The effects of couple stress lubricants and surface roughness on squeeze EHL motion between porous medium layer and elastic ball

Abstract

This paper investigated the effects of porous media layer (PML) and couple stress (CS) lubricants at pure squeeze action in an isothermal elastohydrodynamic lubrication (EHL) point contact with surface roughness (SRN) under constant load condition. The modified transient stochastic Reynolds equation was derived in polar coordinates by means of the Christensen’s stochastic theory and the Stokes’s CS fluid theory. The Gauss-Seidel method (GSM) and the finite difference method (FDM) were both used to solve simultaneously modified transient stochastic Reynolds, force equilibrium, elasticity deformation, and rheology equations. The simulation results revealed that the greater permeability (K) and porous layer thickness (Φ) are, the smaller central pressures (P_c) are, and the smaller film thicknesses (H) are. The times required for achieving maximum central pressures (Pc_max) decrease with increasing K and Φ. Due to squeeze and elastic deformation effects, the slope values of central velocity (V_c) change from negative to positive. The magnitude of V_c and V_min decreased with decreasing K and Φ. The Hcmin of circular type RN are greater than that of radial type RN. The Pc_max of radial type RN are slightly greater than that of circular type RN.

Introduction

Due to self-contained oil reservoir and good low friction features, porous bearings have been extensively used in industry for a long time. They were widely used in clutches, brakes, and so on. The surface roughness (SRN) effect played a very important role in tribology domain, especially when SRN and film thickness have similar order of magnitude. In order to reduce the friction between the contact surfaces and improve the durability of the friction parts, different kinds of lubricating additives are joined to the base lubricant. Their rheological properties belong to non-Newtonian lubricants types. Many micro continuum theories are usually used to investigate rheological characteristics of non-Newtonian lubricants. The high pressure is accompanied by elastic deformation between contact pairs during squeeze process. Therefore, the effects of porous media layer (PML) and couple stress lubricants (CS) on squeeze elastohydrodynamic lubrication (EHL) motion with SRN are worth discussing.

Many tribological simulation methods^1–6 have been presented to explore the effects of SRN. These papers used finite difference method (FDM) and finite element method (FEM) to analyze simultaneously the EHL characteristics of different surface roughness patterns using measured 3D or given stochastic gauss distribution surface roughness. They thought that the rough peak contacts of two surfaces and transient process during start-up are important topics in the future. Over the past years, the generation of the dimple on pure squeeze motion with Newtonian viscous fluid has attracted the attention of many scholars.^7–9 But non-Newtonian lubricants, surface roughness, and porous media layer did not be considered. These effects are very important influence parameters in modern industry. Adding lubricant additives in lubricants can improve the load-carrying capacity and reduce the friction parameter.^10,11 Many micro-continuum theories have been developed to describe the effects of additives. Stokes theory ¹² considered the effects of couple stress, body couple, and asymmetric tensor. This model aims to investigate the effects of particle sizes. It can be applied to additives in lubricants. In recent years, many researchers^13–15 have dedicated their studies on the effect of journal bearing using couple stress lubricant because of the extensive use of additives in lubricants for heavy load applications in industries. Many studies^16–18 investigated the lubrication characteristics of porous journal bearings with different surface roughness patterns using non-Newtonian lubricants, but they did not consider elastic deformation. They believed that the permeability of porous media layer has significant effects for the characteristics of the porous journal bearings. However, the study on the effects of porous media and non-Newtonian lubricant with surface roughness on EHL circular contact at squeeze motion is insufficient. Therefore, these effects on squeeze EHL motion are necessary for further consideration.

This paper explores the effects of the CS lubricant between PML and elastic ball with SRN on squeeze EHL motion for fixed load condition. The finite difference method (FDM), Gaussian elimination method, the explicit method, and Gauss-Seidel iteration method (GSM) were used to calculate simultaneously the effects of the SRN, PML, and CS lubricants for pressure and film thickness distributions at each time step in the EHL region during pure squeeze process, but without asperities contact. To study the SRN behavior, Christensen stochastic model is adopted as it is a randomly varying quantity.

Theoretical analysis

The squeeze film mechanism as shown in Figure 1, a rough elastic sphere is approaching an infinite plate with rough porous medium layer (PML) for constant load condition, and neglect temperature effect. The compressible couple stress lubricant is filled between the ball and the plate. The local film thickness can be considered to include smooth part (h) and random part (δ):

$$\overline{h}=h(t)+\delta (r,\theta ,\xi )$$

(1)

Figure 1.

