Performance Analysis of Thermal and Surface Roughness Effect of Slider Bearings with Unsteady Fluid Film Lubricant Using Finite Element Method

The streamline upwind Petrov-Galerkin (SUPG) ﬁ nite element method was used in this study to investigate the thermal and surface roughness e ﬀ ects on an inclined slider bearing with an unsteady ﬂ uid ﬁ lm. One-dimensional transverse and longitudinal surface roughness models were considered with the supposition that roughness is stochastic and has a Gaussian random distribution. For simplicity of numerical computation, the irregularity caused by the texture of the surface is transformed into a regular domain. The bearing performance of the combined e ﬀ ect is lower than the thermal and surface roughness e ﬀ ects of the one-dimensional longitudinal surface roughness for all modi ﬁ ed Reynolds numbers of nonparallel slider bearings; this means that for nonparallel ( w = 0 4 ) between the surface roughness e ﬀ ect and the combined e ﬀ ect condition, there is a decrease of 13% in load-carrying capacity performance and a minimal change in friction force, respectively. However, in the case of nonparallel one-dimensional transverse type slider bearings, the bearing performance of the thermal e ﬀ ect is lower than the combined and surface roughness e ﬀ ects for all modi ﬁ ed Reynolds numbers, where between the combined e ﬀ ect and the thermal e ﬀ ect condition, there is a reduction of 19% in load-carrying capacity performance and 2% in friction force practically for all changed Reynolds values, respectively. Furthermore, the combined e ﬀ ects at various temperatures have been investigated. As a result, in both longitudinal and transverse models, in the case of the pad temperature being lower than the slider, the load-carrying capacity performance is higher than in other cases for nonparallel slider bearings, whereas when the slider temperature is lower than the pad temperature, the drag frictional force is the leading one in both models. In general, considering surface texture and inertial e ﬀ ects will increase the performance of a slider. The results obtained are displayed using ﬁ gures and tables.


