# Contaminant Particle Motion in Lubricating Grease Flow: A Computational Fluid Dynamics Approach

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## Abstract

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## 1. Introduction

## 2. Method

#### 2.1. Particle Migration Model by Baart et al. [11]

_{p}the particle density, ρ

_{g}the grease density, $U$ the circumferential velocity and r the particle radial position (see Figure 1, Figure 2 and Figure 3). The drag force is described by the Stokes law:

_{p}is the particle mass and a

_{r}the particle acceleration. Considering the leakage flow velocity in the axial direction to be significantly lower than the velocity induced by the rotating axis, only the radial velocity component is considered. For the influence of a leakage flow on a contaminant particle concentration inserted into the domain, please see Westerberg et al. [14]. Further, the particle acceleration is set to zero as the grease viscosity is very high and the contaminant particles are very small, resulting in long migration times. Applying Equations (1) and (2) in Equation (3) and solving for the circumferential velocity component then yields:

_{i}and r

_{o}are the inner and outer cylinders, respectively, and ${U}_{s}$ is the shaft speed. To obtain a solution for the non-Newtonian grease flow, Equation (5) has been modified such that:

#### 2.2. Numerical Modeling of Particle Position in the Grease Pocket

#### Flow and Migration Model

_{c.r}) produced by the rotating shaft and the drag force (F

_{d.r}) of the grease flow. Under these conditions, the particles experience forces in the radial (r) and tangential (z) directions as presented in Section 2.1.

**u**the velocity, ρ the fluid density, p the pressure and

**I**the identity matrix. The term in brackets multiplied with the viscosity is the general expression for the shear rate. For the particle migration model, the field variables change over time as the particle position does, meaning a time-dependent solver is chosen. Within the particle tracing module, the setting to include out-of-plane degrees of freedom is activated to consider the shaft rotation velocity ${U}_{s}$ on the particle model. The particle density and diameter are set according to the values in Table 1. As implemented in Comsol, the Stokes drag force law reads:

**v**is the particle velocity,

**u**the flow velocity and τ the particle velocity response (or relaxation) time:

_{p}is the particle density and d

_{p}the particle diameter. The right-hand side of Equation (9) can easily be shown to equal 6πηa(

**u**−

**v**) by expressing the mass using the density, a being the particle radius. This expression is perhaps better known as the Stokes law. In Comsol (Equation (9)), the centrifugal force is inherently considered, meaning the same governing equations are used as in Baart et al. [11] to describe the particle motion. The particle entrance flow is setup to inject one unique particle when the simulation starts. The initial radial position of the particle is at the rotating shaft (Figure 2), and the particle initial velocity is set from the module results of the grease flow model. The initial vertical position along the shaft boundary is varied depending on the case studied as shown later on in the paper. The housing wall has a frozen boundary condition, meaning the particle position and the velocity remain fixed once the particle has made contact with the housing.

_{phi}, which activates the Comsol swirl flow option. The fluid is described as a single-phase four-parameter Herschel–Bulkley (H-B) fluid with the density and rheological parameters presented in Table 2. The H-B model is considered as it is used in the model by Baart et al. [11]. Comsol enables a user-defined dynamic viscosity using the built-in variable spf.sr representing the shear rate in the flow. In terms of the dynamic viscosity, the four-parameter Herschel–Bulkley rheology model reads η = τ

_{y}/γ + K∙γ

^{n}

^{−1}+ η

_{bo}, where τ

_{y}is the yield stress value, γ the shear rate, K the grease consistency, n the shear thinning (n < 1) (or thickening, n > 1) parameter and η

_{bo}the base oil viscosity. The values for these parameters are presented in Table 2. When modelling the flow numerically, the yield stress is a specific problem, as in the H-B model, it implies that the grease starts to deform (flow) when the shear stress in the flow is greater than the yield stress value; i.e., it is a discontinuous transition from not moving to moving grease. Numerically, we need to describe the transition in order for it to be continuous. Another numerical problem may arise in the limit of very small shear rates as the denominator in the expression for the H-B model will be very small, leading to convergence problems. The expression for the dynamic viscosity has then been described as (recalling that in Comsol spf.sr is the built-in shear rate value):

_{phi}) to represent the shaft velocity ${U}_{s}$. The housing pocket walls are set up with a no slip boundary condition. For avoiding singularities, a pressure point constraint of 0 Pa at the edge (corners in the axisymmetric geometry) of the housing wall is also set.