Geometry of EHL under pure squeeze motion with CS lubricant, PML, and SRN.

According to the Stokes microcontinuum theory ¹² and the usual assumption of EHL applicable to a thin film, the reduced momentum equations and the continuity equation governing the motion of the lubricant given in polar coordinates can be obtained as:

$$\frac{\partial p}{\partial r}=\mu \frac{{\partial }^2{v}_r}{\partial z^2}-\eta \frac{{\partial }^4{v}_r}{\partial z^4}$$

(2)

$$\frac{\partial p}{\partial z}=0$$

(3)

$$\frac{1}{r}\frac{\partial (\rho r{v}_r)}{\partial r}+\frac{\partial \rho v_z}{\partial z}=0$$

(4)

The velocity boundary conditions (BC) at the surfaces of the porous medium layer and elastic ball are:

$$v_r(r,0)=0,\frac{{\partial }^2{v}_r(r,0)}{\partial z^2}=0,v_z(r,0)=-v_z*$$

(5)

$$v_r(r,\overline{h})=0,\frac{{\partial }^2{v}_r(r,\overline{h})}{\partial z^2}=0,v_z(r,\overline{h})=\frac{\partial \overline{h}}{\partial t}$$

(6)

Where $v_z$ * is the modified Darcy velocity component in the porous region. The couple stress lubricant is governed by Darcy’s law, which accounts for the polar effects in the porous region ¹⁶ given by

$$v_z*=\frac{-k}{\mu (1-\beta )}\frac{\partial p*}{\partial z}$$

(7)

where

k is permeability parameter, φ is porous film thickness, $\beta$ is ratio of microstructure size to pore size, $\beta =(\eta /\mu )/k$ .

Due to continuity of the fluid motion in the porous region, the pressure p* satisfies the Laplace equation.

$${\nabla }^{2}p*=0$$

(8)

Integrating equation (2) using BC, the velocity component $v_r$ is:

$$v_r=\frac{1}{\mu }\frac{\mathrm{dp}}{\mathrm{dr}}\{z(z-\overline{h})/2+l^2[1-\cosh \left(\frac{2z-\overline{h}}{2l}\right)/\cosh \left(\frac{\overline{h}}{2l}\right)]\}$$

(9)

where

$l$ is characteristic length of the couple stress fluids, $l=(\eta /\mu {)}^{1/2}$ .

Suppose the porous layer thickness is small. The pressure is continuous (p = p*) at z = 0. Integrating equation (8) across the porous layer thickness using $\partial p*/\partial z=0$ at z = −φ, the equation can be derived as:

$${\frac{\partial p*}{\partial z}|}_{z=0}=-\phi \frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p}{\partial r}\right)$$

(10)

Substituting equation (10) into equation (7), the $v_z*$ at z = 0 can be derived as:

$$v_z*=\frac{k\phi }{\mu (1-\beta )}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p}{\partial r}\right)$$

(11)

Substituting $v_r$ into equation (4) and integrating across the film thickness with equations (5) and (6), the modified transient Reynolds equation can be derived as:

$$\frac{\partial }{\partial r}\{\frac{\rho r}{\mu }\zeta (\overline{h},k,\phi ,\beta ,l)\frac{\partial p}{\partial r}\}=12r\frac{\partial \rho \overline{h}}{\partial t}$$

(12)

where

$$\zeta (\overline{h},k,\phi ,\beta ,l)={\overline{h}}^3-12\overline{h}{l}^2+24l^3\tanh \left(\frac{\overline{h}}{2l}\right)+\frac{12k\phi }{(1-\beta )}$$

(13)

Taking the expected values of both sides of (12) by Christensen ¹ theory, the modified transient stochastic Reynolds equation can be obtained as:

$$\frac{\partial }{\partial r}\{\frac{\rho r}{\mu }E[\zeta (\overline{h})]\frac{\partial E(p)}{\partial r}\}=12r\frac{\partial }{\partial t}[\rho E(\overline{h})]$$

(14)

The expectancy operator is defined by

$$E(\mathrm{\Theta })={\int }_{\delta =-\infty }^{+\infty }(\mathrm{\Theta })\lambda (\delta )d\delta$$

