profiles approach triangle shape. The maximum thickness coefficient of slip layer on fast surface, εLF-max,
2.3. Limiting shear stress in the layered slip lubrication model In the procedure of determining the slip status, limiting shear stress is the reference to determine whether the slip occurs or not. Therefore, the variation of limiting shear stress under different operating conditions has to be considered. In this model, limiting shear stress at each position is related to local pressure and temperature conditions and follows Eq. (1) proposed by Houpert et al. [4], which includes ambient limiting shear stress τ0, limiting shear stress-pressure coefficient γ and limiting shear stress-temperature coefficient β. It was reported that the limiting shear stress may also be related to the interaction between lubricant and solid surface. Li et al. [25] inferred from EHL experiments under zero entrainment velocity conditions that wall slip is related to not only the limiting shear stress of lubricant but also the critical shear stress 4
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Fig. 2. The profile of εLF at the centreline along motion direction at different SRR under the entrainment speeds of: (a) 4 m/s, (b) 10 m/s and (c) 16 m/s (w ¼ 32 N, τ0 ¼ 2.0 MPa, γ ¼ 0.03).
which represents the amplitude of slip, and the area percent ALF, which is defined as the ratio between area where slip occurs and the whole Hertzian contact area, are used to represent the extent of slip on fast surface. εLS-max and ALS are used to represent the extent of slip on slow surface. As shown in Fig. 2, at the entrainment speeds of 4 m/s, 10 m/s and 16 m/s, εLF-max increases with the increase of the slide-to-roll ratio from 0 to 1.8. εLF-max under the slide-to-roll ratios of 1.4 and 1.8 increases with entrainment speed, while under the slide-to-roll ratios of 0.6 and 1.0, εLF-max at 10 m/s is the largest among three speed conditions in Fig. 2. The colour contour plot shown in Fig. 3 evidently indicates the change of slip extent, where the calculated points are plotted as black dots. Firstly, εLF-max increases with increasing slide-to-roll ratio through almost all the speed range. On the other hand, as entrainment speed increases, εLF-max shows different trends at different slide-to-roll ratios. When the slide-to-roll ratio is larger than nearly 1.4, εLF-max increases with the increasing entrainment speed. When the slide-to-roll ratio is larger than 0.4 and smaller than 1.4, εLF-max increases first and then decreases with increasing entrainment speed. When the slide-to-roll ratio is smaller than 0.4, εLF-max is almost zero. From the trend of εLFmax, it can be inferred that slip generally intensifies with the increases of entrainment speed and slide-to-roll ratio, but not monotonically in creases with entrainment speed when slide to roll ratio is less than 1.4 or so under present loading condition. As shown in Fig. 4, ALF generally increases with the increase of slideto-roll ratio. At the entrainment speed of 2 m/s and the slide-to-roll ratio of 1.8, the slip area becomes the largest among all the computation conditions. When the entrainment speed is faster than 16 m/s and the slide-to-roll ratio is larger than 1.6, although εLF-max is the largest as shown in Fig. 3, ALF is relatively small instead compared with the just mentioned slip area ratios at low speed and high slide-to-roll ratio conditions. Slip occurs with different extents under different speeds and sliding conditions. At relatively low speeds, as slide-to-roll ratio increases, slip occurs at smaller slide-to-roll ratio and extends to larger zone but with small εLF-max, which stands for the amplitude of slip. At relatively high speed, as slide-to-roll ratio increases, slip occurs at larger slide-to-roll ratio and intensifies in amplitude but with relatively smaller slip area compared with low speeds cases.
Fig. 3. Colour filled contour plot of εLF-max at different entrainment speeds and slide-to-roll ratios (w ¼ 32 N, τ0 ¼ 2.0 MPa, γ ¼ 0.03). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 4. Colour filled contour plot of ALF at different entrainment speeds and slide-to-roll ratios (w ¼ 32 N, τ0 ¼ 2.0 MPa, γ ¼ 0.03). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
3.2. Slip status under different load conditions As Limiting shear stress and viscosity of lubricant are closely related to applied pressure as expressed by Eq. (1) and Eq. (7), the applied load may significantly affect the slip status. Calculations are conducted under different loading conditions at the slide-to-roll ratios of 0.6, 1.0 and 1.8, 5
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Fig. 5. Variation of εLF-max with entrainment speed under different load conditions, (a) SRR ¼ 0.6, (b) SRR ¼ 1.0 and (c) SRR ¼ 1.8.
Fig. 6. Variation of ALF with entrainment speed under different load conditions, (a) SRR ¼ 0.6, (b) SRR ¼ 1.0 and (c) SRR ¼ 1.8.
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ALF with entrainment speeds at different loads also reduce gradually. Under the loads of 50 N and 100 N in Fig. 6a and 30 N, 50 N and 100 N in Fig. 6b and 30 N and 50 N in Fig. 6c, the variations of ALF in the speed range from 0 to 20 m/s reach 20%–30%. The variations of ALF with entrainment speeds under higher loads are nearly all less than 5%. Therefore, slip area is much larger but changes slowly under higher loads, which should be related to the pressure distributions in the con tact area. As shown in Fig. 7, slip basically occurs around the contact centre where the contact pressures are higher. When ALF is already relatively large, the pressure gradients around the boundary of slip area are steep. Thus, slip area hardly increases in a high rate as the contact pressure rapidly decreases along the outward direction. Therefore, slip status is closely related to load condition, which dominates the pressure condition in the contact. The slip amplitude solidly increases with rising load. The slip area increases and its changing rate slows down with rising load.
Fig. 7. Distributions of pressure and εLF at the centreline along motion direc tion under the loads of 100 N, 200 N and 300 N (ue ¼ 10 m/s,SRR ¼ 1.8).