Introduction
Slider bearings are a type of bearing in which one surface can move over another.They are frequently employed in industrial, agricultural, and automotive applications.Slider bearings can operate at extremely high speeds and can support heavy weights.They are a great option for applications that demand long-term dependability because of their resistance to wear and tear.The geometry of a rough slider bearing is important in many applications, from engineering to the automotive industry.Additionally, the geometry affects the load capacity of the bearing as well as its wear characteristics.Thus, properly understanding the geometry of the bearing is critical for any successful application.
Various bearing types with surface roughness effects have been examined by numerous researchers.Hydrostatic bearings were examined by Lin [1], journal bearings by Guha [2], and slider bearings by Christensen and Tonder [3].For a nonisothermal flow with temperature-dependent density and viscosity in a high-speed slider bearing model, Kumar [4] examined a segregated FEM of the Petrov-Galerkin framework with appropriately SUPG weight functions.Rathish Kumar and Srinivasa Rao [5] examined the use of the SUPG-FEM in the numerical modelling of slider bearings to determine load-carrying support.
Andharia et al. [6] examined how surface roughness affected hydrodynamic slider bearing performance.The bearing surface topography is assumed to be characterized by a stochastic random variable with nonzero mean, variance, and skewness, which is a generalized version of surface roughness.Film shapes such as secant, flat, exponential, and hyperbolic sliders are studied.Alyaqout and Elsharkawy [7] enhance the slider bearing geometry using a thermohydrodynamic bearing model.Thakkar et al. [8] used a stochastic model to account for the impact of surface roughness while examining the behavior of transversely rough narrow-width tapered pad bearings.
A theoretical model was created by Chawla and Bhardwaj [9] to study how couple stress on fluid-lubricated surfaces is affected by surface roughness.Rahman [10] reports that an external transverse magnetic field has been obtained and numerically studied to investigate the effects of unsteady flow and heat transfer in an incompressible laminar, electrically conducting, and non-Newtonian fluid over a nonisothermal stretching sheet with variation in viscosity and thermal conductivity in a porous medium.The unsteady flow of a non-Newtonian fluid over an oscillating vertical porous plate in the presence of a uniform magnetic field has been studied by Ali et al. [11].Using an accurate computer model of the complete set of the Navier-Stokes equations for incompressible flows, the influence of inertial forces in stable and unsteady lubrication films was investigated Sestieri and Piva [12].
The numerical analysis of the time-dependent magnetized micropolar fluid flow over a curved surface was examined by Abbas et al. [13].The impacts of thermal jumped and velocity slip are taken into consideration on the curved surface.
Many scholars have investigated inertial effects, such as Syed and Sarangi [14] on hydrodynamic lubrication with deterministic micro textures, taking the fluid inertia effect into account.Although the Reynolds equation is widely used in thin film lubrication, its application to textured surfaces is not entirely convincing, particularly at moderate to high Reynolds numbers.They have obtained that fluid inertia, which is generally neglected in the Reynolds equation but becomes significant.A perturbation analysis was performed by Ota et al. [15] on a modified Reynolds equation to study the effect of inertia on film rupture in hydrodynamic lubrication.Malvano and Vatta [16] studied the influence of fluid inertia flow on steady laminar lubrication.Prasad et al. [17] examined a theoretical aspect of the hydrodynamic lubrication of two symmetric rollers by power-law fluids.
The thermohydrodynamic analysis of plane slider bearing with roughness was analyzed by Sinha and Adamu [18] using the finite difference method, where surface roughness is assumed to be stochastic and Gaussian randomly distributed.Adamu and Sinha [19] also considered heat conduction through both the pad and slider by taking into account two models of one-dimensional longitudinal and transverse roughness.Singh et al. [20] reported a numerical study of the hydrodynamic lubrication of slider bearings with textured surfaces.Extreme industrial conditions necessitate the use of a bearing that can withstand high-speed operations, heavy loads, and high stiffness, among other things.Alexander Raymand and Jayakaran Amalraj [21] considered the combined effects of fluid inertia forces and non-Newtonian characteristics with the Herschel-Bulkley fluid as a lubricant in an externally pressurized converging thrust bearing.
Recently, Desu Tessema et al. [22] and Tessema et al. [23] using the streamline upwind Petrov-Galerkin (SUPG) finite element method (FEM) examined the performance of slider bearings with the effects of thermal and surface roughness on one-dimensional longitudinal and transverse roughness types under laminar and turbulent conditions.Naduvinamani and Angadi [24] examined the static and dynamic properties of a few stress fluid-lubricated rough, porous Rayleigh step bearings.Naduvinamani and Angadi [25] analyzed the effects of micropolar fluid and roughness on inclined porous slider bearings' dynamic characteristics.Theoretical and numerical analyses were used to investigate the static properties of aerostatic porous journal bearings by Gu et al. [26].Fang et al. [27] discussed transient elastohydrodynamic lubrication under line contact stiffness and damping behaviors.Jamshed et al. [28] used a single-phase optimized entropy analysis to illustrate the thermal efficiency improvement of solar aircraft employing unsteady hybrid nanofluids.Raees et al. [29] examined the transfer of energy in the power-law nanofluid driven to flow along a horizontal wall by magnetization.Overall, our examination of the literature and our knowledge gaps show that SUPG-FEM has not been used to address the performance of slider bearings when surface roughness and coupled heat effects are present along with an unsteady fluid film that takes into account various inertial effects.
Nonetheless, our noteworthy input to this study consists of developing the governing Reynolds equation (found in equations ( 12) and ( 15)) for surface roughness (for onedimensional longitudinal and transverse) with thermal impact for slider bearing.The combined effect of temperature and surface roughness on tilted pad slider bearings with the unsteady fluid flow by SUPG-FEM has received little attention.Thus, the combined effect of temperature and surface roughness on a slider bearing with an unsteady fluid film, considering the inertial effect, will be numerically analyzed in this study using SUPG-FEM.The simplified equations for the flow of lubricants in slider bearings are derived from the governing equations of fluid flow.These equations took into account the viscosity and density of the lubricant and the speed of the bearing, as well as other variables such as the bearing geometry, the applied load, and the temperature of the lubricant.