(15)

Most engineering rough surfaces are Gaussian in nature. ⁵ Choose a polynomial function close to the Gaussian distribution:

$$\lambda (\delta )=\{\begin{array}{c}\frac{35}{32c^7}(c^2-{\delta }^2{)}^3,-c\le \delta \le +c\hfill \\ \multicolumn{1}{c}{0,\mathrm{elsewhere}}\end{array}$$

(16)

The function $\lambda$ terminates at $c=\pm 3\sigma$ . In the current analysis, the radial type RN (RTRN) and circular type RN (CTRN) structures ¹⁹ are worthy of attention. The structures have the long narrow ridges and valleys forms extending in r and $\theta$ directions, respectively. For RTRN, the oil film can be expressed as:

$$\overline{h}=h(t)+\delta (r,\xi )$$

(17)

By means of equations (15) and (16), the dimensionless form of the equation (14) can be derived for RTRN as:

$$\frac{\partial }{\partial X}(\frac{\overline{\rho }X}{\overline{\mu }}E[\overline{\zeta }(\overline{H},\overline{k},\overline{\phi },\beta ,L)]\frac{\partial P}{\partial X})=\frac{8\pi }{W}X\frac{\partial (\overline{\rho }H)}{\partial T}$$

(18)

where

$$\begin{array}{c}E[\overline{\zeta }(\overline{H},\overline{k},\overline{\phi },\beta ,L)]=\frac{35}{32C^7}\hfill \\ {\int }_{-C}^C[{\overline{H}}^3-12\overline{H}{L}^2+24L^3\tanh \left(\frac{\overline{H}}{2L}\right)+\frac{12\overline{k}\overline{\phi }}{(1-\beta )}](C^2-{\overline{\delta }}^2{)}^{3}d\overline{\delta }\hfill \end{array}$$

(19)

For CTRN, the oil film can be expressed as:

$$\overline{h}=h(t)+\delta (\theta ,\xi )$$

(20)

The dimensionless form of the equation (14) can be derived for CTRN as:

$$\frac{\partial }{\partial X}\left(\frac{\overline{\rho }X}{\overline{\mu }}\frac{1}{E[1/\overline{\zeta }(\overline{H},\overline{k},\overline{\phi },\beta ,L)]}\frac{\partial P}{\partial X}\right)=\frac{8\pi }{W}X\frac{\partial (\overline{\rho }H)}{\partial T}$$

(21)

where

$$\begin{array}{c}E[1/\overline{\zeta }(\overline{H},\overline{k},\overline{\phi },\beta ,L)]=\frac{35}{32C^7}\hfill \\ {\int }_{-C}^C{(C^2-{\overline{\delta }}^2)}^3/[{\overline{H}}^3-12\overline{H}{L}^2+24L^3\tanh \left(\frac{\overline{H}}{2L}\right)+\frac{12\overline{k}\overline{\phi }}{(1-\beta )}]d\overline{\delta }\hfill \end{array}$$

(22)

The BC for equations (18) and (21) are:

$$P=0,\mathrm{at}X\to \infty$$

(23a)

$$\frac{\partial P}{\partial X}=0,\mathrm{at}X=0$$

(23b)

$$P\ge 0$$

(23c)

The lubricant viscosity-pressure relation put forward by Roelands et al. ²⁰ can be presented as:

$$\overline{\mu }=\exp \{(\ln {\mu }_0+9.67)[(1+5.1p/{10}^9{)}^{z\prime }-1]\}$$

(24)

The lubricant density-pressure relation put forward by Dowson and Higginson ²¹ were displayed as:

$$\overline{\rho }=1+\frac{0.6p/{10}^9}{1+1.7p/{10}^9}$$

(25)

In EHD point contact, the film thickness can be presented as:

$$\overline{h}={\overline{h}}_0+\frac{r^2}{2R}+\zeta$$

(26)

The ${\overline{h}}_0$ is rigid division, the $r^2/2R$ is geometric profile, the $\zeta$ is elastic distortion. The dimensionless form of the equation (26) can be transformed into:

$${\overline{H}}_i={\overline{H}}_0+\frac{X_i^2}{2}+\overline{\zeta }$$

(27)

The deformation can be calculated as the sum of the deformation contributions from all pressure points j at the discrete points i :

$${\overline{\zeta }}_i=\sum _{j=1}^n{D}_{\mathrm{ij}}{P}_j$$

(28)

The $D_{\mathrm{ij}}$ ^7,9 are influence coefficients.