3.3. Effect of τ0 and γ on slip
and slip statuses at the speed range from 1 m/s to 20 m/s are considered under each load. Fig. 5 shows the maximum thickness coefficient of slip layer on fast surface under different loading conditions from 30 N to 500 N, corre sponding to the Hertzian contact pressures from 492.8 MPa to 1258 MPa respectively, in the speed range from 0 m/s to 20 m/s and with the slideto-roll ratios of 0.6, 1.0 and 1.8. It can be found that εLF-max increases with the increase of applied load at each entrainment speed, and the increase amplitude of εLF-max between two consecutive loads increases with increasing entrainment speed. Although under the load of 32 N as shown in Fig. 3, εLF-max starts to decrease after the entrainment speed exceeds 10 m/s at slide-to-roll ratio of 1.0, εLF-max under higher loads in Fig. 5 solidly increases with rising entrainment speed. As shown in Fig. 5, when the slide-to-roll ratios are 0.6 and 1.0 under the loads larger than 100 N, εLF-max increases near linearly with increasing entrainment speed. When the slide-to-roll ratio is 1.8, εLF-max increases superlinearly with entrainment speed. Thus, heavy loads strengthen εLF-max under each sliding condition and maintain the increasing tendency of εLF-max with rising entrainment speed. Fig. 6 shows the variation of ALF under the same conditions as Fig. 5. Under the load of 30 N, the maximum ALF among all the entrainment speeds is 1.3% at SRR of 0.6, 22.2% at SRR of 1.0 and 36.1% at SRR of 1.8. Under the load of 500 N, the maximum ALF among all the entrain ment speeds reaches 83.9% at SRR of 0.6, 87.6% at SRR of 1.0 and 92.6% at SRR of 1.8. With the increase of load, ALF increases at each entrainment speed for all the three slide-to-roll ratios. Generally, it can be observed that the increase of ALF between two consecutive rising loads gets declined when the applied load is larger than 100 N. Between the load conditions of 400 N and 500 N, ALF in creases only a few percent. With the increase of load, the variations of
The previous sections discussed the slip behaviours under different operation conditions. In fact, the rheological properties of lubricant essentially determine the limiting shear stress, therefore the slip status. This section will discuss the effect of limiting shear stress. There are three parameters in Eq. (1) influencing limiting shear stress, τ0, γ and β, where β is fixed as 1500 K, τ0 and γ are variable as listed in Table 1. 3.3.1. Same τ0 and γ on contacting surfaces The parameter τ0 is generally considered as ambient limiting shear stress of lubricant at normal temperature and pressure. In this section, the effect of surface properties on limiting shear stress is not considered. That is to say, the limiting shear stress of lubricant on fast surface is equal to that on slow surface. Both the slip statuses on fast and slow surfaces are shown in Fig. 8. Coefficient γ is chosen from 0.01 to 0.08. As the results show that no slip occurs when γ equals to 0.05, 0.06, 0.07 and 0.08, therefore, the corresponding results are not given in Fig. 8. The reason of no slip occurring in these cases is that when γ is large enough, limiting shear stress increases sharply with the increasing pressure from outside to the centre of the contact area so that no shear stress in the lubricant is larger than limiting shear stress. As shown in Fig. 8a, the maximum thickness coefficients of slip layers on fast and slow surfaces are almost equal and both of them reduce with the increases of τ0 and γ, as the increases of τ0 and γ directly result in the increase of limiting shear stress. It can be observed from the variations of εmax in Fig. 8 that, the effect of τ0 on slip amplitudes becomes weak with the increase of γ. When γ equals to 0.01, the variation of εmax between the cases τ0 equals to 2.0 MPa and 8.0 MPa is 1.135 � 10 3. This value decreases to 3.97 � 10 5 when γ equals to 0.04. The effect of γ on slip amplitudes also becomes weak with the increase of τ0. At τ0 ¼ 2.0 MPa,
Fig. 8. Slip statuses at both fast and slow surfaces with different τ0 and γ, (a) εLS-max and εLF-max, (b) ALS and ALF (w ¼ 32 N, ue ¼ 18 m/s and SRR ¼ 1.8). 7
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Fig. 9. The slip statuses on both slow and fast surfaces with different pressure-limiting shear stress coefficients from one another (τ0 ¼ 2.0 MPa, ue ¼ 18 m/s and SRR ¼ 1.8).
the increases of τ0 and γ. The effect of τ0 on slip areas gets weak with the increase of γ, while the effect of γ on slip areas also gets weak with the increase of τ0. Besides, the slip area on fast surface is slightly larger than slow surface once slip occurs, which is related to the thermal properties of the contact surfaces. Because the conductivity of steel (1.44 � 10 5 m2/s) is much larger than glass (4.99 � 10 7 m2/s), the temperature of glass ring is higher than the steel ball in the contact area; as a result, the limiting shear stress of lubricant is smaller on the fast ring surface, and slip will be more intense on the ring surface. 3.3.2. Different τ0 and γ on contacting surfaces It has been stated in some molecular dynamics simulations [32–35] and experiments [36,37] that slip effect of confined lubricant is influ enced by solid wall-fluid interactions. The solid wall-lubricant in teractions vary with surface properties, and consequently, the limiting shear stresses could be different from one another. Thus, slip status should also vary with different contact surfaces. In this section, both τ0 and γ on one surface can be different from that at another surface. Fig. 9 shows contour plots of the maximum thickness coefficients of the slip layers and the slip area percentages on both the slow and fast sur faces with different γS and γ F. Both γS and γ F are set from 0.01 to 0.06 with interval of 0.005. As shown in Fig. 9, εmax and A on both surfaces decrease with the increases of γS and γF. In detail, if γ on one surfaces increases and keeps unchanged on another surface, the situation will be different. For instance, in the cases where γ S < 0.02, when γ F is smaller than 0.02, both γ S and γ F will influence εmax and A; however, when γF exceeds 0.02, εmax and A are barely influenced by γF but greatly influenced by γS. Similar trend can be found when γ F < 0.02 and γ S ranges from 0.01 to 0.06. Compared to the cases when γ S < 0.02, εmax and A are larger when γF < 0.02, which suggests that slip is more intense when γ is smaller on fast surface than on slow surface.
Fig. 10. The impact of limiting shear stress-pressure coefficient on the maximum thickness coefficients of slip layer on slow surface.
the variation of εmax is 1.57 � 10 3 when γ increases from 0.01 to 0.04; however, when τ0 ¼ 8.0 MPa, the variation of εmax is only 4.65 � 10 4 when γ increases from 0.01 to 0.04. Fig. 8 also shows that, in respect to the whole ranges of γ and τ0, slip is almost completely suppressed when γ ¼ 0.04 which does not reach the middle of its range; while slip still cannot be ignored when τ0 takes its maximum value of 8.0 MPa. As the different lubricants may have different value of γ and τ0, therefore, the results imply that the lubricant with larger γ will lead to a relatively weaker slip compared with the lubricant with larger τ0. The slip area percentages shown in Fig. 8b have the similar changing trend to εmax. Both the slip areas on fast and slow surfaces decrease with 8
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Fig. 11. The slip statuses on both slow and fast surfaces with different ambient limiting shear stresses from one another (γ ¼ 0.03, ue ¼ 18 m/s and SRR ¼ 1.8).
Fig. 12. Slip statuses of lubricants with various rheological properties (w ¼ 100 N, ue ¼ 10 m/s, SRR ¼ 1.8 and ϕ ¼ 0.0342 K-1).
Based on Fig. 9a, the impact of γ on εLS-max can be divided into four zones as shown in Fig. 10. In zone 1 and zone 3, γ S or γF dominates the variation of εLS-max respectively. In zone 1, γS is smaller than γ F, while in zone 3, γF is smaller than γS in most cases except for a small area near zone 2. It indicates that in zone 1 and zone 3, εLS-max is mainly affected by the surface on which γ is smaller. Meanwhile, γ on fast surface seems to have larger effect on slip status compared to that on slow surface. In zone 2, γ S and γ F have comparable impacts on slip. The change of either γ S or γ F will lead to the variation of εLS-max. In zone 4, γ S and γ F are relatively large, and the limiting shear stresses at both surfaces are large enough to prevent slip at fast surface. Fig. 10 only shows the case of εLSmax, but the deductions can be extended to εLF-max, ALS and ALF. The contour plots of slip statuses with different τ0-LS and τ0-LF are shown in Fig. 11. εLS-max, εLF-max, ALS and ALF increase with the decrease
of τ0-LF. Both εmax and A are less influenced by τ0-LS rather than τ0-LF. With the increase of τ0-LS, εLS-max, εLF-max and ALF slightly increase while ALS decreases. 3.4. Discussions In this model, slip occurs once the calculated shear stress of the lubricant on the surface exceeds limiting shear stress, and it is insepa rable with the rheological properties of lubricant. Last sections have discussed the effect of limiting shear stress on slip. Another important property, the viscosity of lubricant expressed by Eq. (7) is also a critical parameter in calculation of lubricant shear stress. Both of the limiting shear stress and the viscosity expressed by Eq. (1) and Eq. (7) increase with the increase of pressure and the decrease of temperature, and thus 9
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the pressure and temperature coefficients in Eq. (1) and Eq. (7) become also critical factors during the determination of slip in the numerical solution process. Slip status with various α, β and γ are explored and shown in Fig. 12, while ϕ is set as 0.0342 K-1 for all the cases. It can be found that both εLFmax and ALF increase with the increase of β in all the cases. This is because limiting shear stress tends to decrease faster with increasing temperature for higher β, which makes itself the weak one in the com parison with calculated shear stress in lubricant. For the same γ, εLF-max and ALF increase with the increase of α. For the same α, εLF-max and ALF increase with the decrease of γ. In addition, it can be inferred that slip will not occur if β is small enough or ϕ is large enough. It also can be inferred that slip will not occur if α is small enough and γ is large enough. In general, the operating conditions are the external factors that provide loading conditions and shearing on lubricant for slip to occur, while the rheological properties of lubricant internally determines the slip behaviour under various operation conditions, especially the re sponses of slip to pressure, temperature and shearing variations.