Governing Equations
The modified Reynolds lubrication equation was developed using the averaged inertia method as given in Hooke [30].This is the most common method for obtaining the extended form of the Reynolds equation, which includes fluid inertia effects.The initial system of equations is replaced with an approximate system of equations obtained by averaging the inertial terms across the fluid film thickness.To derive the governing equation, the following assumptions are considered: the fluid is Newtonian lubricant; the fluid film thickness is significantly less than other bearing dimensions; in fluid flow, the inertial effect is considered; body force is negligible; the flow is laminar; and there is no slip at the bearing surfaces.
Following the above assumptions, the following equations are obtained: Momentum equation: Continuity equation: Eqs. ( 1) and ( 2) are combined to generate the general Reynolds equation for unsteady fluid flow lubricant by applying the boundary conditions u = U and v = 0 on the moveable slider, u = v = 0 on the stationary pad, and substituting ρ by ρ av and μ by μ av , respectively.
The generalized equation of Reynolds has a form of ∂ ∂x

3
where is the mean velocity and H t is the lubricant film thickness.
Along with the lubricating presumption mentioned above, the following were taken into account for the equation of energy: (1) Conduction terms other than those across the fluid film are negligible (2) Constant-specific heat and thermal conductivity are assumed The equation of energy becomes as follows: The following equations indicate temperature-dependent density and viscosity: where λ is thermal expansion coefficient and ρ and ρ a are the density at temperature T and ambient temperature T a correspondingly.
where μ a and μ are the viscosities at temperature T a and T, respectively, at ambient pressure and β is the temperatureviscosity coefficient.An isothermal HD lubrication of rough surface bearings of stochastic theory with rapidly varying quantity developed by Christensen and Tonder [31] and Christensen [32], a Reynolds type equation in the average pressure to rough 3 Journal of Applied Mathematics surface bearings was formulated by taking the film thickness as a stochastic process.
The thickness of the lubricant film considered in this work on the slider bearing has a geometry of two parts: where h 0 is the nominal (smooth) part, which measures the large-scale part of the film geometry, including any long wavelength disturbances, and δ is a randomly varying quantity with a zero mean, which arises due to the surface roughness measured from the nominal level.In order to deal with a one-dimensional rough surface, this paper uses the following two additional assumptions introduced by Christensen and Tonder [31] and Christensen [32] in order to establish a foundation for the use of stochastic theory: (i) The variance of the pressure gradient of unsteady fluid flow film in the roughness direction is negligible (ii) The variance of unit flow of unsteady fluid flow film is negligible in the direction perpendicular to the roughness.In this analysis, the assumptions of Christensen and Tonder [31] and Christensen [32] will be extended to the HD lubrication of rough surface bearings with temperature-dependent ρ and μ by imposing the following extra assumption for velocity and temperature variables due to Sinha and Adamu [18].Temperature, velocity, and U m magnitudes associated with roughness are negligible in comparison to the corresponding general magnitudes in the bearing.As a result, the variances of ρ, μ, temperature gradients ∂T/∂y, and velocity gradients ∂u/∂y in the direction of roughness are negligible According to assumption (ii), the magnitude of the average temperature T av and U m associated with roughness is negligible.As a result, the magnitudes of ρ av and μ av associated with roughness are small.The theory is applied to longitudinal and transverse roughness patterns.In the longitudinal roughness model, roughness is assumed to take the form of long, narrow furrows and ridges in the sliding surface (x-direction).As a result, the film thickness can be described as a function of the form.
Similarly, the roughness in the transverse roughness model is assumed to take the form of long, narrow ridges and furrows running perpendicular to the sliding direction.
Hence, the film thickness becomes part of the form: Taking the expected values on both sides of the Reynolds equation for unsteady fluid flow film given in Eq. ( 3), E h s is the expectancy operator defined by and f h s is the probability density distribution for the stochastic variable h s .Using the aforementioned assumptions (i) and (ii), as well as some of the details given in Sinha and Adamu [18], the modified Reynolds equation for unsteady fluid flow film of longitudinal roughness is as follows: ∂ ∂x Following assumption (ii), approximations of u and ∂ ρu /∂x, v and ∂ ρu /∂y, μ and ∂u/∂y, ρ and u, and ρ and v are independent random variables, and taking expected values in both sides of Eq. ( 1) and Eq. ( 2), we can get Taking assumptions (i) and (iii) into account, the transformed energy equation will be The transverse roughness modified Reynolds equation for unsteady fluid flow film becomes ∂ ∂x Transverse surface roughness, momentum, and continuity equations will take the same form as one-dimensional longitudinal surface roughness equations of unsteady fluid flow film.

