# Thermal and Surface Roughness Effects on Performance Analysis of Slider Bearing via Petrov–Galerkin Finite Element Method

Authors: Girma Desu Tessema; Getachew Adamu Derese; Awoke Andargie Tiruneh
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, the performance of slider bearing with impacts of temperature and roughness of surface on 1D longitudinal and transverse roughness type is investigated using the Streamline Upwind Petrov–Galerkin (SUPG)-finite element method (FEM). It is considered that the roughness is stochastic and Gaussian in random distribution. It is also considered that viscosity and density depend upon temperature. The domain’s irregularity caused by surface roughness is changed into a regular domain for numerical computation purposes. To determine the capacity of load-carrying and pressure distribution, the continuity, momentum, modified Reynolds, and energy equations are decoupled and solved by the SUPG-FEM. It can be demonstrated that in the case of the longitudinal model of nonparallel slider bearing, the load-bearing bearing capacity and drag force of friction due to the casei of the combined effects are lower than those attributable to the influence of surface roughness. Nevertheless, the thermal case’s impact on the 1D transverse type capacity of load-carrying capability and drag friction force is below that of the combined and surface roughness impact instances. The reverse is true for parallel slider bearings, however, for a 1D transverse type, the load-carrying ability is not particularly impressive.


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## Body

## 1. Introduction

The plane slider thrust hydrodynamic bearing (HD) is one of the most common hydrodynamic bearing that is to support rotating shafts.It is widely used in many types of machinery due to its durability, stability, and high capacity of load-carrying. It is believed that load-carrying performance is influenced by so many factors, such as surface roughness, heat conduction through the stationary and moving solids, and viscosity and density of the fluid.Many scholars, such as Lewicki [1], Young [2], and Lebeck [3] have experimentally investigated the effect of thermal on the capacity of the load carrying over parallel slider bearings.Zienkiewicz [4] examine that pad and slider at different temperatures and analytically showed for a parallel slider bearing that if the slider temperature is less than the pad temperature then a suctional effect, which may lead to a drastic fall in load-carrying capacity, is possible.Christensen and Tonder [5] analyzed three different models of hydrodynamic lubrication of an inclined slider bearing with rough surfaces. The first model is associated with longitudinal and one-dimensional roughness, the second is related to one-dimensional transverse roughness, and the third deals with the case of uniform, isotropic roughness. Using a stochastic approach, they developed a modified Reynolds equation applicable to each of these models. These were subsequently used to analyze the behavior of a fixed pad slider bearing with no side leakage. Using the same approach Christensen [6] considered two types of roughness: longitudinal and transverse. Christensen et al. [7] derived a general form of the Reynolds equation using the same approach.Many researchers have studied different types of bearings with roughness effect such as the study of hydrostatic bearings by Lin [8], study of journal bearing by Guha [9], and the slider bearing by Christensen and Tonder [10]. A segregated FEM of the Petrov–Galerkin framework with suitably SUPG weight functions for a nonisothermal flow with temperature dependent density and viscosity in a high speed slider bearing has been carried out by Kumar [11]. A numerical simulation of slider bearing of load-carrying support using the SUPG-FEM has been studied by Rathish Kumar and Rao [12].Sinha and Adamu [13, 14] analyzed the thermo hydrodynamic analysis of an infinitely long tilted pad slider bearing with roughness and also considering heat conduction through both the pad and slider by the finite difference method, by taking two models of one-dimensional longitudinal and transverse roughness on the assumption that roughness is assumed to be stochastic and Gaussian randomly distributed.The effect of surface roughness on the performance of hydrodynamic slider bearings is being investigated by Andharia et al. [15]. The bearing surface topography is considered to be described by a generalized form of surface roughness characterized by a stochastic random variable with a nonzero mean, variance, and skewness. Various film shapes are studied, including flat, exponential, secant, and hyperbolic slider. To improve the geometry of slider bearings, Sulaiman and Abdallah [16] utilize a Thermo hydrodynamic bearing model. Thakkar et al. [17] investigate the behavior of a transversely rough narrow width tapered pad bearing, taking into account the stochastic model for the effect of surface roughness.Naduvinamani and Siddangouda [18] analyzed the combined effects of surface roughness and couple stresses on squeeze film lubrication between porous circular stepped plates. This theory has been widely used to investigate the effects of couple stresses on the performance of a different type of fluid film bearings, such as slider bearings Lu et al. [19, 20]. Rao and Agarwal [21] taking the same parameter as Naduvinamani and Siddangouda [18] conducted a theoretical study and analyzed the comparison of porous structures of a parameter on the performance of slider bearing. Surface roughness is mathematically modeled by a stochastic random variable. The globular sphere model of Kozeny–Carman and Irmay’s capillary fissures model have been subjected to investigations.Recently, Siddangouda et al. used [22] Christensen’s stochastic theory for the lubrication of rough surfaces, to derive an extended stochastic Reynolds-type equation and to investigate the effect of surface roughness on the static properties of an inclined plane slider-bearing lubricated with Rabinowitsch fluid. They are considered the two forms of one-dimensional roughness patterns (longitudinal and transverse). In order to generate an equation for pressure, load-carrying capacity, frictional force, and coefficient of friction. The effects of deterministic roughness and small elastic deformation of the surface on flow rates, coefficient of friction, and load capacity in Rayleigh Step bearing under thin-film lubrication were analyzed by Kumar et al. [23]. In addition, Kumar et al. [24] studied the effect of stochastic roughness on the performance of a Rayleigh step bearing operating under Thermo-Elastohydrodynamic lubrication (Thermo-EHL). The shear flow factor along with a pressure correction factor has been introduced in the generalized Reynolds equation of the bearing. The mathematical model obtained has been solved by applying Progressive Mesh Densification (PMD) as an efficient method for iteration.Nevertheless, the combined impact of temperature and surface roughness on tilted pad slider bearings by SUPG-FEM has not been that much analyzed. Thus, in this study, the combined impact of temperature and surface roughness on a tilted pad slider bearing will be numerically analyzed using SUPG-FEM.