Due to the constant load must be maintained all the time during the squeeze process, the force balance equation can be presented as:

$$3{\int }_0^{\infty }\mathrm{PX}\mathrm{dX}=1$$

(29)

Results and discussion

To discuss the effects of the CS lubricant between PML and elastic ball with SRN on squeeze EHL motion for fixed load condition. The modified transient stochastic Reynolds equation (18) and (21) with their boundary condition (23a)–(23c), film thickness equation (27), viscosity-pressure relation (24), density-pressure relation (25), and force equilibrium equation (29) must be solved simultaneously. The flow chart for the solution procedure is shown in Figure 2. The finite difference method (FDM), Gaussian elimination method, the explicit method, and Gauss-Seidel iteration method were used to calculate the numerical solutions of film thickness profiles (H) and pressure distributions (P) at each time step on pure squeeze motion, but without asperities contact. The computational parameters used in this research are listed in Table 1. The maximum analysis area initially selected was Xmax = 16.0. Since more than half of the analysis area was cavitation, the Xmax was reduced to half of its initial area, and so on, until Xmax = 2.0. The grid consists of 401 nodes uniformly distributed in the computational domain. The typical problem with $W=1.048\times {10}^{-8}$ , $G=3500$ , $h_{00}=1000\mathrm{nm}$ , L = 0.08, $\overline{\phi }$  = 0.08, K = 0.0127, and C = 0.08 was solved.

Figure 2.

Flow diagram of computational procedure.

Table 1.

Computational data.

Inlet viscosity of lubricant, Pa s	0.04
Inlet density of lubricant, kg/m3	846
Pressure viscosity coefficient (α), 1/GPa	15.91
Pressure-viscosity index (Roelands)	0.4836
G (material parameter)	3500
Elastic modulus of balls, GPa	200
Poisson’s ratio of ball	0.3
Equivalent radius of elastic ball, m	0.02
Density of balls, kg/m3	7850

Figures 3 and 4 show the relative change in the P and H for an elastic sphere approaching a rough porous medium layer with CS lubricant under fixed load condition for different K (0.0127, 0.0191) and $\overline{\phi }$ (0.08, 0.16) values, respectively. The P is very flat with a relatively thick film thickness, but as the H decreases, the P becomes steeper. The peak pressure is always at the center. Figure 3 shows the central pressure (P_c) gradually increases as the central thickness (H_c) decreases (from t = 0.60 ms to t = 5.78 ms, K = 0.0191) until H_c decreases to 0.19 at t = 5.78 ms. After t = 5.78 ms, the peak pressure decreases as the H decreases (from t = 5.78 ms to t = 7.20 ms, K = 0.0191) until the minimum film thickness (H_min) decreases to 0.002 at t = 7.20 ms without asperities contact. Due to the constant load must be maintained (equation (29)), the pressure distribution increases at central region, but the pressure distribution decreases outside central region. The position of the H_min is at the center (X = 0). When elastic deformation occurs, the dimple produces at central region, the position of the H_min is moved to near the periphery of Hertz contact (X = 1). Due to lubricant leaks from H_min position, the dimple becomes smaller at the final period. The lubricants are squeezed into porous layer due to squeeze motion. The greater the K is, the smaller the H is, and the smaller the P is at central region, but the greater the P is outside central region due to the constant load must be maintained. Figure 4 shows the P_c gradually increases as the H_c decreases (from t = 0.6 ms to t = 3.82 ms, $\overline{\phi }$  = 0.16) until H_c decreases to 0.22 at t = 3.82 ms. After t = 3.82 ms, the peak pressure decreases as the H decreases (from t = 3.82 ms to t = 5.38 ms, $\overline{\phi }$  = 0.16) until the minimum film thickness (H_min) decreases to 0.002 at t = 5.38 ms without asperities contact. The position of the H_min is moved out of the center (X = 0). The greater the $\overline{\phi }$ is, the smaller the H is, and the smaller the P is at central region, but the greater the P is outside central region. The pressure and film thickness decrease with increasing permeability parameter (k) and porous layer thickness (φ). The pressure distributions are less concentrated on the central region with PML. Therefore, PML has damper and oil storage effects.