References [1] Smith FW. Lubricant behavior in concentrated contact—some rheological problems. Tribol Trans 1960;3:18–25. https://doi.org/10.1080/ 05698196008972381. [2] Crook AW. The lubrication of rollers .4. Measurements of friction and effective viscosity. Philos Trans R Soc Lond A 1963;255:281–312. https://doi.org/10.1098/ rsta.1963.0005. [3] Johnson KL, Cameron R. Fourth paper: shear behaviour of elastohydrodynamic oil films at high rolling contact pressures. Proc Inst Mech Eng 1967;182:307–30. https://doi.org/10.1243/PIME_PROC_1967_182_029_02. [4] Houpert L, Flamand L, Berthe D. Rheological and thermal effects in lubricated E.H. D. Contacts. J Tribol 1981;103:526. https://doi.org/10.1115/1.3251731. [5] Wikstr€ om V, H€ oGlund E. Investigation of parameters affecting the limiting shear stress-pressure coefficient: a new model incorporating temperature. J Tribol 1994; 116:612–20. https://doi.org/10.1115/1.2928889. [6] Bair S, Winer WO. The high pressure high shear stress rheology of liquid lubricants. J Tribol Trans ASME 1992;114:1–13. https://doi.org/10.1115/1.2920862. [7] Ndiaye S-N, Martinie L, Philippon D, Devaux N, Vergne P. A quantitative frictionbased approach of the limiting shear stress pressure and temperature dependence. Tribol Lett 2017;65:149. https://doi.org/10.1007/s11249-017-0929-2. [8] Bair S, Qureshi F, Winer WO. Observations of shear localization in liquid lubricants under pressure. J Tribol 1993;115:507–13. https://doi.org/10.1115/1.2921667. [9] Ponjavic A, Wong JSS. The effect of boundary slip on elastohydrodynamic lubrication. RSC Adv 2014;4:20821–9. https://doi.org/10.1039/c4ra01714e. [10] Medina-Banuelos EF, Marin-Santibanez BM, Perez-Gonzalez J, Malik M, Kalyon DM. Tangential annular (Couette) flow of a viscoplastic microgel with wall slip. J Rheol 2017;61:1007–22. https://doi.org/10.1122/1.4998177. [11] Jeffreys S, di Mare L, Liu X, Morgan N, Wong JSS. Elastohydrodynamic lubricant flow with nanoparticle tracking. RSC Adv 2019;9:1441–50. https://doi.org/ 10.1039/c8ra09396b. [12] Xu J, Yang C, Sheng Y-J, Tsao H-K. Apparent hydrodynamic slip induced by density inhomogeneities at fluid-solid interfaces. Soft Matter 2015;11:6916–20. https:// doi.org/10.1039/C5SM01627D. [13] Porras-Vazquez A, Martinie L, Vergne P, Fillot N. Independence between friction and velocity distribution in fluids subjected to severe shearing and confinement. Phys Chem Chem Phys 2018;20:27280–93. https://doi.org/10.1039/ C8CP04620D. [14] Savio D, Falk K, Moseler M. Slipping domains in water-lubricated microsystems for improved load support. Tribol Int 2018;120:269–79. https://doi.org/10.1016/j. triboint.2017.12.030. [15] Voeltzel N, Giuliani A, Fillot N, Vergne P, Joly L. Nanolubrication by ionic liquids: molecular dynamics simulations reveal an anomalous effective rheology. Phys Chem Chem Phys 2015;17:23226–35. https://doi.org/10.1039/C5CP03134F. [16] Barrat JL, Bocquet L. Large slip effect at a nonwetting fluid-solid interface. Phys Rev Lett 1999;82:4671–4. https://doi.org/10.1103/PhysRevLett.82.4671. [17] Alghalibi D, Lashgari I, Brandt L, Hormozi S. Interface-resolved simulations of particle suspensions in Newtonian, shear thinning and shear thickening carrier fluids. J Fluid Mech 2018;852:329–57. https://doi.org/10.1017/jfm.2018.532. [18] Fillot N, Berro H, Vergne P. From continuous to molecular scale in modelling elastohydrodynamic lubrication: nanoscale surface slip effects on film thickness and friction. Tribol Lett 2011;43:257–66. https://doi.org/10.1007/s11249-0119804-8. [19] Echeverri Restrepo S, van Eijk MCP, Ewen JP. Behaviour of n-alkanes confined between iron oxide surfaces at high pressure and shear rate: a nonequilibrium molecular dynamics study. Tribol Int 2019;137:420–32. https://doi.org/10.1016/ j.triboint.2019.05.008. [20] Gattinoni C, Heyes DM, Lorenz CD, Dini D. Traction and nonequilibrium phase behavior of confined sheared liquids at high pressure. Phys Rev E 2013;88:471–90. https://doi.org/10.1103/PhysRevE.88.052406. [21] Fu Z, Guo F, Wong PL. Non-classical elastohydrodynamic lubricating film shape under large slide-roll ratios. Tribol Lett 2007;27:211–9. https://doi.org/10.1007/ s11249-007-9227-8. [22] Wang P, Reddyhoff T. Wall slip in an EHL contact lubricated with 1-dodecanol. Tribol Int 2016. https://doi.org/10.1016/j.triboint.2016.10.034. [23] Hili J, Olver AV, Edwards S, Jacobs L. Experimental investigation of elastohydrodynamic (EHD) film thickness behavior at high speeds. Tribol Trans 2010;53:658–66. https://doi.org/10.1080/10402001003658326. [24] Zhang YG, Wang WZ, Zhang SG, Zhao ZQ. Experimental study of EHL film thickness behaviour at high speed in ball-on-ring contacts. Tribol Int 2017;113: 216–23. https://doi.org/10.1016/j.triboint.2017.02.040. [25] Li X, Guo F, Wong P. Study of boundary slippage using movement of a post-impact EHL dimple under conditions of pure sliding and zero entrainment velocity. Tribol Lett 2011;44:159–65. https://doi.org/10.1007/s11249-011-9834-2. [26] Ehret P, Dowson D, Taylor CM. On lubricant transport conditions in elastohydrodynamic conjuctions. Proc R Soc Lond A 1998;454:763–87. https://doi. org/10.1098/rspa.1998.0185. [27] Zhang Y, Wen S. An analysis of elastohydrodynamic lubrication with limiting shear stress: Part II — load influence. Tribol Trans 2002;45:211–6. https://doi.org/ 10.1080/10402000208982542. [28] Ståhl J, Bo OJ. A lubricant model considering wall-slip in EHL line contacts. J Tribol 2003;125:523–32. https://doi.org/10.1115/1.1537750. [29] Kumar P, Anuradha P. Elastohydrodynamic lubrication model with limiting shear stress behavior considering realistic shear-thinning and piezo-viscous response. J Tribol 2014;136:021503. https://doi.org/10.1115/1.4026111.