## 2. Basic Equations

The geometry of a rough, infinitely long slider bearing is shown below in Figure1. When compared to the fluid film thickness’s height h, it is expected that the bearing’s length is quite large.Figure 1
Geometry of one-dimensional roughness slider bearing.The developed equations to express the flow of lubricants in bearings result from simplifications of the governing equations of fluid flow.Reynold equation of lubricant pressure in the classical theory of lubrication was formulated by Osborne Reynold over a century ago. The Reynolds lubrication equation is obtained on the following assumptions given in Hooke [25].(i)
The fluid is Newtonian lubricant(ii)
The fluid film thickness is significantly less than other bearing dimensions(iii)
Inertia of the fluid is negligible(iv)
Body force of the fluid is negligible(v)
The lubricant is incompressible(vi)
The fluid flow is Laminar and no slip at the bearing surfacesConsidering the prior suppositions, we derive the governing equations.


### 2.1. Equations of Momentum and Continuity Respectively

(1)∂∂yμ∂u∂y=dpdx,(2)∂∂xρu+∂∂yρv=0.To obtain the general Reynolds equation by combining Eqns, (1) and (2) using the boundary conditions u=U,v=0 on the slider, and u=v=0 on the pad. Replacing ρ by ρav and μ by μav gives, the generalized modified Reynolds equation:(3)∂∂xρavHt36μavdpdx=U∂∂xρavHt,where Ht is the thickness of the lubricant film.In addition to the above-given lubrication assumptions, the following two assumptions are considered:(1)
Conduction terms are negligible other than those across the fluid film(2)
Specific heat and thermal conductivity are constant


### 2.2. Equation of Energy Becomes

(4)ρcu∂T∂x+v∂T∂y=k∂2T∂y2+μ∂u∂y2+u∂p∂x.Density and Viscosity are assumed to be temperature dependent and expressed by the following equations:(5)ρ=ρ01−λT−Ta,ρav=ρa1−λTav−Ta,where λ is the coefficient of thermal expansion and the subscript a indicates an ambient condition, ρ and ρa are the density at temperature T and ambient temperature Ta, respectively.(6)μ=μ0exp−βT−Ta,μav=μaexp−βTav−Ta,where μ and μa are the viscosity at temperature T and Ta, respectively, and β is the temperature-viscosity coefficient at ambient pressure.In the stochastic theory of an isothermal HD lubrication of rough surface bearings with rapidly varying quantity developed by Christensen and Tonder [5], and Christensen [6], a Reynolds-type equation in the mean pressure as applicable to rough surface bearings was formulated by considering the film thickness as a stochastic process, an ergodic (stationary).The lubricant film thickness is considered as the geometry of two parts:(7)Htx,z,ϵ=h0x+δx,z,ϵ,where delta is a randomly variable quantity with zero mean that results from the surface roughness measured from the nominal level and h0 is the nominal (smooth) part that measures the large-scale part of the film geometry including any long wavelength disruptions. According to this idea, the film thickness must be of a certain type in order for the Reynolds equation to be applied correctly. This prerequisite is fundamental and has little to do with considering film thickness to be a stochastic process. To obtain a basis for the application of stochastic theory, as the detail is given in Christensen and Tonder [5, 6] and Sinha and Adamu [13].The magnitude of the average temperatureTav associated to roughness is negligible when compared to the general average temperature Tav of the bearing. As a result, the magnitudes of ρav and μav associated to roughness are also small relative to the general magnitudes of ρav and μav in the bearing, respectively. The theory is applied to two types of one-dimensional roughness patterns: longitudinal and transverse. In consideration of the longitudinal roughness model, the roughness is assumed to have the form of long, furrows, and narrow ridges in the sliding (x-direction). Thus, the film thickness can be described as a function of the following form:(8)Htx,z,ϵ=h0x+δz,ϵ.In a similar way, in the transverse roughness model, the roughness is assumed to have the form of long, narrow ridges, and furrows running in the direction perpendicular to the sliding. Thus, the thickness of the film can be described as a function of the following form:(9)Htx,z,ϵ=h0x+δx,ϵ.Taking the expected values in both sides of the Reynolds equation, obtained that(10)∂∂xEρavHt36μavdpdx=U∂∂xEρavHt,where Es is the expectancy operator defined by the following equation:(11)Es=∫−∞∞sfsds.and fs is the probability density distribution for the stochastic variables.


## 2.1. Equations of Momentum and Continuity Respectively

(1)∂∂yμ∂u∂y=dpdx,(2)∂∂xρu+∂∂yρv=0.To obtain the general Reynolds equation by combining Eqns, (1) and (2) using the boundary conditions u=U,v=0 on the slider, and u=v=0 on the pad. Replacing ρ by ρav and μ by μav gives, the generalized modified Reynolds equation:(3)∂∂xρavHt36μavdpdx=U∂∂xρavHt,where Ht is the thickness of the lubricant film.In addition to the above-given lubrication assumptions, the following two assumptions are considered:(1)
Conduction terms are negligible other than those across the fluid film(2)
Specific heat and thermal conductivity are constant