## 3. Results and Discussion

#### 3.1. Velocity Field and Velocity Profiles in the Wide and Narrow Grease Pocket

#### 3.2. Particle Migration in the Wide Grease Pocket

#### 3.3. Particle Migration in the Narrow Grease Pocket

**u**= u

_{ϕ}(r)

**a**

_{ϕ}, where

**a**

_{ϕ}is the unit vector in the circumferential direction. This is in every aspect correct for the wide channel (ideal 1D flow) as the lateral boundaries will not influence the flow, and hence, the velocity field only has one component (ϕ), which is only dependent on the radial coordinate. The µPIV measurements used for the wide pocket migration modeling enable measurements of the velocity at a plane determined by the focal plane of the microscope. This means that the motion/flow in and out of the plane is not captured, which in turn yields that the velocity profile used in the migration model by Baart et al. [11] for the wide channel implies a migration model valid in this particular plane; physically, this can be viewed as the radial component contribution to the migration. Important to point out is that this indeed is a valid and relevant model, but it can be considered as a lowest order solution, as the full velocity field is not solved, meaning the contribution from components other than the radial component are not included. Continuing with the migration results from the numerical model, it is clear that in the narrow pocket, the migration time is several orders of magnitude smaller than in the wide pocket. Reconnecting to the velocity profile in the narrow pocket versus the corresponding profile in the wide pocket (Figure 4), it may seem contradictory with a faster migration in the narrow pocket as the velocity is lower. However, the velocity profiles in Figure 4 are valid for the location of the actual plane as denoted in the figure (dashed red line in Figure 3); i.e., the vertical distance from the lateral wall. With gradients in the flow in other spatial directions, the particle will be subject to a flow field dependent also on the z-coordinate, inducing drag force components in other directions than the radial direction. In Figure 7, the particle positions (r- and z-components) for different start positions in the wide pocket are shown. Interestingly, for other vertical start positions than between the lateral walls (F2 = 1 mm; F2 represents the plane for which the µPIV measurements have been made), the particle does not reach the stationary housing before hitting the lateral wall as shown by the z-coordinate of the particle. Additionally, with a no-slip boundary condition (

**u**= 0), the particle is stuck at the lateral wall once reaching it. Gradients in the flow induced by the lateral walls can hence be concluded to influence the particle migration except at the mid-plane between the lateral boundaries (z = 1 mm). With a parabolic velocity distribution (Figure 4), the maximum velocity is at the mid-plane between the lateral walls, and with that start position being the only position from which the particle manages to reach the stationary housing, it can be argued that that is the reason for the shorter migration time in the narrow pocket compared to the results by Baart et al. [11]. Furthermore, the velocity profile from µPIV measurements used by Baart et al. cannot be from a plane as far into the flow as 1 mm due to the limited transparency of the grease. According to Baart et al., the location of the plane is approximately 0.2 mm from the lateral wall, represented by the transparent window in the DRS geometry. This supports the observation of a longer migration time from the model by Baart et al. as the velocity is significantly lower, so close to the lateral boundary as 0.2 mm. As shown in Figure 7, the trend is clear with the particle hitting the lateral wall earlier as the initial distance from the lateral wall decreases. Relating this to the model by Baart et al., their model indeed is relevant for comparisons with the numerical results, as the physics close to the wall is not resolved in the numerical model. There, the rheology of the grease is assumed to be homogeneous throughout the whole grease pocket. However, as shown, e.g., by Westerberg et al. [4], there is a heterogeneous region present close to the solid boundaries where the shear rate is high, enabling particle migration due to, e.g., wall slip or shear banding caused by a locally different grease rheology and phase composition.

