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\DOI{10.5802/crmeca.347}
\datereceived{2025-03-15}
\daterevised{2025-07-24}
\datererevised{2025-10-16}
\dateaccepted{2026-01-20}
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\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Hydrodynamic lubrication study of plain journal bearings
lubricated with couple stress fluids: surface texture effects}

\alttitle{\'{E}tude de la lubrification hydrodynamique des paliers
lisses lubrifi\'{e}e \`{a} l'aide de fluide a couple de contrainte :
effets de texturation des surfaces}

\author{\firstname{Atika} \lastname{Kabouya}\CDRorcid{0009-0008-0648-636X}\IsCorresp}
\address{Electro Mechanics Department, Electromechanical Systems
Laboratory, Faculty of Technology, University Badji Mokhtar, 12, PO
Box, Annaba, 23000, Algeria}
\email[A. Kabouya]{atika.kabouya@univ-annaba.dz}

\author{\firstname{Ali} \lastname{Belhamra}\CDRorcid{0009-0008-6050-0927}}
\addressSameAs{1}{Electro Mechanics Department, Electromechanical
Systems Laboratory, Faculty of Technology, University Badji Mokhtar,
12, PO Box, Annaba, 23000, Algeria}
\email[A. Belhamra]{belhamraali@yahoo.fr}

\begin{abstract}
This study investigates the static characteristic of textured bearings
lubricated with oils containing viscosity additives, thus influencing
their tribological performance. The texture model adopted is that of
the positive sinusoidal half-wave, a configuration that enables
lubrication to be optimized by modifying the distribution of the fluid
film. The lubricant used has a non-Newtonian rheological behavior, and
is modeled as a couple stress fluids, taking into account the effects
of fluid microstructure on flow.

The results obtained are analyzed by comparing them with those of a
plain bearing lubricated by a conventional Newtonian fluid, enabling
the influence of the lubricant's texturing and properties on bearing
performance to be assessed. It is shown that the combination of the
surface texturing of the pad and lubrication with polar fluids has
significant effects on the static characteristics of the journal
bearing, such as: load-carrying capacity, friction factor, attitude
angle, and side leakage flow. It is also found that the sensitivity of
these effects increases with the couple stress parameter and the
eccentricity ratio.
\end{abstract}

\begin{altabstract}
Cette \'{e}tude porte sur le comportement statique d'un palier
textur\'{e} lubrifi\'{e} \`{a} l'aide d'huiles contenant des additifs
de viscosit\'{e}, influen\c{c}ant ainsi ses performances tribologiques.
Le mod\`{e}le de texture adopt\'{e} est celui de demi-onde
sinuso\"{i}dale positive, une configuration permettant d'optimiser la
lubrification en modifiant la r\'{e}partition du film fluide. Le
lubrifiant utilis\'{e} pr\'{e}sente un comportement rh\'{e}ologique non
newtonien et est mod\'{e}lis\'{e} comme un fluide \`{a} couples de
contraintes, en consid\'{e}rant les effets de la microstructure du
fluide sur l'\'{e}coulement.

Les r\'{e}sultats obtenus sont analys\'{e}s en les comparants \`{a}
ceux d'un palier lisse lubrifi\'{e} par un fluide newtonien classique,
ce qui permet d'\'{e}valuer l'influence de la texturation et des
propri\'{e}t\'{e}s du lubrifiant sur les performances du palier. Il est
d\'{e}montr\'{e} que la combinaison de la texturation de surface du
coussinet et la lubrification \`{a} l'aide de fluides polaires a des
effets significatifs sur les caract\'{e}ristiques statiques du palier,
tels que : la portance hydrodynamique, le nombre de frottement, l'angle
de calage et le d\'{e}bit. Il est \'{e}galement constat\'{e} que la
sensibilit\'{e} de ces effets augmente avec le param\`{e}tre de couple
de contrainte et l'excentricit\'{e} relative.
\end{altabstract}

\keywords{\kwd{Hydrodynamic lubrication}\kwd{Hydrodynamic
bearing}\kwd{Couple stress fluid}\kwd{Non-Newtonian fluid}\kwd{Surface
texture}}

\altkeywords{\kwd{Lubrification hydrodynamique}\kwd{Palier
hydrodynamique}\kwd{Fluide \`{a} couples de contraintes}\kwd{Fluide non
Newtonien}\kwd{Texturation de surface}}

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\section{Introduction}\label{sec1}