4. Conclusion A parametric study on slip status in point contact is conducted based on layered oil slip model, which explores the effects of operation con ditions and lubricant properties on slip status. The results show that the sliding condition, load condition and rheological properties of lubricant influence the slip status in contact area. 1. Overall, both slip amplitude and slip area increase with rising slideto-roll ratio. With rising entrainment speed, slip amplitude basically increases while slip area does not change monotonically. 2. Both slip amplitude and slip area increase with rising load. More over, the increasing rate of slip amplitude increases with rising load, while the increasing rate of slip area decreases with rising load. 3. When the effect of surface property on the limiting shear stress of lubricant is not considered, both the slip amplitude and the slip area decrease with the increase of limiting shear stress. 4. When the limiting shear stress on one surface is unequal to another, both slip amplitude and slip area are mainly affected by the ambient limiting shear stress on the faster surface and the smaller of the two pressure-limiting shear stress coefficients on two contacting surfaces. 5. The variation characteristics of viscosity and limiting shear stress of lubricant essentially determine the slip behaviours under various operation condition. Author contribution Yaoguang Zhang: Conceptualization, Methodology, Data Curation, Visualization, Writing- Original draft preparation. Wenzhong Wang: Conceptualization, Resources, Writing- Reviewing and Editing, Super vision, Funding acquisition. Ziqiang Zhao: Investigation, Data Cura tion, Project administration. He Liang: Investigation, Resources, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The work was supported by the National Natural Science Foundation of China [grant numbers 51675046, U1637205], and National Key R&D Program of China [2018YFB2000604].
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[30] Zhang YG, Wang WZ, Liang H, Zhao ZQ. Layered oil slip model for investigation of film thickness behaviours at high speed conditions. Tribol Int 2019;131:137–47. https://doi.org/10.1016/j.triboint.2018.10.035. [31] Ma�ckowiak S, Heyes DM, Dini D, Bra� nka AC. Non-equilibrium phase behavior and friction of confined molecular films under shear: a non-equilibrium molecular dynamics study. J Chem Phys 2016;145:164704. https://doi.org/10.1063/ 1.4965829. [32] Savio D, Fillot N, Vergne P, Zaccheddu M. A model for wall slip prediction of confined n-alkanes: effect of wall-fluid interaction versus fluid resistance. Tribol Lett 2012;46:11–22. https://doi.org/10.1007/s11249-011-9911-6. [33] Ta DT, Tieu AK, Zhu HT, Kosasih B. Thin film lubrication of hexadecane confined by iron and iron oxide surfaces: a crucial role of surface structure. J Chem Phys 2015;143:164702. https://doi.org/10.1063/1.4933203.
[34] Jabbarzadeh A, Atkinson JD, Tanner RI. Wall slip in the molecular dynamics simulation of thin films of hexadecane. J Chem Phys 1999;110:2612–20. https:// doi.org/10.1063/1.477982. [35] Thompson PA, Troian SM. A general boundary condition for liquid flow at solid surfaces. Nature 1997;389:360–2. https://doi.org/10.1038/38686. [36] Guo F, Wong PL, Geng M, Kaneta M. Occurrence of wall slip in elastohydrodynamic lubrication contacts. Tribol Lett 2009;34:103–11. https://doi.org/10.1007/ s11249-009-9414-x. [37] Wong PL, Li XM, Guo F. Evidence of lubricant slip on steel surface in EHL contact. Tribol Int 2013;61:116–9. https://doi.org/10.1016/j.triboint.2012.12.009.
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Slip status in lubricated point-contact based on layered oil slip lubrication model Yaoguang Zhang, Wenzhong Wang *, He Liang, Ziqiang Zhao School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Slip extent Limiting shear stress Rheological properties Layered slip model
Obvious decrease of film thickness is found in recent experimental tests of high-speed elastohydrodynamic lubrication (EHL) point contact, which shows a close correlation with interface slip. This work investigates the effects of operation conditions and rheological properties of lubricant on slip behaviours based on a layered oil slip model. The results show that slip extent generally increases with the aggravation of sliding, the increase of applied load and the decrease of limiting shear stress; furthermore, the slip extent will also be affected by the contacting surface properties. Essentially, as internal factors, the rheological properties of lubricant may determine the slip behaviour under various operation conditions.
1. Introduction Oil film under elastohydrodynamic lubrication (EHL) usually suffers from high contact pressure and high shear stress, especially in high speeds conditions while sliding occurs. Many widely used machine el ements such as rolling bearing and gears usually work at such conditions that the lubricant will not behave as Newtonian fluid. Along with the increase of shear rate, the shear stress in the lubricant will increase at first, and usually approach limiting shear stress eventually. Therefore, the limiting shear stress of lubricant would inevitably have an effect on EHL film behaviour when the components are operated under high speeds condition. Limiting shear stress was found to be a property of lubricant in traction measuring experiments. The earlier studies on lubricant rheology at high pressure were conducted in two-disc or four-disc ma chines for EHL contact [1–3]. Results under sliding condition show the phenomenon that the traction coefficient no longer increases with the increase of shear rate after having reached a limit value, which is similar to plastic yield of solid material. Thus, limiting shear stress was intro duced to characterize this property of lubricant. Like viscosity, the limiting shear stress decreases with the increase of temperature, while increases with the applied pressure. Subsequently, Houpert et al. [4] proposed Eq. (1) to approximate limiting shear stress considering pres sure and temperature effects through fitting the traction curves obtained from two-disc machine. Other models that considering both the effects of pressure and temperature were also established based on impacting
ball apparatus [5], high pressure viscometer [6], or falling viscometer [7]. � � �� 1 1 τL ¼ ðτ0 þ γPÞexp β (1) Ti T0 When limiting shear stress is reached, usually at high shear rate condition, lubricant flow behaviour may change. Velocity variations across the lubricant film and near the solid surface are deemed to be related to limiting shear stress, such as shear banding, wall slip and plug slip. Shear banding was observed at the shear rate around 10 1 by Bair et al. [8] in the oil with large viscosity confined between parallel plates with separation distance of 150 μm. In their observed results, shear banding was found in an inclination angle to solid surface, but does not extend to the surface. By contrast, wall slip occurs at the interface be tween lubricant and surface when the velocity of the lubricant near the surface is unequal to the velocity of the surface or there is very large velocity gradient in lubricant near the surface, as captured in experi ments [9–11] and found in molecular dynamic simulations [12–16]. It can be found that the significant change of speed occurs in the fluid layer next to the surface as the density and viscosity of liquids adjacent to solid surfaces are distinct from the bulk ones as manifested in references such as Ref. [12], and low viscosity can be found in this boundary layer depending on the surface properties and interaction between surface and fluid. Slip occurs at the interface between the solid phase and fluid phase, and takes place in the lubricant layer below or on the order of molecular size as shown in the flow profiles in Refs. [14,17,18]. Slip is closely related to the limiting shear stress of lubricant, and the operation
* Corresponding author. E-mail address: [email protected] (W. Wang). https://doi.org/10.1016/j.triboint.2019.106104 Received 4 September 2019; Received in revised form 10 November 2019; Accepted 4 December 2019 Available online 6 December 2019 0301-679X/© 2019 Elsevier Ltd. All rights reserved.