21
where is the modified Reynolds number Boundary conditions are as follows: p ⊛ = 0 at x ⊛ = 0 and p ⊛ = 0 at x ⊛ = 1, u ⊛ = 1, and v ⊛ = 0, on the moveable slider, and u ⊛ = v ⊛ = 0 on the stationary pad of the bearing.
The following boundary conditions for the temperature of unsteady fluid flow are used for the energy equation: T ⊛ = T s on the moveable slider, T ⊛ = T p on the stationary pad, and T ⊛ = T i at the inlet of slider x ⊛ = 0 , where For computation purposes, the irregular domain of the slider bearing for unsteady fluid flow lubricant is transformed into a regular geometric domain, according to Sinha and Adamu [18].Assuming the roughness on the pad and runner to be identical random distributions δ 1 = δ 2 , the following linear transformation was chosen: The decoupled main governing equations of unsteady lubricant of ( 17), ( 18), ( 19), (20), and (21) become where The transformed dimensionless temperature-dependent viscosity and density equations are as follows: where T ⊛ av = 1 0 T ⊛ dy † .The following initial essential boundary conditions are tokens for this article: The dimensionless load-carrying capacity performance W ⊛ and the drag force F ⊛ are obtained from the following equations, respectively:

Finite Element Formulation
In order to obtain the weak form of the aforementioned equations for finite element formulation, we first form the weighted integral equations for ( 26), ( 27), (29), and (30) and apply integration by parts to the terms with second derivatives.Bilinear rectangular elements are used for governing.
Let the geometry of the domain Ω be divided into N e bilinear rectangular elements, and

34
where N e discretized number of rectangular elements, Ω e denotes the interior domain of an element, and Γ e is the boundary of rectangular element Ω e .
The elemental weak form of the governing Eqs. ( 26), ( 27), ( 29), (30), and ( 31) is as follows: 35 36 Journal of Applied Mathematics 38 where N i is the weight (or test) for one-and twodimensional functions.To generalize the finite element formulation, it is assumed that the same order of polynomials is used to approximate velocity, temperature, average temperature, and U ⊛ m unknowns and different for pressure unknown weight function, i.e., where U j , V j , P j , T j , T av , and U m are nodal unknown values and ⋔ j are shape (basis) functions of space variables only that are used to construct the approximate solutions.nel is the pressure, velocity, and temperature number of nodes over an element, i.e., nel = 2 for P, T av , and U m and nel = 4 for U, V, and T. In consideration of GFEM, the selection of weight functions is the same as the shape functions N i = ⋔ j .
Due to the node-to-node oscillatory solutions of the Galerkin finite element method (GFEM), instead, streamline upwind Petrov-Galerkin (SUPG) was used for U, V, and T. The weighted residual formulation of the above equation is obtained by substituting the approximation equation ( 40) into (35), (36), (37), (38), and (39).Thus, the weak integral form of P and the SUPG decoupled governing equations for U, V, and T are as follows:

45
where N i is the SUPG weight (or test) function expressed as k is the SUPG upwind parameter calculated using elemental dimensions and elemental unknown to the center of quadrilateral elements.The most detailed explanation is given by Brooks and Hughes [33].Since we are using a linear shape and weight function, terms of T ⊛ av and U ⊛ m in (41) and (42) that have a second-order derivative go to zero.