## 2.2. Equation of Energy Becomes

(4)ρcu∂T∂x+v∂T∂y=k∂2T∂y2+μ∂u∂y2+u∂p∂x.Density and Viscosity are assumed to be temperature dependent and expressed by the following equations:(5)ρ=ρ01−λT−Ta,ρav=ρa1−λTav−Ta,where λ is the coefficient of thermal expansion and the subscript a indicates an ambient condition, ρ and ρa are the density at temperature T and ambient temperature Ta, respectively.(6)μ=μ0exp−βT−Ta,μav=μaexp−βTav−Ta,where μ and μa are the viscosity at temperature T and Ta, respectively, and β is the temperature-viscosity coefficient at ambient pressure.In the stochastic theory of an isothermal HD lubrication of rough surface bearings with rapidly varying quantity developed by Christensen and Tonder [5], and Christensen [6], a Reynolds-type equation in the mean pressure as applicable to rough surface bearings was formulated by considering the film thickness as a stochastic process, an ergodic (stationary).The lubricant film thickness is considered as the geometry of two parts:(7)Htx,z,ϵ=h0x+δx,z,ϵ,where delta is a randomly variable quantity with zero mean that results from the surface roughness measured from the nominal level and h0 is the nominal (smooth) part that measures the large-scale part of the film geometry including any long wavelength disruptions. According to this idea, the film thickness must be of a certain type in order for the Reynolds equation to be applied correctly. This prerequisite is fundamental and has little to do with considering film thickness to be a stochastic process. To obtain a basis for the application of stochastic theory, as the detail is given in Christensen and Tonder [5, 6] and Sinha and Adamu [13].The magnitude of the average temperatureTav associated to roughness is negligible when compared to the general average temperature Tav of the bearing. As a result, the magnitudes of ρav and μav associated to roughness are also small relative to the general magnitudes of ρav and μav in the bearing, respectively. The theory is applied to two types of one-dimensional roughness patterns: longitudinal and transverse. In consideration of the longitudinal roughness model, the roughness is assumed to have the form of long, furrows, and narrow ridges in the sliding (x-direction). Thus, the film thickness can be described as a function of the following form:(8)Htx,z,ϵ=h0x+δz,ϵ.In a similar way, in the transverse roughness model, the roughness is assumed to have the form of long, narrow ridges, and furrows running in the direction perpendicular to the sliding. Thus, the thickness of the film can be described as a function of the following form:(9)Htx,z,ϵ=h0x+δx,ϵ.Taking the expected values in both sides of the Reynolds equation, obtained that(10)∂∂xEρavHt36μavdpdx=U∂∂xEρavHt,where Es is the expectancy operator defined by the following equation:(11)Es=∫−∞∞sfsds.and fs is the probability density distribution for the stochastic variables.


## 3. Longitudinal Roughness

As the detail is expressed by Sinha and Adamu in [13], the derived modified Reynolds equation for 1D longitudinal roughness becomes the following equation:(12)∂∂xρ¯av1μ¯avh3+3hσ2dp¯dx=6U∂∂xρ¯avh.According to Sinha and Adamu in [13] μ, ∂u/∂y, ρ, u, and v are independent random variables. The expected value of equations (1) and (2), becomes the following equation:(13)∂∂yμ¯∂u¯∂y=dp¯dx,(14)∂∂xρ¯u¯+∂∂yρ¯v¯=0.The expected value for the energy equation according to Sinha and Adamu in [13] obtained as follows:(15)ρ¯cu¯∂T¯∂x+v¯∂T¯∂y=k∂2T¯∂y2+μ¯∂u¯∂y2+u¯∂p¯∂x.


## 4. Transverse Roughness

Using the above-given assumptions a, b, and c, in addition to the details is given by Sinha and Adamu [13], the modified Reynolds equation for transverse roughness becomes the following equation:(16)∂∂xρ¯avμ¯avh3−6hσ2dp¯dx=6U∂∂xρ¯avh1−3σ2h2.The momentum, continuity, and energy equations for transverse roughness will have similar forms as longitudinal roughness model one. As a unit flow is assumed to be of negligible variance, one can approximate the unit heat flow in the slider direction to be of negligible variance. As a result, the temperature gradient in the slider direction becomes a random variable with negligible variance. Therefore, the transverse roughness energy equation will also have the same form as the one obtained in the longitudinal roughness case.The following nondimensional variables according to Sinha and Adamu [13] have been used:(17)u⊛=u¯U,v⊛=v¯BUhi,x⊛=xB,y⊛=yhi,T⊛=T¯Ta,Tav⊛=T¯avTa,μ⊛=μ¯μa,μav⊛=μ¯avμa,ρav⊛=ρ¯avρa,ρ⊛=ρ¯ρa,p⊛=p¯hi2μaUB,Pe=ρaUchi2kB,σ⊛=σhi,p¯i⊛=p¯ihi2μaUB,h⊛=hhi,β⊛=βTa,λ⊛=λTa,PrEc=μaU2kTa,δ⊛=δhi.+ρav⊛β⊛dTav⊛dx†h⊛3+3h⊛σ⊛2d⋔jdx†dΩPj=∫Ωe6Niμav⊛ρav⊛dh⊛dx†−λ⊛h⊛dTav⊛dx†dΩ,General nondimensional governing equations expressed in (12–16) can be rewritten, as follows, respectively.Generalized modified Reynolds equation for longitudinal roughness model:(18)∂∂x⊛ρav⊛h⊛3+3h⊛σ⊛26μav⊛dp⊛dx⊛=∂∂x⊛ρav⊛h⊛.Generalized modified Reynolds equation for transverse roughness model:(19)∂∂x⊛ρav⊛h⊛3−6h⊛σ⊛26μav⊛dp⊛dx⊛=∂∂x⊛ρav⊛h⊛1−3σ⊛2h⊛2,(20)∂∂y⊛μ⊛∂u⊛∂y⊛=dp⊛dx⊛,(21)∂∂x⊛ρ⊛u⊛+∂∂y⊛ρ⊛v⊛=0,(22)ρ⊛u⊛∂T⊛∂x⊛+v⊛∂T⊛∂y⊛=1Pe∂2T⊛∂y⊛2+PrEcPeμ⊛∂u⊛∂y⊛2+PrEcPeu⊛dp⊛dx⊛,where(23)ρ⊛=ρ0⊛1−λ⊛T⊛−1.0,μ⊛=exp−β⊛Tav⊛−1.0,ρav⊛=ρ01−λ⊛Tav⊛−1.0,μav⊛=exp−β⊛Tav⊛−1.0,h⊛=1−x⊛1−w,wherew=h0hi≤1.