The current trend in modern engineering is to reduce weight and operate
at supersonic speeds. These conditions have an effect on the
characteristics of journal bearings, and may lead to exceeding the
safety limits of these components. For precise prediction of bearing
performance, it is necessary to develop numerical simulations that
consider the impact of parameters neglected by Reynolds~\cite{1} in
classical lubrication theory, such as the surface state and the
rheology of lubricant. So, the development of micro texturing
processes~\cite{2,3}, the means of control and the development of
computational and numerical tools have allowed the introduction of new
methods of characterization and modeling to study the effect of surface
topography on the tribological performance of a hydrodynamic contact.
These elements allowed~\cite{4} to demonstrate the potential of
micro-irregularities to generate local hydrodynamic pressure, which
allows significantly increasing of the load-carrying capacity. Since
then, several studies have been conducted in order to understand the
physical mechanisms responsible for the increase in load capacity. Some
studies attribute this elevation to the creation of a local pressure
peak in the cavities~\cite{5,6}, so the succession of texture
judiciously arranged allows the construction of an evolutionary
pressure field (global effect) due to the continuity of the local
phenomenon. Therefore, the texture can be considered as a convergent
zone~\cite{7}, where the importance of optimizing the texture geometry
to maximize their impact on the static characteristics of hydrodynamic
contacts. A second theory attributes the positive impact of textures to
convective inertia effects~\cite{8}, while other authors show that
inertia effects have a negative influence on hydrodynamic
performance~\cite{9}.

In 2016 Gropper et~al.~\cite{10} conducted a comparative study of the
results of research work concerning surface texturing in the
hydrodynamic regime and they conclude that surface texturing allows the
improvement of performances such as: load capacity, minimum lubricating
film thickness, friction and wear.

Four years later, Vencl et~al.~\cite{11} provided a review retracing the
evolution of research that considers diverse surface textures and
concluded that: the advantages of surface texturing can be better
utilized when the surface texture design is optimized and applied to
specific contact pairs. More recent studies are focused on optimizing
the dimensions, shapes and positions of textures. For example,
Dhiman~\cite{12} and Dhiman~\cite{13} studied the effect of positive
sinusoidal half-wave and positive sinusoidal full-wave textures on the
performance of a porous journal bearing by developing a model based on
the Reynolds model and Darcy's law. It shows that partial texturing
gives a better improvement in journal bearing characteristics compared
to full texturing, and the optimized texture location was observed at
an angle between 36\textdegree\ and 108\textdegree\ from the center
line for the first texture type, and from 45\textdegree\ to
120\textdegree\ for the second type. Similarly, Hamdavi et~al.~\cite{14}
and Tauviqirrahman et~al.~\cite{15} have shown that partial texturing
has positive effects on the pressure and load carrying capacity of
journal bearing with low relative eccentricity.

In 2017, Guo~\cite{16} studied the effect of spherical and cylindrical
type texturing on an axial three-groove bearing, he concludes that
texturing has a significant effect on the pressure in the lubricating
film and the load carrying capacity when placed in the convergent zone
of the journal bearing. One year later, Wang et~al.~\cite{17} studied
the effect of texture geometry, and he concludes that the convex
spherical texture is better suited for plain bearings than the concave
spherical texture. In another context, the incorporation of spherical
textures at different locations (convergence zone and maximum pressure
zone) of non-porous and porous journal bearings provides significant
performance improvement~\cite{18}. A very recent study shows that the
stability of a bearing can be increased by using cylindrical textures
and that increasing the depth of the dimples has a significant effect
on the critical mass~\cite{19}. Partial texturing of a bearing
lubricated with micro polar and pseudo plastic fluid is very
advantageous. It allows the bearing load to be increased and the
coefficient of friction to be reduced~\cite{20}. 

The above studies are based on the assumption that the lubricant is
Newtonian. However, practical lubrication applications show that oils
containing viscosity additives have non-Newtonian rheological behavior.
Their viscosity decreases when the shear rate to which they are
subjected increases. During flow forces appear perpendicular to the
shear plane~\cite{21}, these forces could be proportional to the square
of the rate of shear. Rosenberg~\cite{22} has shown experimentally that
the minimum film thickness of a journal bearing lubricated by
polymerized oil is more important than that measured with the pure
mineral oils having the same viscosity. Therefore, classical continuum
theory, which dismisses particle size, cannot accurately describe the
flow of such fluids. Various models have been proposed in the
literature to remedy this restriction and describe the behaviour of
these complex fluids, known as couple stress fluids. These models
incorporate stress moments in addition to conventional surface
forces~\cite{23,24,25}. Of these theories, Stokes~\cite{23} theory is
preferred in many studies because of its mathematical simplicity. It
allows taking into account the moving particle size and the couple
stress due to the presence of viscosity additives. Several authors have
used it to study problems of hydrodynamic
lubrication~\cite{26,27,28,29}. It was demonstrated, that the static
behavior of the journal bearing is significantly influenced by
increasing the couple stress parameter. Similarly, Abass and
Mahdi~\cite{30} showed that non-Newtonian fluids are beneficial for
elliptical journal bearings with micro protrusions. However, the use of
polar fluids to reduce the temperature in the contact was shown by Wang
et~al.~\cite{31}.