Treatment of the Solution.
Figure 2 shows a 25 × 25 grid discretization mesh for finite element for numerical computation for temperature and velocity, whereas for pressure, average temperature, and U m , only the bottom one-dimensional line of the x-axis is used in this article.Equation (43) includes nodal unknowns and time derivatives.Therefore, it is not a set of algebraic equations but instead a set of ODEs.First, SUPG-FEM was developed using a semidiscrete formulation, and the Crank-Nicolson schemes were applied to solve the system of equations for time-dependent equation.Because the remaining equations (41), (42), (44), and (45) contain nonlinear terms, the formulated system of algebraic equations is solved iteratively.
The approximated SUPG-FEM solution obtained for the above equation is to an accuracy of tolerance Tol , where the error is calculated as and ψ j are values of the unknown nodal variables at coordinate points.Tol = 10 −3 , 10 −4 , and 10 −5 tolerance were used for iterations performed.However, on approximate solution, there is no significant difference for all different values of tolerance.MATLAB software version 2021 is used to obtain the result by developing MATLAB code.Algorithm 1 shows the general steps to solve the governing equation to obtain the result.

Results and Discussion
The tilted slider plane bearing parameters for timedependent fluid flow lubricant in this article appear to be functions of the dimensionless parameters P e , P r , and E c as well as γ ⊛ , η ⊛ , w, Re ⊛ , and σ ⊛ .The following parameter Step 1. Initialization.
Step 3. Calculate u ⊛new using p ⊛new , u ⊛old , v ⊛old and T ⊛old and Eq.(43).
Step 4. Calculate v ⊛new using u ⊛new , v ⊛old and T ⊛old and Eq.(44).
Step 5. Calculate T ⊛new using p ⊛new , u ⊛new , v ⊛new and T ⊛old and Eq.(45).
Step 6. Test the convergence rate.
Step 7. Repeat steps 2-6 till convergence is obtained for all spatial variables.
Step 8. Calculate load carrying capacity performance W ⊛ and friction force F ⊛ using Eq.(33).
Algorithm 1: General steps to solve the governing equation.8 Journal of Applied Mathematics values were selected in accordance with the Lebeck consideration stated in [34].Some other constants are given in detail in Sinha and Adamu [18].The findings on pressure distribution, load-bearing capacity performance, drag friction force, velocity, and temperature performance have been investigated.
The results obtained are illustrated using tables and graphs.To ensure domain discretization grid independence, the numerical simulations were run on various grid systems with 10 by 10, 20 by 20, and 25 by 25 grid points.The load-carrying capacity performance of various grids is compared and displayed in Figure 3.This leads to the conclusion that the grid scheme of 25 by 25 produces a grid-independent solution.Finally, the bearing performance is calculated from equation ( 33) using the trapezoidal method.Figure 4 illustrates the one-dimensional longitudinal distribution of pressure performance caused by the above conditions (C 1 , C 2 , and C 3 ) for various modified Reynolds numbers, considering unsteady fluid flow lubricant.For all conditions, the load-carrying capacity performance (LCCP) is in the sequence of conditions (C 1 ) < conditions (C 2 ) < conditions (C 3 ) at time t = 0 01 for each fixed value of x † .One can deduce from this that the bearing's LCCP increased as the inertial term effect Re ⊛ increased for t = 0 01 constant time.Thus, increasing the inertial term effect value obstructs the    11 Journal of Applied Mathematics outlet, and the pattern of the graph of pressure distribution on the plane slider bearing is in good agreement with Sinha and Adamu [18], Kumar [4], Rathish Kumar and Srinivasa Rao [5] Desu Tessema et al. [22], and Naduvinamani and Angadi [24].
Figure 5 depicts the pressure distribution of the longitudinal model type for different modified Reynolds numbers due to the fluid inertial term effect at w = 0 4, of temperature T i = T p = T s = 1 0, and condition (C 1 ).From this, one can observe that as the modified Reynolds number increases, the LCCP of the plane slider bearing increases.Furthermore, again from Tables 1-3, as the modified Reynolds number increases, the corresponding drag force increases.This result is in good agreement with Malvano and Vatta [16] and Sestieri and Piva [12].