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Nomenclature A cf h kf P SRR T T0 u ue v W x y z
α
β γ δ
slip area ratio specific heat of fluid film thickness heat conductivity of fluid pressure slide-to-roll ratio temperature ambient temperature speed in x direction average speed (uF þ uS)/2 speed in y direction load coordinate along the motion direction coordinate perpendicular to the motion direction coordinate through the film thickness pressure-viscosity coefficient
ε η η0 θ λ
ρ τ0 τL ϕ F LF LS max S
conditions lubricant sustains. Echeverri [19] conducted the nonequi librium molecular dynamics (NEMD) study of n-alkanes, and the slip was proved to be related to the length of n-alkanes. The results of the NEMD work by Gattinoni et al. [20] show that the phase behaviour of lubricant in confined gap is depending on various operation conditions, and slip occurs under high pressure and shearing conditions. Consequently, slip may cause the formation of dimple [21,22], or large sliding under tractive rolling mode which was detected in ball-on-disc [23] and ball-on-ring [24] experimental investigations. Affected by the properties of contact surfaces, slip could occur at one or both surfaces under pure sliding condition as discussed in the experi mental work by Li et al. [25]. Theoretical models were established for lubrication analysis with slip. Ehret et al. [26] proposed a slip coefficient to express the degree of slip, which is proportional to the pressure, their model well reproduced the dimples observed in experiments; however, the detailed slip status in the contact area, such as slip area and the degree of slip in different velocity conditions is not solely related to pressure. Therefore, slip should relate to the shearing condition and rheological properties of lubricant. Zhang et al. [27] developed a slip model with the assumption
limiting shear stress-temperature coefficient limiting shear stress-pressure coefficient thermal expansion coefficient the thickness coefficient of slip layer viscosity of lubricant viscosity of lubricant at ambient pressure and temperature viscosity coefficient of slip layer coefficient in modified Reynolds equation density ambient limiting shear stress limiting shear stress temperature-viscosity coefficient fast surface slip layer of lubricant near the fast surface slip layer of lubricant near the slow surface maximum value slow surface
that wall slip occurs when shear stress at the surface exceeds limiting shear stress. Load effects on lubricant flow, friction and so on are ana lysed in their isothermal line contact model. Their strategy they used to determine slippage is adopted in some researches [28–30], where the isothermal expression of limiting shear stress was used to investigate the wall slip in line contact. In the recent work by Zhang et al. [30], a layered oil slip model considering the slip and thermal effect is proposed to explore the mechanism of film thickness behaviours at high speed. The film thickness distribution obtained show good agreement with experimental results; moreover, local slip status and velocity profile through film thickness in contact area are obtained. However, in-depth theoretical investigations of slip effect are still required at present. In this work, an comprehensive analysis of slip status in ball-on-ring contact is conducted based on the layered oil slip model in Ref. [30]. The factors that may affect slip behaviour including sliding condition, load conditions and rheological properties of lubricant are parametrically discussed in detail by characterizing the extent of slip through slip amplitude and slip area.
Fig. 1. Schematic diagram of layered slip lubrication model, adapted from Ref. [30]. 2
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adjacent layers including solid surfaces.
Table 1 Lubricant properties and operation conditions. Parameters
2.1. Governing equations
Values
Viscosity at 25 C, η0 Viscosity-pressure coefficient, α Viscosity-temperature coefficient, ϕ Ambient limiting shear stress (LSS), τ0 LSS-pressure coefficient, γ LSS-temperature coefficient, β Load, w Hertzian contact pressure, PH Entrainment speed, u Slide-to-roll ratio, SRR �
0.048 Pa⋅s 15.0E-9 Pa 1 0.0342 K-1 2.0 to 8.0 MPa 0.01 to 0.15 1500 K 30 to 500 N 492.8 to 1258 MPa 0 to 20 m/s 0 to 2.0
As shown in Fig. 1, the shear stress equilibrium equations in each layer across the film can be obtained based on Navier-Stokes equation as follows, 8 ∂p η ∂2 u > > > ¼ z 2 ½ð1 εLF Þh; h � > > ∂x θ ∂x2 > > > < 2 ∂p ∂u (2) z 2 ½εLS h; ð1 εLF Þh � ¼η 2 > ∂ x ∂x > > > > 2 > > > ∂p η ∂ u : z 2 ½0; εLS h� ¼ ∂x θ ∂x2
2. Brief review of layered oil slip model
Then, based on continuity conditions of the shear stress and velocity at each interface of lubricant layers, the velocity distributions are ob tained:
A lubrication model with the consideration of slip effect was pro
� 8 ∂p 2 2 2 > > ðθ 1ÞðεLF þ εLS 1ÞuF þ uS θ θ ∂x h ðθ 1Þ εLF εLS þ εLS εLF > > ¼ u LF > > 2η½ðθ 1ÞðεLF þ εLS Þ þ 1 � ðθ 1ÞðεLF þ εLS Þ þ 1 > > > > > > > � � � ∂p ∂p > > θ h ðθ 1Þ ε2LS ε2LF þ 2εLF þ 1 θ z2 > > θðuS uF Þ > ∂ x ∂x > z z þ > > 2η½ðθ 1ÞðεLF þ εLS Þ þ 1 � h½ðθ 1ÞðεLF þ εLS Þ þ 1 � 2η > > > > > > > � � � ∂p 2 < 2 ðθ 1ÞðεLF uS þ εLS uF Þ þ uS εLS ∂xh ðθ 1Þ ðθ 1Þ εLF εLS εLF 2εLF þ εLS 1 uLM ¼ > > 2η½ðθ 1ÞðεLF þ εLS Þ þ 1 � ðθ 1ÞðεLF þ εLS Þ þ 1 > > > > > > > � � ∂p � ∂p 2 > > > h ðθ 1Þ ε2LS ε2LF þ 2εLF þ 1 z > ðuS uF Þ > ∂ x > z z þ ∂x > > > h½ðθ 1Þð 2 ε þ ε Þ þ 1 � η ½ðθ 1Þð ε þ ε Þ þ 1 � 2 η LF LS LF LS > > > > > > � � ∂p � ∂p > > > θ h ðθ 1Þ ε2LS ε2LF þ 2εLF þ 1 θ z2 > θðuS uF Þ : ∂ x uLS ¼ uS z z þ ∂x 2η½ðθ 1ÞðεLF þ εLS Þ þ 1 � h½ðθ 1ÞðεLF þ εLS Þ þ 1 � 2η
posed by authors in recent work [30]. As shown in Fig. 1, slip may occur at either or both contacting surfaces. Slip status of each position in contact area is determined by the local rheological and operating con ditions. Limiting shear stress is considered as the criteria to determine whether slip occurs or not. Once the shear stress of lubricant at one surface exceeds limiting shear stress, there will be a presumed thin lubricant slip layer with much lower viscosity between the middle lubricant and surface. The viscosity of slip layer is 1/θ of middle layer, and the ratio between the thicknesses of slip layer and whole film thickness is ε. Velocity is continuous at the interfaces between any
(3)
Therefore, the modified Reynolds equation with consideration of slip can be obtained by substituting Eq. (3) into the mass flow conservation equation, � � � � ∂ ρh3 ∂p ∂ ρh3 ∂p ∂½ρhðλS uS þ λF uF Þ � λP þ λP ¼ ∂x ∂y 12η ∂x 12η ∂y ∂x þ
3
∂½ρhðλS vS þ λF vF Þ � ∂y
(4)
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b
where
i
h
�2 ðεLS þ εLF Þ5 ε2LS þ εLS εLF λP ¼ þ ðεLS þ εLF Þ2 ½ðθ 1ÞðεLS þ εLF Þ þ 1 � 3 ε2LF
ðεLS þ εLF
h 1Þ ðεLS þ εLF Þ4
12εLS εLF ðεLS þ εLF ÞðεLS þ εLF
2ðεLS þ εLF Þ3 þ 4ε2LS þ 4ε2LF
1Þ θ
ðεLS þ εLF Þ2 i 4εLS εLF
ðεLS þ εLF Þ2 λS ¼ λF ¼
(5)
ε2LS
ε2LF þ 2εLF ε2LF ε2LS þ εLS εLF þ 2ðεLS þ εLF Þ 2ðεLS þ εLF Þ½ðθ 1ÞðεLS þ εLF Þ þ 1 �
ε2LF
ε2LS þ 2εLS 2ðεLS þ εLF Þ
ε2LF
ε2LS þ εLS εLF 1ÞðεLS þ εLF Þ þ 1 �
2ðεLS þ εLF Þ½ðθ
etween oil and solid. The smaller one among the limiting shear stress of lubricant and the critical shear stress at oil-solid interface should be regarded as the limiting shear stress actually adopted in the determi nation of slip. Thus, the coefficients in Eq. (1) of lubricant at one contact surface may be different from that at another surface in some occasions.