### 4.1. Boundary Conditions

p¯⊛=0 at x⊛=0 and p¯⊛=0 at x⊛=1, u⊛=1,v⊛=0, on the slider, and u⊛=v⊛=0 on the pad of the bearing.For the energy equation the following temperature boundary conditions are used.T⊛=Ts on the slider, T⊛=Tp on the pad, and T⊛=Ti at inlet x⊛=0, whereCase iTp=Ti=TsCase iiTp>Ts>TiCase iiiTs>Tp>Ti


## 4.1. Boundary Conditions

p¯⊛=0 at x⊛=0 and p¯⊛=0 at x⊛=1, u⊛=1,v⊛=0, on the slider, and u⊛=v⊛=0 on the pad of the bearing.For the energy equation the following temperature boundary conditions are used.T⊛=Ts on the slider, T⊛=Tp on the pad, and T⊛=Ti at inlet x⊛=0, whereCase iTp=Ti=TsCase iiTp>Ts>TiCase iiiTs>Tp>Ti


## 5. Transformation of the Problem

To simplify the numerical computation, the irregular domain is transformed to a regular geometric domain as expressed in Sinha and Adamu [13], following that the lower boundary of the fluid is mapped to y†=0 and the upper boundary of the fluid is mapped to y†=1. The newly transformed coordinate system consists of two spatial variables x† and y†.Since the mapping is essentially with respect to they⊛-axis, x⊛=x† is chosen. Assuming the roughness on the pad and the roughness on the runner to be identical random distributions δ1=δ2, the following linear transformation was chosen:(24)y⊛=y†h⊛+δ1⊛,0≤y†≤1.By using the above-given transformations, the governing equations (18)–(22) become(25)ρav⊛h⊛3+3h⊛σ⊛2d2p⊛dx†2+−λ⊛∂Tav⊛∂x†h⊛3+3h⊛σ⊛2+ρav⊛β⊛∂Tav⊛∂x†h⊛3+3h⊛σ⊛2dp⊛dx†=6μav⊛ρav⊛dh⊛dx⊛−λ⊛h⊛∂Tav⊛∂x†,(26)−λ⊛∂Tav⊛∂x†h⊛3−6h⊛σ⊛2dp⊛dx†+ρav⊛β⊛∂Tav⊛∂x†h⊛3−6h⊛σ⊛2dp⊛dx†+ρav⊛h⊛3−6h⊛σ⊛2d2p⊛dx†2=6μav⊛ρav⊛1+3σ⊛2h⊛2dh⊛dx†−λ⊛h⊛−3σ⊛2h⊛2∂Tav⊛∂x†,(27)1h⊛2μ⊛−β⊛∂T⊛∂y†∂u⊛∂y†+∂2u⊛∂y†2=dp⊛dx†,(28)−λ⊛u⊛∂T⊛∂x†+ρ⊛∂u⊛∂x†+λ⊛h⊛y⊛dh⊛dx†u⊛∂T⊛∂y†−ρ⊛h⊛y⊛dh⊛dx†∂u⊛∂y†−λ⊛h⊛v⊛∂T⊛∂y†+ρ⊛h⊛∂v⊛∂y†=0,(29)ρ⊛u⊛∂T⊛∂x†−1h⊛y†dh⊛dx†∂T⊛∂y†+1h⊛v⊛∂T⊛∂y†=1peh⊛2∂2T⊛∂y†2+prEcμ⊛peh⊛2∂2u⊛∂y†22+prEcu⊛pedp⊛dx†.The detail has been given in Sinha and Adamu [13].The corresponding nondimensional density and viscosity relationships in the new coordinate system are(30)ρ⊛=1.0−λ⊛T⊛−1.0.μ⊛=exp−β⊛T⊛−1.0,(31)ρav⊛=1.0−λ⊛Tav⊛−1.0.μav⊛=exp−β⊛Tav⊛−1.0,where Tav⊛=∫01T⊛dy†.The corresponding boundary conditions are(32)u⊛=1−y†andv⊛=0atx†=0.The nondimensional load-carrying capacityW⊛ and the friction force F⊛ are determined from the following equations, respectively:(33)W⊛=Whi2μaUB2=∫01p⊛dx†,F⊛=FhiμaUB=∫01μ⊛∂u⊛∂y†y†=0dx†.