At an inclination of w = 0 4, T i = T p = T s = 1 0, and for surface roughness and thermal combined effect, Figure 6 displays the pressure distribution for an unsteady fluid flow lubricant of a plane slider bearing for various modified Reynolds numbers.This suggests that the current study is superior to earlier work when compared to the steady laminar flow of the pressure distribution, which is carried out in Sinha and Adamu [18] and Desu Tessema et al. [22].In comparison to other methods that are described in the literature, our method generally has a better LCCP.
The LCCP and drag force of one-dimensional longitudinal roughness for various inertial term effect values corresponding to the stated conditions are displayed in Tables 1-3.We see an almost 5% improvement in load-carrying capacity performance between condition (C 1 ) and condition (C 2 ) in all inertial term effect values at w = 0 4. Similarly, Tables 1 and 2 demonstrate that there is a drag force of less than 1% at w = 0 4. Furthermore, we observed a 13% decrement in LCCP and a negligible change in drag force for all Re ⊛ values between condition (C 3 ) and condition (C 1 ) at w = 0 4 in Tables 1 and 3.This happens because the density and viscosity of the lubricant's fluid are reduced because of the condition (C 1 ) effect.As a result, the LCCP is reduced.Even if the LCCP of a parallel slider bearing at w = 1 is small, a better LCCP is generated with condition (C 1 ) and condition (C 2 ) in comparison to condition (C 3 ) due to fluid expansion as the lubricant temperature rises.
A one-dimensional longitudinal surface roughness model assumes that the texture has the appearance of furrows, long narrow ridges, and valleys in the sliding direction (x-axis), allowing for fast lubricant flow and resulting in a decrease in pressure distribution, implying a reduction in load-carrying capacity (LCC).
Figures 7 and 8 show the surface plot of temperature for one-dimensional longitudinal surface roughness in the cases    In parallel plane slider bearings, if the temperature of surface of the pad bearing is higher than the slider temperature of surface, a suction action may occur, according to Zienkiewicz [35].The study of the surface thermal impact on the plane slider and pad at a certain temperature of (T s < T p or T s > T p ) on the LCC and frictional force of a plane slider bearing is of excessively realistic relevance.In practice, the fixed surface is typically warmer than the moving surface, according to Pinkus and Sternlicht [36].Another critical scenario in a slider bearing application is that the inlet lubricant temperature is lower than the slider and pad temperatures.Based on this, the combined thermal and surface roughness effects (C 1 ) of the following temperature boundary conditions have been investigated with a modified Reynolds number Re ⊛ = 0 05: Case 1. T i = T p = 1 2 and T s = 1 3.
Drag forces and LCCP for one-dimensional longitudinal and transverse surface roughness are shown in Figure 9 at Re ⊛ = 0 05 of inertial term effect.The inlet temperature is equal to or less than the temperatures of the slider (movable) and pad (stationary) of plane slider bearing.
When the temperature of the pad (stationary) and slider (movable) is different, there is an 18% difference in loading performance between Case 2 and Case 1 for w = 0 4 as shown in Figure 9(a).For w = 1, Case 1's LCCP is slightly better than Case2's.In the case of drag force, there is a significant difference between Case 2 and Case 1 as we observe from Figure 9(c).
These figures show that if the inlet temperature is lower than that of the pad and slider of plane slider bearing, Case 4's drag force of friction and LCCP are lower than Case 3's for each inclination parameter.According to these various factors, if the slider temperature is higher than the pad temperature, the temperature may drop due to the inertial term effect, increasing the LCCP and drag friction force.
For a better understanding, the temperature surface contour plots of Case 1 and Case 2 for w = 0 4 are shown in Figures 10 and 11.Case 2's load-carrying capacity is lower than Case 1's because Case 2 has total higher average temperature than Case 1.
The performance of LCCP for different (δ ⊛ ) values is shown in Table 4 at a modified inertial value of Re ⊛ = 1 0. We can see from this table that as δ ⊛ increases, the bearing's LCCP decreases.This is because the surface roughness 13 Journal of Applied Mathematics (asperity) increases, allowing for faster lubricant flow and a decrease in pressure distribution for the one-dimensional longitudinal roughness model; however, it is the converse for the one-dimensional transversal roughness model.