λP, λS, and λF are only dependent on ε and θ and various with different slip cases. The modified Reynolds equation for the case that slip does not occur can be obtained by setting εLS as 0 and εLF as 0 successively. The modified Reynolds equation for the cases that slip occurs at slow or fast surfaces can be obtained by setting εLS as 0 or εLF as 0 respectively. The detailed derivations of the equation and parameters involved can be referred to Ref. [30]. Energy equation of oil film for considering the thermal effect is expressed as � � �� �2 � �2 � � � ∂T ∂T ∂2 T ∂p ∂p ∂u ∂v ρc f u þ v þ (6) ¼ kf 2 þ δT u þ v þη ∂x ∂y ∂z ∂x ∂y ∂z ∂z
3. Results and discussions In the following numerical analysis, the point contact is considered formed between a glass ring with inner radius of 51.4 mm and a steel ball of 25.4 mm in diameter. The elastic moduli of steel and glass are 206 GPa and 88 GPa respectively. The Poisson’s ratios of steel and glass are 0.3 and 0.215. The initial temperature is 25 � C. The viscosity coef ficient of slip layer θ is set as 10000. In order to study the slip effect in EHL point contact, numerical calculations based on layered slip model are conducted under various operating conditions with different rheological properties of lubricant. Related parameters are listed in Table 1. For the variable parameters that are concerned factors in following discussion, their values are given in a range. τ0 is equal to 2.0 MPa and γ is equal to 0.03 in Sessions 3.1 and 3.2. The change of lubricant viscosity with pressure and tempera ture follows Eq. (7). � � � α � η ¼ η0 exp ðlnη0 þ 9:67Þ 1 þ 5:1 � 10 9 P 5:1�10 9 ðlnη0 þ9:67Þ 1 ϕðT
The thermal effect of slip layers is neglected in the thermal calcula tion due to the negligible thickness of slip layers. u and v in Eq. (6) are the velocity distributions in the middle layer of oil. 2.2. Determination of slip Slip status at each location should be determined before solving the governing equations in each iteration. In layered slip lubrication model, on which surface slip occurs depends on whether shear stress at the surface exceeds limiting shear stress. The thickness of slip layer repre sents the amplitude of slip, which depends on the theoretical shear stress condition at each position. When the thickness of slip layer gets thicker, the slip layer with much lower viscosity will sustain more velocity variation so that the shearing of lubricant in the middle layer is weakened. The thickness of slip layer is determined by the extent to which the shear stress on the surface is greater than limiting shear stress. The detailed processes to determine the slip status can be inferred in Ref. [30].
� T0 Þ
(7)
3.1. Slip status under different sliding conditions A matter worth of attention to delve into is the occurring condition of slip. Some results of molecular dynamic simulations [13,31] have indicated that slip occurs at the conditions within an approximate range of pressure and shear rate. However, the contact models in molecular dynamic simulation usually consist of two parallel solid walls with confined lubricant in between, which have large discrepancy with practical point contact EHL. Thus, in point contact EHL, the occurrence and escalation of slip at each position should be different. In EHL contact, the shear rate is usually at very high level when sliding occurs. Therefore, sliding condition directly influences the shear stress at the interface of lubricant and contact surface and the status of slip. Fig. 2 shows the slip extents at the centreline along the motion direction at different slide-to-roll ratios with the entrainment speeds of 4 m/s, 10 m/s and 16 m/s. In these figures, the profile of εLF looks like trapezoid shape when entrainment speed is 4 m/s and slide-to-roll ratio is 1.0, while all the other profiles approach triangle shape. The maximum thickness coefficient of slip layer on fast surface, εLF-max,
2.3. Limiting shear stress in the layered slip lubrication model In the procedure of determining the slip status, limiting shear stress is the reference to determine whether the slip occurs or not. Therefore, the variation of limiting shear stress under different operating conditions has to be considered. In this model, limiting shear stress at each position is related to local pressure and temperature conditions and follows Eq. (1) proposed by Houpert et al. [4], which includes ambient limiting shear stress τ0, limiting shear stress-pressure coefficient γ and limiting shear stress-temperature coefficient β. It was reported that the limiting shear stress may also be related to the interaction between lubricant and solid surface. Li et al. [25] inferred from EHL experiments under zero entrainment velocity conditions that wall slip is related to not only the limiting shear stress of lubricant but also the critical shear stress 4
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Fig. 2. The profile of εLF at the centreline along motion direction at different SRR under the entrainment speeds of: (a) 4 m/s, (b) 10 m/s and (c) 16 m/s (w ¼ 32 N, τ0 ¼ 2.0 MPa, γ ¼ 0.03).
which represents the amplitude of slip, and the area percent ALF, which is defined as the ratio between area where slip occurs and the whole Hertzian contact area, are used to represent the extent of slip on fast surface. εLS-max and ALS are used to represent the extent of slip on slow surface. As shown in Fig. 2, at the entrainment speeds of 4 m/s, 10 m/s and 16 m/s, εLF-max increases with the increase of the slide-to-roll ratio from 0 to 1.8. εLF-max under the slide-to-roll ratios of 1.4 and 1.8 increases with entrainment speed, while under the slide-to-roll ratios of 0.6 and 1.0, εLF-max at 10 m/s is the largest among three speed conditions in Fig. 2. The colour contour plot shown in Fig. 3 evidently indicates the change of slip extent, where the calculated points are plotted as black dots. Firstly, εLF-max increases with increasing slide-to-roll ratio through almost all the speed range. On the other hand, as entrainment speed increases, εLF-max shows different trends at different slide-to-roll ratios. When the slide-to-roll ratio is larger than nearly 1.4, εLF-max increases with the increasing entrainment speed. When the slide-to-roll ratio is larger than 0.4 and smaller than 1.4, εLF-max increases first and then decreases with increasing entrainment speed. When the slide-to-roll ratio is smaller than 0.4, εLF-max is almost zero. From the trend of εLFmax, it can be inferred that slip generally intensifies with the increases of entrainment speed and slide-to-roll ratio, but not monotonically in creases with entrainment speed when slide to roll ratio is less than 1.4 or so under present loading condition. As shown in Fig. 4, ALF generally increases with the increase of slideto-roll ratio. At the entrainment speed of 2 m/s and the slide-to-roll ratio of 1.8, the slip area becomes the largest among all the computation conditions. When the entrainment speed is faster than 16 m/s and the slide-to-roll ratio is larger than 1.6, although εLF-max is the largest as shown in Fig. 3, ALF is relatively small instead compared with the just mentioned slip area ratios at low speed and high slide-to-roll ratio conditions. Slip occurs with different extents under different speeds and sliding conditions. At relatively low speeds, as slide-to-roll ratio increases, slip occurs at smaller slide-to-roll ratio and extends to larger zone but with small εLF-max, which stands for the amplitude of slip. At relatively high speed, as slide-to-roll ratio increases, slip occurs at larger slide-to-roll ratio and intensifies in amplitude but with relatively smaller slip area compared with low speeds cases.