## 6. Finite Element Formulation

To derive the weak form of the above equations as a start-up first form the weighted integral equations (25)–(29) and apply integration by parts to the terms with second derivatives, using bilinear rectangular elements.LetΩ be divided into Ne bilinear rectangular elements and Ω¯=Ω∪Γ.(34)∪e=1NeΩ¯e=Ω¯and∩e=1NeΩ¯e=∅,where Ne number of rectangular element, Ωe denotes the interior domain of an element and Γe be the boundary of Ωe.The resulting elemental weak form of equations (25)–(29) will be(35)∫Ωe−ρav⊛h⊛3+3h⊛σ⊛2dNidx†dp⊛dx†+Ni−λ⊛∂Tav⊛∂xh⊛3+3h⊛σ⊛2+ρav⊛Niβ⊛∂Tav⊛∂x†h⊛3+3h⊛σ⊛2dp⊛dx†dΩ=∫ΩeNiμav⊛ρav⊛dh⊛dx†−λ⊛h⊛∂Tav⊛∂x†dΩ,(36)∫Ωe−ρav⊛h⊛3−6h⊛σ⊛2dNidx†dp⊛dx†−Niλ⊛∂Tav⊛∂x†h⊛3−6h⊛σ⊛2−ρav⊛β⊛∂Tav⊛∂x†h⊛3−6h⊛σ⊛2dp⊛dx†dΩ=∫Ωe6Niμav⊛ρav⊛1+σ⊛2h⊛2dh⊛dx†−λ⊛h⊛−3σ⊛2h⊛∂Tav⊛∂x†dΩ,(37)∫Ωe−μ⊛h⊛2∂Ni∂y†∂u⊛∂y†−μ⊛βNi∂T⊛∂y†∂u⊛∂y†−Nidp⊛dx†dΩ=0,(38)∫ΩeNi−λ⊛u⊛∂T⊛∂x†+ρ⊛∂u⊛∂x†+λ⊛h⊛y⊛dh⊛dx†u⊛∂T⊛∂y†−ρ⊛h⊛y⊛dh⊛dx†∂u⊛∂y†−λ⊛v⊛∂T⊛∂y⊛+ρ⊛∂v⊛∂y†dΩ=0,(39)∫Ωeρ⊛Niu⊛∂T⊛∂x⊛−y†h⊛dh⊛dx†∂T⊛∂y†+v⊛h⊛∂T⊛∂y†−1peh⊛2∂Ni∂y⊛∂T⊛∂y⊛−PrEcμ⊛PeNi∂u⊛∂y⊛2dΩ−∫ΩePrEcu⊛PeNidp⊛dx⊛dΩ=0,where Ni is weight function. To generalize the formulation it is assumed that the same order of polynomials are used to approximate velocity, pressure, and temperature unknowns, i.e.,(40)u⊛x†,y†≈∑j=1neluj⋔jx†,y†,v⊛x†,y†≈∑j=1nelvj⋔jx†,y†,P⊛x†≈∑j=1nelPj⋔jx†,T⊛x†,y†≈∑j=1nelTj⋔jx†,y†,Where Uj,Vj,Pj and Tj are nodal unknown values and ⋔j are shape (basis) functions that are used to construct the approximate solution. nel is velocity, pressure, and temperature nodes over an element, i.e., nel=2 for P and nel=4 for U,V and T. For GFEM consideration, the selection of weight functions are the same as the shape functions Ni=⋔j.Due to the node-to-node oscillatory solutions of the Galerkin finite element method (GFEM), instead removed Streamline Upwind Petrov–GalerkinSUPG was used. The weighted residual formulation of the above-given equation is obtained by substituting the approximation equation (40) into (35)–(39). Thus, the weak form of P and the SUPG decoupled equations for U,V, and T are shown as follows:(41)∑j=1nel∫Ωe−ρav⊛h⊛3+3h⊛σ⊛2dNidx†d⋔jdx†+Ni−λ⊛dTav⊛dx′h⊛3+3h⊛σ⊛2+ρav⊛β⊛dTav⊛dx†h⊛3+3h⊛σ⊛2d⋔jdx†dΩPj=∫Ωe6Niμav⊛ρav⊛dh⊛dx†−λ⊛h⊛dTav⊛dx†dΩ,∑j=1nel∫Ωe−ρavg⊛h⊛3−6h⊛σ⊛2dNidx†d⋔jdx†+Ni−λ⊛dTav⊛dx†h⊛3−6h⊛σ⊛2+ρav⊛β⊛dTav⊛dx†h⊛3−6h⊛σ⊛2d⋔jdx†dΩPj=∫Ωe6Niμav⊛ρav⊛1+σ⊛2h⊛2dh⊛dx†−λ⊛h⊛−3σ⊛2h⊛∂Tav∂x†dΩ,∑j=1nel∫Ωe−N˜iβ∂⋔k∂y†∂⋔j∂y†∑k=1nelTk−∂Ni∂y†∂⋔k∂y†dΩUj=∫Ωeh⊛2μ⊛d⋔kdx†∑k=1nelPkN˜idΩ,∑j=1nel∫Ωe−⋔zh⊛λ⊛∂⋔j∂y†+ρ⊛h⊛∂⋔j∂y†N˜idΩVj=∫Ωeλ⊛u0⊛∂⋔j∂x†∑k=1nelTk−ρ⊛∂⋔z∂x†∑k=1nelUk+λ⊛h⊛y⊛dh⊛dx†u0⊛∂⋔z⊛∂y†∑k=1nelTk−ρ⊛h⊛y†dh⊛dx†∂⋔z⊛∂y†∑k=1nelUkN˜idΩ,∑j=1nel∫ΩeN˜iρ⊛u0∂⋔j∂x†−u0y†h⊛dh⊛dx†∂⋔j∂y†+v0∂⋔j∂y†+1Pe∂Ni∂y†∂⋔j∂y†dΩTj=∫ΩePrEcμ⊛Pe∂⋔z∂y†∑k=1nelUk2+PrEcPeu0∂⋔z∂x†∑k=1nelPkN˜idΩ,where N˜i is the SUPG weight function expressed as follows:(42)N˜i=⋔i+Si,Si=k¯u˜j⋔i,ju.k˜ is the upwind parameter calculated using elemental dimensions and elemental velocity to the center of quadrilateral elements. The detailed explanation is given by Brooks and Hughes [26].


### 6.1. Treatment of the Solution

The system of equations with a finite number of unknowns was developed and solved by SUPG-FEM. The formulated equations contain nonlinear terms. Due to this nonlinearity, the formulated algebraic system of equations is solved using a directive iterative approach.The approximated solution obtained is to an accuracy of toleranceTol, where the error is calculated as follows:(43)max∣ϕznew−ϕzprev∣∣ϕznew∣<Tol,and ϕz are values of the unknown variables at nodal points.The iteration was carried out withTol=10−3, 10−4 and 10−5. However, there is no significant difference in the approximation. The outcome was accomplished by developing a MATLAB code for MATLAB software version 2021.


### 6.2. General Algorithm

The following algorithm is used to approximate the solution of all field variables.


## 6.1. Treatment of the Solution

The system of equations with a finite number of unknowns was developed and solved by SUPG-FEM. The formulated equations contain nonlinear terms. Due to this nonlinearity, the formulated algebraic system of equations is solved using a directive iterative approach.The approximated solution obtained is to an accuracy of toleranceTol, where the error is calculated as follows:(43)max∣ϕznew−ϕzprev∣∣ϕznew∣<Tol,and ϕz are values of the unknown variables at nodal points.The iteration was carried out withTol=10−3, 10−4 and 10−5. However, there is no significant difference in the approximation. The outcome was accomplished by developing a MATLAB code for MATLAB software version 2021.


## 6.2. General Algorithm

The following algorithm is used to approximate the solution of all field variables.