U-Velocity
Profile.Plotting of a one-dimensional longitudinal roughness U-velocity profile with different boundary condition U values of the slider for condition (C 1 ) at x = 0 02 is shown in Figure 12.This indicates that the full U-velocity increases as the slider's U value increases.
Table 5 displays the influence of U-velocity for condition (C 1 ).It is evident that raising the U value enhances the ability to withstand load and drag forces.This is because, in addition to the effects of inertial forces and surface roughness, an increasing U value accelerates the fluid in the sliding direction.The fluid's viscosity increases with even a small increase in temperature, which enhances the bearing capacity.
Table 6 shows the bearing performance for the combined effect (C 1 ) at different time discretization.As anyone can observe from the table, as time discretization increases, unsteady fluid flow lubricant bearing performance rises also.4.3.One-Dimensional Transverse Roughness.With unsteady fluid flow taken into consideration, Figure 13 shows a onedimensional unsteady fluid flow lubricant transverse roughness distribution of pressure performance produced by the aforementioned conditions (C 1 , C 2 , and C 3 ) for different inertial values.At a time discretization of t = 0 01, the LCCP is in the sequence of conditions (C 2 ) < conditions (C 1 ) < conditions (C 3 ) for each fixed value of x † .This indicates that the LCCP of the bearing grew at t = 0 01 constant time as the fluid inertial term's influence increased.Consequently, increasing the inertial fluid effect causes pressure to rise and obstructs fluid movement.Moreover, under all circumstances, surface roughness has a bigger impact than both combined and thermal impacts.
Regarding the time-dependent pressure equation's slider and pad at different temperatures.For a fixed modified Reynolds number value of Re ⊛ = 0 5, the distribution of pressure in Figure 9(b) has a slightly higher LCCP than the one-dimensional longitudinal distribution but otherwise follows the same pecking order.The drag force results for onedimensional longitudinal roughness in Cases 1-4 are almost the same as those for one-dimensional transverse roughness, as shown in Figure 9(d).With that exception, Cases 4 and 3 are reversed.
The LCCP and drag force for unsteady fluid flow lubricant of one-dimensional transverse roughness for different fluid inertial terms corresponding to the aforementioned conditions are shown in Tables 7-9.At an inclination of w = 0 4, there is a nearly 19% difference in LCCP between condition (C 1 ) and condition (C 2 ).Similarly, at w = 0 4, there is a 2% difference in drag force for all modified Reynolds numbers at Re ⊛ .
Moreover, for all Re ⊛ values between condition (C 3 ) and condition (C 1 ) at w = 0 4, we observed a nearly 13% difference in LCCP and a 2% difference in drag force, as shown in Tables 7 and 9.Because of the combined effect, the density and viscosity of the lubricant's fluid are reduced.The LCCP is reduced as a result.Tables 8 and 9 demonstrate that for w = 0 4, there is a significant difference in LCCP between condition (C 3 ) and condition (C 2 ).There is also a 26% difference in value for transverse frictional force over all modified Reynolds number values between condition (C 3 ) and condition (C 2 ).Even if a parallel slider bearing's LCCP is small at w = 1, there is an LCCP generated in condition (C 1 ) and condition (C 2 ) in comparison to condition (C 3 ) due to fluid expansion as the lubricant temperature rises.
Due to the plane slider bearing nature of our consideration, when there is transverse roughness for unsteady fluid flow lubricant, the fluid typically only travels in the x-direction and must pass through a series of constrictions at the flow gap, or fluid film domain area.In addition to the inertial term effect, this will impede the flow.The widening due to valleys of surface asperities, on the other hand, will ease the flow.