Fig. 3. Colour filled contour plot of εLF-max at different entrainment speeds and slide-to-roll ratios (w ¼ 32 N, τ0 ¼ 2.0 MPa, γ ¼ 0.03). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 4. Colour filled contour plot of ALF at different entrainment speeds and slide-to-roll ratios (w ¼ 32 N, τ0 ¼ 2.0 MPa, γ ¼ 0.03). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
3.2. Slip status under different load conditions As Limiting shear stress and viscosity of lubricant are closely related to applied pressure as expressed by Eq. (1) and Eq. (7), the applied load may significantly affect the slip status. Calculations are conducted under different loading conditions at the slide-to-roll ratios of 0.6, 1.0 and 1.8, 5
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Fig. 5. Variation of εLF-max with entrainment speed under different load conditions, (a) SRR ¼ 0.6, (b) SRR ¼ 1.0 and (c) SRR ¼ 1.8.
Fig. 6. Variation of ALF with entrainment speed under different load conditions, (a) SRR ¼ 0.6, (b) SRR ¼ 1.0 and (c) SRR ¼ 1.8.
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ALF with entrainment speeds at different loads also reduce gradually. Under the loads of 50 N and 100 N in Fig. 6a and 30 N, 50 N and 100 N in Fig. 6b and 30 N and 50 N in Fig. 6c, the variations of ALF in the speed range from 0 to 20 m/s reach 20%–30%. The variations of ALF with entrainment speeds under higher loads are nearly all less than 5%. Therefore, slip area is much larger but changes slowly under higher loads, which should be related to the pressure distributions in the con tact area. As shown in Fig. 7, slip basically occurs around the contact centre where the contact pressures are higher. When ALF is already relatively large, the pressure gradients around the boundary of slip area are steep. Thus, slip area hardly increases in a high rate as the contact pressure rapidly decreases along the outward direction. Therefore, slip status is closely related to load condition, which dominates the pressure condition in the contact. The slip amplitude solidly increases with rising load. The slip area increases and its changing rate slows down with rising load.
Fig. 7. Distributions of pressure and εLF at the centreline along motion direc tion under the loads of 100 N, 200 N and 300 N (ue ¼ 10 m/s,SRR ¼ 1.8).
3.3. Effect of τ0 and γ on slip
and slip statuses at the speed range from 1 m/s to 20 m/s are considered under each load. Fig. 5 shows the maximum thickness coefficient of slip layer on fast surface under different loading conditions from 30 N to 500 N, corre sponding to the Hertzian contact pressures from 492.8 MPa to 1258 MPa respectively, in the speed range from 0 m/s to 20 m/s and with the slideto-roll ratios of 0.6, 1.0 and 1.8. It can be found that εLF-max increases with the increase of applied load at each entrainment speed, and the increase amplitude of εLF-max between two consecutive loads increases with increasing entrainment speed. Although under the load of 32 N as shown in Fig. 3, εLF-max starts to decrease after the entrainment speed exceeds 10 m/s at slide-to-roll ratio of 1.0, εLF-max under higher loads in Fig. 5 solidly increases with rising entrainment speed. As shown in Fig. 5, when the slide-to-roll ratios are 0.6 and 1.0 under the loads larger than 100 N, εLF-max increases near linearly with increasing entrainment speed. When the slide-to-roll ratio is 1.8, εLF-max increases superlinearly with entrainment speed. Thus, heavy loads strengthen εLF-max under each sliding condition and maintain the increasing tendency of εLF-max with rising entrainment speed. Fig. 6 shows the variation of ALF under the same conditions as Fig. 5. Under the load of 30 N, the maximum ALF among all the entrainment speeds is 1.3% at SRR of 0.6, 22.2% at SRR of 1.0 and 36.1% at SRR of 1.8. Under the load of 500 N, the maximum ALF among all the entrain ment speeds reaches 83.9% at SRR of 0.6, 87.6% at SRR of 1.0 and 92.6% at SRR of 1.8. With the increase of load, ALF increases at each entrainment speed for all the three slide-to-roll ratios. Generally, it can be observed that the increase of ALF between two consecutive rising loads gets declined when the applied load is larger than 100 N. Between the load conditions of 400 N and 500 N, ALF in creases only a few percent. With the increase of load, the variations of
The previous sections discussed the slip behaviours under different operation conditions. In fact, the rheological properties of lubricant essentially determine the limiting shear stress, therefore the slip status. This section will discuss the effect of limiting shear stress. There are three parameters in Eq. (1) influencing limiting shear stress, τ0, γ and β, where β is fixed as 1500 K, τ0 and γ are variable as listed in Table 1. 3.3.1. Same τ0 and γ on contacting surfaces The parameter τ0 is generally considered as ambient limiting shear stress of lubricant at normal temperature and pressure. In this section, the effect of surface properties on limiting shear stress is not considered. That is to say, the limiting shear stress of lubricant on fast surface is equal to that on slow surface. Both the slip statuses on fast and slow surfaces are shown in Fig. 8. Coefficient γ is chosen from 0.01 to 0.08. As the results show that no slip occurs when γ equals to 0.05, 0.06, 0.07 and 0.08, therefore, the corresponding results are not given in Fig. 8. The reason of no slip occurring in these cases is that when γ is large enough, limiting shear stress increases sharply with the increasing pressure from outside to the centre of the contact area so that no shear stress in the lubricant is larger than limiting shear stress. As shown in Fig. 8a, the maximum thickness coefficients of slip layers on fast and slow surfaces are almost equal and both of them reduce with the increases of τ0 and γ, as the increases of τ0 and γ directly result in the increase of limiting shear stress. It can be observed from the variations of εmax in Fig. 8 that, the effect of τ0 on slip amplitudes becomes weak with the increase of γ. When γ equals to 0.01, the variation of εmax between the cases τ0 equals to 2.0 MPa and 8.0 MPa is 1.135 � 10 3. This value decreases to 3.97 � 10 5 when γ equals to 0.04. The effect of γ on slip amplitudes also becomes weak with the increase of τ0. At τ0 ¼ 2.0 MPa,
Fig. 8. Slip statuses at both fast and slow surfaces with different τ0 and γ, (a) εLS-max and εLF-max, (b) ALS and ALF (w ¼ 32 N, ue ¼ 18 m/s and SRR ¼ 1.8). 7
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Fig. 9. The slip statuses on both slow and fast surfaces with different pressure-limiting shear stress coefficients from one another (τ0 ¼ 2.0 MPa, ue ¼ 18 m/s and SRR ¼ 1.8).