## 4. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the DRS geometry developed by Green et al. [12] to visualize velocity profiles in the grease pocket using micro Particle Image Velocimetry (µPIV). The setup enables measurements of the three flow components. F1–3 and B1–3 symbolize possible measurement planes. The thick red arrow indicates the direction of the leakage particle motion [14] (not covered in the present study), while the dashed black line is the rotating shaft, and the solid black lines are the stationary housing. The solid black lines also denote the lateral boundaries.

**Figure 2.**Comsol 2D axisymmetric geometry model of the narrow pocket. The blue boundaries represent the stationary boundaries.

**Figure 3.**NLGI2 grease velocity field, wide pocket (

**a**) and narrow pocket (

**b**). The red dashed lines illustrate the planes at which the velocity profiles in Figure 4 are. The white dot is the starting position for the particle in the migration simulation.

**Figure 4.**NLGI2 grease velocity profile for the narrow and wide pocket with a shaft speed velocity of 1 m/s. The velocity profiles for the narrow pocket (positions 0.2 mm and 1 mm) were also validated by Westerberg et al. [14].

**Figure 5.**Particle migration in the NLGI00, NLGI1 and NLGI2 greases. The dashed lines are the results from Baart et al.’s model [11] and the solid lines the numerical results. The initial position for the particle for the numerical model is shown by the white dot in Figure 3b. Blue: NLGI2, black: NLGI1, and red: NLGI00.

**Figure 7.**Particle motion of the NLGI2 grease (r- and z-components) in the narrow grease pocket for different start locations along the rotating shaft, calculated from the lateral boundary. One millimeter equals a start position at the mid-plane between the two lateral boundaries.

**Figure 8.**Number of particle revolutions (in degrees) due to the circumferential velocity as a function of time (hours). Dashed line: wide pocket. Solid line: narrow pocket.

**Table 1.**Test parameters from Baart et al. [11].

a (µm) | ρ_{p} (kg/m^{3}) | ${\mathit{U}}_{\mathit{s}}$ (m/s) | Temperature Test (°C) |
---|---|---|---|

7 | 2100 | 1 | 25 |

**Table 2.**Grease rheological parameters based on the three-parameter Herschel–Bulkley rheological model. From Baart et al. [11]. The parameters are (from left to right): the yield stress, grease consistency, shear thinning exponent, base oil viscosity and density. α is the fitting parameter in Equation (6).

NLGI ^{1} grade | τ_{yield} (Pa) | K (Pa.s) | n | ƞ_{bo} (Pa.s) | ρ_{g} (kg/m^{3}) | α (m^{−1}) |
---|---|---|---|---|---|---|

00 | 15 | 12 | 0.63 | 0.89 | 890 | −1000 |

1 | 260 | 61 | 0.42 | 0.49 | 910 | −2000 |

2 | 500 | 8.2 | 0.63 | 0.25 | 930 | −3000 |

^{1}The National Lubricating Grease Institute.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Westerberg, L.-G.; Farré-Lladós, J.; Sarkar, C.; Casals-Terré, J.
Contaminant Particle Motion in Lubricating Grease Flow: A Computational Fluid Dynamics Approach. *Lubricants* **2018**, *6*, 10.
https://doi.org/10.3390/lubricants6010010

**AMA Style**

Westerberg L-G, Farré-Lladós J, Sarkar C, Casals-Terré J.
Contaminant Particle Motion in Lubricating Grease Flow: A Computational Fluid Dynamics Approach. *Lubricants*. 2018; 6(1):10.
https://doi.org/10.3390/lubricants6010010

**Chicago/Turabian Style**

Westerberg, Lars-Göran, Josep Farré-Lladós, Chiranjit Sarkar, and Jasmina Casals-Terré.
2018. "Contaminant Particle Motion in Lubricating Grease Flow: A Computational Fluid Dynamics Approach" *Lubricants* 6, no. 1: 10.
https://doi.org/10.3390/lubricants6010010