Conclusion
In this work, time-dependent fluid flow pressure and momentum equation in slider bearings were evaluated via FEM to analyze the bearing characteristics (LCCP and drag force) of the one-dimensional longitudinal and transverse models.In both types of models for nonparallel slider bearings, the achievement of the bearing characteristics improves.However, for w = 1 (parallel), the performance of the bearing is negligible.Furthermore, in the case of condition (C 1 ) taking the thermal of the slanted pad slider as higher than the thermal of the pad, in both models, LCCP rises for nonparallel bearing, whereas it goes to zero for parallel w = 1 .The opposite is true for drag force in both model types.As the modified Reynolds number increases, the inertial terms become increasingly important, and the geometrical configuration of the film differs significantly from that obtained using the elementary lubrication theory.This is due to fluid inertial term effects from unsteady fluid flow in addition to the surface roughness effect.This demonstrates the significance of taking surface texture and inertial effects into account due to unsteady fluid flow lubricant.One-dimensional transversal models typically have a pressure distribution that is larger than that of one-dimensional longitudinal slider bearings in the situation of unsteady fluid flow lubrication.

Symbols δ:
Random distribution of roughness μ: Viscosity of the lubricant μ av : Average viscosity across the film ρ: Density of the lubricant ρ av : Average density across the film σ 2 : Variation of roughness B: Bearing width E: Expected value operator E c : Eckert number h: Nominal film thickness h 0 : Nominal film thickness at outlet h i : Nominal film thickness at inlet H t : Height of rough surface k: Thermal conductivity of lubricant P: Lubricant pressure p e : Peclet number p i : Inlet pressure p r : Prandtl number T: Lubricant temperature T av : Average temperature across the film U: Velocity of the moving surface v: Velocity in the direction of the y-coordinate w: h 0 /h i W ⊛ : Load-carrying capacity of the bearing x † , y † : Transformed coordinate system.

Figure 1
Figure 1 hereunder illustrates the geometry of a rough slider bearing.Comparing the length of the bearing in the direction orthogonal to the xy-plane to the height of the fluid film thickness h, a fairly long length is assumed.The simplified equations for the flow of lubricants in slider bearings are derived from the governing equations of fluid flow.These equations took into account the viscosity and density of the lubricant and the speed of the bearing, as well as other variables such as the bearing geometry, the applied load, and the temperature of the lubricant.

Figure 1 :
Figure 1: Geometry of one-dimensional transverse roughness of slider bearing.

Figure 2 :
Figure 2: Regular domain discretization system of grid.

Figure 5 :
Figure 5: Pressure distributions at w = 0 4 and for different values of Re ⊛ .

Figure 6 :
Figure 6: Pressure distributions at T i = T p = T s = 1 0, w = 0 4 and for different value of Re ⊛ (longitudinal roughness).

Figure 13 :
Figure 13: Pressure distributions at T i = T p = T s = 1 0 and w = 0 4 for different value of Re ⊛ (transversal roughness).

Table 1 :
Longitudinal roughness W ⊛ and F ⊛ for condition(C 1 ) at different values of Re ⊛ and w.

Table 2 :
Longitudinal roughness W ⊛ and F ⊛ for condition(C 2 ) at different values of Re ⊛ and w.

Table 3 :
Longitudinal roughness W ⊛ and F ⊛ for condition(C 3 ) at different values of Re ⊛ and w.

Table 5 :
Effect of U-velocity on bearing performance for C 1 at w = 0 4 and Re ⊛ = 1 0.

Table 6 :
Performance of bearing for C 1 at w = 0 4 and Re ⊛ = 1 0 for different time discretization.

Table 9 :
Transversal roughness W ⊛ and F ⊛ for condition(C 3 ) at various value of Re ⊛ and w.

Table 7 :
Transversal roughness W ⊛ and F ⊛ for condition(C 1 ) at various value of Re ⊛ and w.

Table 8 :
Transversal roughness W ⊛ and F ⊛ for condition(C 2 ) at various value of Re ⊛ and w.