the increases of τ0 and γ. The effect of τ0 on slip areas gets weak with the increase of γ, while the effect of γ on slip areas also gets weak with the increase of τ0. Besides, the slip area on fast surface is slightly larger than slow surface once slip occurs, which is related to the thermal properties of the contact surfaces. Because the conductivity of steel (1.44 � 10 5 m2/s) is much larger than glass (4.99 � 10 7 m2/s), the temperature of glass ring is higher than the steel ball in the contact area; as a result, the limiting shear stress of lubricant is smaller on the fast ring surface, and slip will be more intense on the ring surface. 3.3.2. Different τ0 and γ on contacting surfaces It has been stated in some molecular dynamics simulations [32–35] and experiments [36,37] that slip effect of confined lubricant is influ enced by solid wall-fluid interactions. The solid wall-lubricant in teractions vary with surface properties, and consequently, the limiting shear stresses could be different from one another. Thus, slip status should also vary with different contact surfaces. In this section, both τ0 and γ on one surface can be different from that at another surface. Fig. 9 shows contour plots of the maximum thickness coefficients of the slip layers and the slip area percentages on both the slow and fast sur faces with different γS and γ F. Both γS and γ F are set from 0.01 to 0.06 with interval of 0.005. As shown in Fig. 9, εmax and A on both surfaces decrease with the increases of γS and γF. In detail, if γ on one surfaces increases and keeps unchanged on another surface, the situation will be different. For instance, in the cases where γ S < 0.02, when γ F is smaller than 0.02, both γ S and γ F will influence εmax and A; however, when γF exceeds 0.02, εmax and A are barely influenced by γF but greatly influenced by γS. Similar trend can be found when γ F < 0.02 and γ S ranges from 0.01 to 0.06. Compared to the cases when γ S < 0.02, εmax and A are larger when γF < 0.02, which suggests that slip is more intense when γ is smaller on fast surface than on slow surface.
Fig. 10. The impact of limiting shear stress-pressure coefficient on the maximum thickness coefficients of slip layer on slow surface.
the variation of εmax is 1.57 � 10 3 when γ increases from 0.01 to 0.04; however, when τ0 ¼ 8.0 MPa, the variation of εmax is only 4.65 � 10 4 when γ increases from 0.01 to 0.04. Fig. 8 also shows that, in respect to the whole ranges of γ and τ0, slip is almost completely suppressed when γ ¼ 0.04 which does not reach the middle of its range; while slip still cannot be ignored when τ0 takes its maximum value of 8.0 MPa. As the different lubricants may have different value of γ and τ0, therefore, the results imply that the lubricant with larger γ will lead to a relatively weaker slip compared with the lubricant with larger τ0. The slip area percentages shown in Fig. 8b have the similar changing trend to εmax. Both the slip areas on fast and slow surfaces decrease with 8
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Fig. 11. The slip statuses on both slow and fast surfaces with different ambient limiting shear stresses from one another (γ ¼ 0.03, ue ¼ 18 m/s and SRR ¼ 1.8).
Fig. 12. Slip statuses of lubricants with various rheological properties (w ¼ 100 N, ue ¼ 10 m/s, SRR ¼ 1.8 and ϕ ¼ 0.0342 K-1).
Based on Fig. 9a, the impact of γ on εLS-max can be divided into four zones as shown in Fig. 10. In zone 1 and zone 3, γ S or γF dominates the variation of εLS-max respectively. In zone 1, γS is smaller than γ F, while in zone 3, γF is smaller than γS in most cases except for a small area near zone 2. It indicates that in zone 1 and zone 3, εLS-max is mainly affected by the surface on which γ is smaller. Meanwhile, γ on fast surface seems to have larger effect on slip status compared to that on slow surface. In zone 2, γ S and γ F have comparable impacts on slip. The change of either γ S or γ F will lead to the variation of εLS-max. In zone 4, γ S and γ F are relatively large, and the limiting shear stresses at both surfaces are large enough to prevent slip at fast surface. Fig. 10 only shows the case of εLSmax, but the deductions can be extended to εLF-max, ALS and ALF. The contour plots of slip statuses with different τ0-LS and τ0-LF are shown in Fig. 11. εLS-max, εLF-max, ALS and ALF increase with the decrease
of τ0-LF. Both εmax and A are less influenced by τ0-LS rather than τ0-LF. With the increase of τ0-LS, εLS-max, εLF-max and ALF slightly increase while ALS decreases. 3.4. Discussions In this model, slip occurs once the calculated shear stress of the lubricant on the surface exceeds limiting shear stress, and it is insepa rable with the rheological properties of lubricant. Last sections have discussed the effect of limiting shear stress on slip. Another important property, the viscosity of lubricant expressed by Eq. (7) is also a critical parameter in calculation of lubricant shear stress. Both of the limiting shear stress and the viscosity expressed by Eq. (1) and Eq. (7) increase with the increase of pressure and the decrease of temperature, and thus 9
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the pressure and temperature coefficients in Eq. (1) and Eq. (7) become also critical factors during the determination of slip in the numerical solution process. Slip status with various α, β and γ are explored and shown in Fig. 12, while ϕ is set as 0.0342 K-1 for all the cases. It can be found that both εLFmax and ALF increase with the increase of β in all the cases. This is because limiting shear stress tends to decrease faster with increasing temperature for higher β, which makes itself the weak one in the com parison with calculated shear stress in lubricant. For the same γ, εLF-max and ALF increase with the increase of α. For the same α, εLF-max and ALF increase with the decrease of γ. In addition, it can be inferred that slip will not occur if β is small enough or ϕ is large enough. It also can be inferred that slip will not occur if α is small enough and γ is large enough. In general, the operating conditions are the external factors that provide loading conditions and shearing on lubricant for slip to occur, while the rheological properties of lubricant internally determines the slip behaviour under various operation conditions, especially the re sponses of slip to pressure, temperature and shearing variations.
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4. Conclusion A parametric study on slip status in point contact is conducted based on layered oil slip model, which explores the effects of operation con ditions and lubricant properties on slip status. The results show that the sliding condition, load condition and rheological properties of lubricant influence the slip status in contact area. 1. Overall, both slip amplitude and slip area increase with rising slideto-roll ratio. With rising entrainment speed, slip amplitude basically increases while slip area does not change monotonically. 2. Both slip amplitude and slip area increase with rising load. More over, the increasing rate of slip amplitude increases with rising load, while the increasing rate of slip area decreases with rising load. 3. When the effect of surface property on the limiting shear stress of lubricant is not considered, both the slip amplitude and the slip area decrease with the increase of limiting shear stress. 4. When the limiting shear stress on one surface is unequal to another, both slip amplitude and slip area are mainly affected by the ambient limiting shear stress on the faster surface and the smaller of the two pressure-limiting shear stress coefficients on two contacting surfaces. 5. The variation characteristics of viscosity and limiting shear stress of lubricant essentially determine the slip behaviours under various operation condition. Author contribution Yaoguang Zhang: Conceptualization, Methodology, Data Curation, Visualization, Writing- Original draft preparation. Wenzhong Wang: Conceptualization, Resources, Writing- Reviewing and Editing, Super vision, Funding acquisition. Ziqiang Zhao: Investigation, Data Cura tion, Project administration. He Liang: Investigation, Resources, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The work was supported by the National Natural Science Foundation of China [grant numbers 51675046, U1637205], and National Key R&D Program of China [2018YFB2000604].
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