## 7. Results and Discussion

The slider-bearing parameters in this paper appear to be functions of the dimensionless parametersPe,Pr, and Ec as well as γ⊛, η⊛, w, andσ⊛. The following are the parameter values that were selected in accordance with the Lebeck consideration stated in [27]:U=20m/s,hi=0.00005m,B=0.1m,c=1,926J/kgK,Ta=310K,ρa=897.1kg/m3, μa=0.0174Pas,β=0.035K,λ=0.0012K,k=0.132W/mK. The simulations were run for a range of values for the parameters w,λ⊛,β⊛ and σ⊛ with fixed values for Pe, Pr, and Ec. Only via, ho, where, ho, is the minimum film thickness as determined by Gururajan and Prakash [28], can one experience the effects of roughness. When ho≫3σ, the effect of roughness is insignificant. However, if ho≈3σ, (within the HD limit, i.e., ho>3σ), this could have a major impact on the bearing performance. For reasons of comparison, the parameter roughness of σ⊛ is assumed to be fixed at 0.04, which is 10% and 4% of the minimum film thickness for w=0.4 and w=1.0, respectively. Furthermore, the constants β⊛=10 and λ⊛=0.4 are assumed to be fixed for the same.The pressure distribution, load-bearing capacity, drag friction force, velocity, and temperature performance findings have been examined.Tables and graphs are used to present the solutions. The numerical simulations were run on various grid systems using 10 by 10, 20 by 20, and 25 by 25 grid points to ensure the grid independence of the results. In Figure2, the load-carrying capacity performance for various grids has been compared and displayed. This leads to the conclusion that the 25 by 25 grid scheme produces a grid-independent solution.Figure 2
Performance of load-carrying capacity vs inclination on different grid system.In the result that follows, emphasis has been given to the following conditions:C1: the combined effect of thermal and surface roughnessC2: thermal effect on smooth surfaceC3: pure surface roughness effect (with constant viscosity and density)


### 7.1. 1D Longitudinal Roughness

The distribution of pressure performance caused by the combined impact of temperature and surface texture, the thermal impact, and the pure roughness impact are shown in Figure3. Indicating that load-carrying performance capacity is in the same order as the corresponding condition of pressure, the values of the load carrying for each fixed x are in the order of pressure owing to (C1< C2 < C3). This leads one to the conclusion that taking into account the effects of temperature and surface roughness on load-carrying capacity will improve the bearing design.Figure 3
For longitudinal roughness pressure distributions atTi=Tp=Ts=1.0 and w=0.4.In Figures3 and 4, the performance of load-carrying capacity and friction force of longitudinal and transverse roughness is presented for corresponding to the above conditions. As anyone can easily be seen that there is a decrement (5%) in load-carrying capacity between the surface roughness effect C3 and combined effect C2 with inclination parameter w=0.4. In a similar way, there is a significant (14%) decrement of friction force between C1 and C3 conditions for w=0.4.Figure 4
Dimensionless load-carrying capacities and friction force, for both longitudinal and transverse roughness. (a) Inclination. (b) Inclination. (c) Inclination. (d) Inclination.
(a)(b)(c)(d)This is due to the fluid viscosity and density of the lubricant being reduced in the case of the combined effect. Accordingly, the corresponding capacity of load carrying is decreased. Furthermore, betweenC1 and C2 there is a difference of 3% in the capacity of load carrying and no change in friction force observed for the inclination of w=0.4.In the case of the longitudinal model roughness type, the texture is taken to have the appearance of furrows, long narrow ridges, and valleys in the sliding direction (x-axis), which allows the fast flow of lubricant and consequences in the decrease of pressure distribution, that implies a reduction in the capacity of load-carrying.In consideration of parallel slider bearing atw=1, even if it is small, there is a load-carrying capacity generated in C1 and C2 in comparison with C3, due to fluid expansion as temperature rises in the lubricant. There is no significant variation on the value of temperatures in the middle of C1 and C3 as one can observe from Figures 5 and 6.Figure 5
Contour plot of temperature, due toC2 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (longitudinal roughness).Figure 6
Contour plot of temperature, due toC2 at δ=0.0,w=0.4 and Ti=Tp=Ts=1.0 (longitudinal roughness).


### 7.2. u-Velocity and v-Velocity Profile

Figures7 and 8 show U-velocity profiles for longitudinal and transverse roughness, respectively, and Figures 9 and 10 show the v-velocity profiles for longitudinal and transverse roughness, respectively in the fluid film region of the bearing over combined effect (thermal and roughness consideration). As one can observe from the Figures velocity profile distributions for transverse roughness are similar to longitudinal roughness in both velocity distributions.Figure 7
Contour plot of u-velocity, due toC1 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (longitudinal roughness).Figure 8
Contour plot of u-velocity, due toC1 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (transverse roughness).Figure 9
Contour plot of v-velocity, due toC1 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (longitudinal roughness).Figure 10
Contour plot of v-velocity, due toC1 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (transverse roughness).Zienkiewicz [4] examined sliders and pads at various temperatures and indicates that in parallel slider bearing, if the surface temperature of the pad bearing is higher than the slider temperature, then a suction action may occur. The load-carrying ability may be significantly reduced as a result. Zienkiewicz also restricted his findings to smooth Parallel surfaces in his 1957 paper.According to Pinkus’ mention of the Pinkus 1961 Theory, the study of the thermal effect on the slider and pad at a certain temperature of (Tp>Ts or Tp<Ts) on the load-carrying capacity and friction force of a sliding bearing is of excessively realistic relevance. According to Pinkus and Sternlicht [29], the fixed surface is typically warmer than the moving surface in practice. Consideration of the inlet lubricant temperature being lower than the slider and pad temperatures is another crucial scenario in a slider-bearing application. On the basis of this, the following temperature boundary conditions have been examined for combined thermal and surface roughness effects:Case i:Tp=Ti=1.2andTs=1.3Case ii:Ts=Ti=1.2andTp=1.3Case iii:Ts=1.3,Ti=1.2andTp=1.4Case iv:Ti=1.2,Ts=1.4andTp=1.3In Caseii and Case i, for w=0.4, there is a difference of 12% in the performance of load when the temperature of the pad and slider is different. In contrast, for w=1, Case i’s performance of load is somewhat higher than Case ii’s. There is a significant difference between Case ii and Case i in the case of friction force.Figure11 shows the load-carrying capacity and friction forces for the corresponding scenarios where the intake temperature is lower than the slider and pad temperatures. These figures show that, for each inclination parameter, Case iv’s friction force and load-carrying capacity are lower than Case iii’s if the inlet temperature is lower than that of the pad and slider. According to these various circumstances of consideration, if the slider temperature is higher than the pad temperature, the temperature may decrease, increasing the load-carrying capacity and friction force.Figure 11
Dimensionless load capacities and friction forces for different cases. (a) Inclination. (b) Inclination. (c) Inclination. (d) Inclination.
(a)(b)(c)(d)For a greater grasp, the contour of the temperature of Casesi and Cases ii for w=0.4 are presented in Figures 12 and 13. The load-carrying capacity of Case i is lower than that of Case ii as a result of Case i’s greater average temperature than Case ii.Figure 12
Contour line for casei at w=0.4 (longitudinal roughness).Figure 13
Contour line for caseii at w=0.4 (longitudinal roughness).The load-carrying capacity for various values ofβ is shown in Table 1, assuming Pi=0. It is clear that lubricants with low coefficients of viscosity β perform better in terms of a slider bearing’s ability to carry loads. (31) states that decreasing “β” (for a fixed value of T⊛) raises the viscosity of the fluid lubricant. As a result, lubricants with high viscosity values have greater thermal effect thresholds, which is connected to their ability to carry heavy loads.Table 1
Load-carrying capacity for different value ofβ, at w=0.4 and σ=0.04.
βw⊛100.889680.908160.921840.929620.930000.9314Similar to this, Table2 demonstrates that the performance of the slider bearing’s load-carrying capacity increases when the lubricant viscosity coefficient β lowers, even when the w value varies.Table 2
Load-carrying capacity for different value ofβ, at w=1.0 and σ=0.04.
βw⊛100.893380.910860.923540.930320.930700.9320