Figure 3.

P and H versus time using two different K.

Figure 4.

P and H versus time using two different φ.

Figure 5 illustrates the P_c, H_c, and H_min versus time for different K. The P_c increases rapidly to a maximum value with time at the early period, and then decreases slowly toward to the 1.0 (Hertzian pressure, P_hertz) with time at the later period. The H decreases rapidly with time at the early period and then decreases slowly with time at the later period. The larger the K is, the smaller the P_c is, and the smaller the H is. The times need to achieve the Pc_max and the P_hertz decrease as the K increases.

Figure 5.

P _c and H versus time using circular RN model for different K.

Figure 6 illustrates the P_c, H_c, and H_min versus time for different $\overline{\phi }$ . The P_c increased rapidly to a maximum value with time at the early period, and then decreased slowly toward to the 1.0 (P_hertz) with time at the later period. The H decreases rapidly with time at the early period and then decreases slowly with time at the later period. The greater the $\overline{\phi }$ is, the smaller the P_c is, and the smaller the H is. The times need to achieve the Pc_max and the P_hertz decrease as the $\overline{\phi }$ increases.

Figure 6.

P _c and H versus time using circular RN model for different $\overline{\phi }$ .

Figure 7 illustrates the central normal squeeze velocity (V_c) and the minimum normal squeeze velocity (V_min) versus different K. at the early period, the magnitude of the V_c and V_min decreased rapidly with time. Due to squeeze and elastic deformation effects, the slope values of the V_c change from negative to positive. The magnitude of the V_c and V_min decreased with decreasing K. The magnitude of V_c is smaller than the magnitude of V_min at first half period. The magnitude of V_c is greater than the magnitude of V_min at second half period.

Figure 7.

V versus T using circular RN model for different K.

Figure 8 illustrates the V_c and the V_min for different φ, at the early period, the magnitude of the V_c and V_min decreased rapidly with time. Due to squeeze and deformation effects, the slope values of the V_c change from negative to positive. The magnitude of the V_c and V_min decreased with decreasing $\overline{\phi }$ . The magnitude of V_c is smaller than the magnitude of V_min at first half period. The magnitude of V_c is greater than the magnitude of V_min at second half period.

Figure 8.

V versus T using circular RN model for different $\overline{\phi }$ .

Figure 9 illustrates the Pc_max and the Hc_min versus $\overline{\phi }$ using the CTRN and the RTRN. The larger the $\overline{\phi }$ is, the smaller the Pc_max is, and the smaller the Hc_min is. The Pc_max of the RTRN are slightly larger than that of the CTRN under the same $\overline{\phi }$ condition. The Hc_min of the CTRN are greater than that of the RTRN under the same $\overline{\phi }$ condition.

Figure 9.

Pc_max and Hc_min versus φ for different RN type.

Figure 10 illustrates the Pc_max and the Hc_min versus K using the CTRN and the RTRN. The larger the K is, the smaller the Pc_max is, and the smaller the Hc_min is. The Pc_max of the RTRN are slightly greater than that of the CTRN under the same K condition. The Hc_min of the CTRN are greater than that of the RTRN under the same K condition. The CTRN has oil resistance effect. The RTRN has oil guide effect.

Figure 10.

Pc_max and Hc_min versus K for different RN type.

Conclusion

A numerical method was developed to calculate the effects of the CS lubricants between PML and elastic ball with SRN on squeeze EHL motion for fixed load condition. The conclusions include:

1. The larger the K and $\overline{\phi }$ are, the smaller the H is, and the smaller the P is at central region, but the greater the P is outside central region.

2. The larger the K and $\overline{\phi }$ are, the smaller the P_c is, and the smaller the H is. The times need to achieve the Pc_max and the P_hertz decrease as the K and $\overline{\phi }$ increase.

3. Due to squeeze and elastic deformation effects, the slope values of the V_c change from negative to positive. The magnitude of the V_c and V_min decreased with decreasing K and $\overline{\phi }$ .

4. The larger the K and $\overline{\phi }$ are, the smaller the Pc_max is, and the smaller the Hc_min is.

5. For the same K and $\overline{\phi }$ conditions, the Hc_min of the CTRN are larger than that of the RTRN. The Pc_max of the RTRN is slightly larger than that of the CTRN.