### 7.3. 1D Transverse Roughness

The pressure distributions along the entirex-axis caused by various situations and their effects are shown in Figure 14. Pressure distribution in the following order: due to C2, followed by C1, and then C3. According to Figure 4, the pressure distribution (load) follows the same hierarchy as the longitudinal example. For w=0.4, the friction force recorded between C3 and C1 is 5 percent different from C3, and 34 percent different from C1. The difference between C2 and C1 in both parameters, however, is negligible. In consideration of this, it may be said that the roughness effect has a greater bearing than the thermal impact.Figure 14
Transverse roughness pressure distributions atTi=Tp=Ts=1.0 and w=0.4.From Figure4, it can be shown that for w=0.4, the difference in load-carrying capacity between C3 and C2 is 9%. This could be as a result of the transverse roughness pattern, which consists of valleys and ridges running in the direction of sliding (x-direction), which obstructs the fluid flow and increases pressure. In fact, a negligible decrease in load-carrying capability is seen for a parallel bearing w=1. This is because the asperities of the roughness, whose nominal height is substantially bigger than their thickness, have no impact on the unit flow of lubricant. This suggests that a parallel slider bearing’s properties may be more influenced by temperature than surface texture.Pertaining to a slider and a pad of varying temperatures. As shown in Figure11, the findings obtained for longitudinal roughness in cases i−iv are almost identical to those found in the transverse roughness scenario.


## 7.1. 1D Longitudinal Roughness

The distribution of pressure performance caused by the combined impact of temperature and surface texture, the thermal impact, and the pure roughness impact are shown in Figure3. Indicating that load-carrying performance capacity is in the same order as the corresponding condition of pressure, the values of the load carrying for each fixed x are in the order of pressure owing to (C1< C2 < C3). This leads one to the conclusion that taking into account the effects of temperature and surface roughness on load-carrying capacity will improve the bearing design.Figure 3
For longitudinal roughness pressure distributions atTi=Tp=Ts=1.0 and w=0.4.In Figures3 and 4, the performance of load-carrying capacity and friction force of longitudinal and transverse roughness is presented for corresponding to the above conditions. As anyone can easily be seen that there is a decrement (5%) in load-carrying capacity between the surface roughness effect C3 and combined effect C2 with inclination parameter w=0.4. In a similar way, there is a significant (14%) decrement of friction force between C1 and C3 conditions for w=0.4.Figure 4
Dimensionless load-carrying capacities and friction force, for both longitudinal and transverse roughness. (a) Inclination. (b) Inclination. (c) Inclination. (d) Inclination.
(a)(b)(c)(d)This is due to the fluid viscosity and density of the lubricant being reduced in the case of the combined effect. Accordingly, the corresponding capacity of load carrying is decreased. Furthermore, betweenC1 and C2 there is a difference of 3% in the capacity of load carrying and no change in friction force observed for the inclination of w=0.4.In the case of the longitudinal model roughness type, the texture is taken to have the appearance of furrows, long narrow ridges, and valleys in the sliding direction (x-axis), which allows the fast flow of lubricant and consequences in the decrease of pressure distribution, that implies a reduction in the capacity of load-carrying.In consideration of parallel slider bearing atw=1, even if it is small, there is a load-carrying capacity generated in C1 and C2 in comparison with C3, due to fluid expansion as temperature rises in the lubricant. There is no significant variation on the value of temperatures in the middle of C1 and C3 as one can observe from Figures 5 and 6.Figure 5
Contour plot of temperature, due toC2 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (longitudinal roughness).Figure 6
Contour plot of temperature, due toC2 at δ=0.0,w=0.4 and Ti=Tp=Ts=1.0 (longitudinal roughness).


## 7.2. u-Velocity and v-Velocity Profile

Figures7 and 8 show U-velocity profiles for longitudinal and transverse roughness, respectively, and Figures 9 and 10 show the v-velocity profiles for longitudinal and transverse roughness, respectively in the fluid film region of the bearing over combined effect (thermal and roughness consideration). As one can observe from the Figures velocity profile distributions for transverse roughness are similar to longitudinal roughness in both velocity distributions.Figure 7
Contour plot of u-velocity, due toC1 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (longitudinal roughness).Figure 8
Contour plot of u-velocity, due toC1 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (transverse roughness).Figure 9
Contour plot of v-velocity, due toC1 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (longitudinal roughness).Figure 10
Contour plot of v-velocity, due toC1 at δ=0.04,w=0.4 and Ti=Tp=Ts=1.0 (transverse roughness).Zienkiewicz [4] examined sliders and pads at various temperatures and indicates that in parallel slider bearing, if the surface temperature of the pad bearing is higher than the slider temperature, then a suction action may occur. The load-carrying ability may be significantly reduced as a result. Zienkiewicz also restricted his findings to smooth Parallel surfaces in his 1957 paper.According to Pinkus’ mention of the Pinkus 1961 Theory, the study of the thermal effect on the slider and pad at a certain temperature of (Tp>Ts or Tp<Ts) on the load-carrying capacity and friction force of a sliding bearing is of excessively realistic relevance. According to Pinkus and Sternlicht [29], the fixed surface is typically warmer than the moving surface in practice. Consideration of the inlet lubricant temperature being lower than the slider and pad temperatures is another crucial scenario in a slider-bearing application. On the basis of this, the following temperature boundary conditions have been examined for combined thermal and surface roughness effects:Case i:Tp=Ti=1.2andTs=1.3Case ii:Ts=Ti=1.2andTp=1.3Case iii:Ts=1.3,Ti=1.2andTp=1.4Case iv:Ti=1.2,Ts=1.4andTp=1.3In Caseii and Case i, for w=0.4, there is a difference of 12% in the performance of load when the temperature of the pad and slider is different. In contrast, for w=1, Case i’s performance of load is somewhat higher than Case ii’s. There is a significant difference between Case ii and Case i in the case of friction force.Figure11 shows the load-carrying capacity and friction forces for the corresponding scenarios where the intake temperature is lower than the slider and pad temperatures. These figures show that, for each inclination parameter, Case iv’s friction force and load-carrying capacity are lower than Case iii’s if the inlet temperature is lower than that of the pad and slider. According to these various circumstances of consideration, if the slider temperature is higher than the pad temperature, the temperature may decrease, increasing the load-carrying capacity and friction force.Figure 11
Dimensionless load capacities and friction forces for different cases. (a) Inclination. (b) Inclination. (c) Inclination. (d) Inclination.
(a)(b)(c)(d)For a greater grasp, the contour of the temperature of Casesi and Cases ii for w=0.4 are presented in Figures 12 and 13. The load-carrying capacity of Case i is lower than that of Case ii as a result of Case i’s greater average temperature than Case ii.Figure 12
Contour line for casei at w=0.4 (longitudinal roughness).Figure 13
Contour line for caseii at w=0.4 (longitudinal roughness).The load-carrying capacity for various values ofβ is shown in Table 1, assuming Pi=0. It is clear that lubricants with low coefficients of viscosity β perform better in terms of a slider bearing’s ability to carry loads. (31) states that decreasing “β” (for a fixed value of T⊛) raises the viscosity of the fluid lubricant. As a result, lubricants with high viscosity values have greater thermal effect thresholds, which is connected to their ability to carry heavy loads.Table 1
Load-carrying capacity for different value ofβ, at w=0.4 and σ=0.04.
βw⊛100.889680.908160.921840.929620.930000.9314Similar to this, Table2 demonstrates that the performance of the slider bearing’s load-carrying capacity increases when the lubricant viscosity coefficient β lowers, even when the w value varies.Table 2
Load-carrying capacity for different value ofβ, at w=1.0 and σ=0.04.
βw⊛100.893380.910860.923540.930320.930700.9320


## 7.3. 1D Transverse Roughness

The pressure distributions along the entirex-axis caused by various situations and their effects are shown in Figure 14. Pressure distribution in the following order: due to C2, followed by C1, and then C3. According to Figure 4, the pressure distribution (load) follows the same hierarchy as the longitudinal example. For w=0.4, the friction force recorded between C3 and C1 is 5 percent different from C3, and 34 percent different from C1. The difference between C2 and C1 in both parameters, however, is negligible. In consideration of this, it may be said that the roughness effect has a greater bearing than the thermal impact.Figure 14
Transverse roughness pressure distributions atTi=Tp=Ts=1.0 and w=0.4.From Figure4, it can be shown that for w=0.4, the difference in load-carrying capacity between C3 and C2 is 9%. This could be as a result of the transverse roughness pattern, which consists of valleys and ridges running in the direction of sliding (x-direction), which obstructs the fluid flow and increases pressure. In fact, a negligible decrease in load-carrying capability is seen for a parallel bearing w=1. This is because the asperities of the roughness, whose nominal height is substantially bigger than their thickness, have no impact on the unit flow of lubricant. This suggests that a parallel slider bearing’s properties may be more influenced by temperature than surface texture.Pertaining to a slider and a pad of varying temperatures. As shown in Figure11, the findings obtained for longitudinal roughness in cases i−iv are almost identical to those found in the transverse roughness scenario.


## 8. Concluding Remark

The pressure performance distribution of an HD slider lubrication bearing with a rough surface is numerically evaluated in this study using SUPG-FEM. When 1D longitudinal and 1D transverse roughness models are used, the performance of load-carrying capacity is at its peak atw=0.4 in both directions. However, it almost reaches zero for parallel slider bearings. Surface roughness is taken into account in both models, which results in an increase in frictional force, load-carrying capacity, and pressure distribution for nonparallel slider bearings. In addition, it can be demonstrated that, for each surface roughness model, when the temperature of the slider is lower than that of the pad, the capacity of load-carrying and frictional force for slider bearings increase for a nonparallel bearing while dropping for a parallel bearing w=1. This leads one to the conclusion that the texture’s roughness has a significant impact on nonparallel slider bearings. Contrarily, the effects of C1 have a greater impact on a parallel slider bearing than those of C3.



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*Source: 4216136-2022-12-16.xml*