This study investigates how surface texturing and couple stress fluid
influence the hydrodynamic performance of journal bearings. Although
surface texturing has been extensively studied with Newtonian fluids
and couple stress models have demonstrated potential in representing
non Newtonian lubricants, their combined impact remains largely
unexplored. 

The coupled effects of texture geometry and polar fluid behavior on
global system performance are examined through numerical analysis, with
a focus on pressure generation, Load-carrying capacity, friction
factor, attitude angle, and side leakage flow.

\section{Problem mathematical formulation}\label{sec2}
\subsection{Geometric model}\label{ssec21}

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Journal bearing section.}
\end{figure}

Figure~\ref{fig1} show section of a cylindrical bearing. In this case,
the lubricant film thickness $h_{0}$ (m) in the $y$ direction for a
given angle $\theta$ is calculated by using this formula:
{\begin{equation}\label{eq1}
h_{0} = C [1 + \varepsilon \cos \theta]
\end{equation}}\unskip
with 
\begin{itemize}
\item $C = (R_{c}-R_{a})$ is the radial clearance, $R_{c}$ and $R_{a}$
are the radius of the bearing and journal respectively,
\item $\tilde{\varepsilon}$ is the relative eccentricity
($\tilde{\varepsilon}=e/C$) varying from 0 to 1,
\item $\theta$ is the circumferential coordinate, measured
trigonometrically from the line connecting the centers
$O_{a}$ and $O_{c}$.
\end{itemize}
This angular convention is used not only to locate textures on the
surface of the bearing, but also to analyse pressure distribution and
identify areas of convergence/divergence of the lubricating film based
on the position around the shaft.

In this study, the positive sinusoidal half-wave profile was chosen for
its mathematical simplicity, continuity and ability to provide a
parametric representation of textured surfaces. It also allows clear
parametric study, by independently controlling the amplitude and
density of the asperity.

For this type of texture the lubricant film thickness $h$ can be
approached through the use of the formula proposed by Kango
et~al.~\cite{32}:
{\begin{eqnarray}
h &=& h_{0} - \delta \label{eq2} \Seqnsplit
\delta &=& -\left[\frac{2*a_{p}}{\uppi}\right]
\left[\displaystyle\sum _{q=2,4,6\ldots \infty}
\dfrac{\cos (q*B)}{q^{2}-1}\right] + 
\left[\dfrac{a_{p}}{\uppi}\right] + 
\dfrac{a_{p}}{2} \sin (B) \label{eq3}
\end{eqnarray}}\unskip
with
\begin{itemize}
\item In the $\theta$ direction: $B = {\uppi R\theta }/{\lambda}$ and 
$\lambda = {2\uppi R}/{n_{p}}$  
\item In the $\tilde{z}$ direction: $B = {\uppi z}/{\lambda}$ and 
$\lambda = {L}/{n_{p}}$  
\item $\tilde{a}_{p} = {a_{p}}/{C}$ 
\end{itemize}
where, $a_{p}$, $n_{p}$ and $\lambda$ represent respectively:
amplitude, number and wavelength of asperity, and $\delta$ is the film
thickness variation caused by the presence of texture.

Figure~\ref{fig2} show the lubricant film profile obtained by using the
chosen model.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Lubricant film profile obtained for different
values of asperity amplitude.}
\end{figure}

\subsection{Governing equation}\label{ssec22}

The theoretical solution of a hydrodynamic lubrication problem requires
the determination of the pressure field inside the lubricant film. The
model used is based on the Stokes micro-continuum theory~\cite{23}, the
momentum equations and the continuity equation of an incompressible
fluid with couple stress are:
{\begin{eqnarray}
\rho \frac{D\vec {V}}{Dt} &=& -\vec{\nabla }p +
\rho \vec{A} + \frac{1}{2} \vec{\nabla} X (\rho \vec {E}) +
\mu \vec {\nabla ^{2}}\vec {V} - \eta \vec {\nabla ^{4}}\vec {V}
\label{eq4} \Seqnsplit
\vec{\nabla}\cdot \vec {V} &=& 0, \label{eq5}
\end{eqnarray}}\unskip
where the vectors $\vec{V}$, $\vec{A}$ and $\vec{E}$ represent the
velocity field, body force per unit mass, and body couple per unit
mass, respectively; $\rho$ represent the fluid density, $P$ is the
pressure, $\mu$ is the shear viscosity and $\eta$ is a material
constant responsible for the couple stress fluid property.

In case of steady state incompressible flow, when the body forces and
body moments are ignored, and making the usual assumption of
hydrodynamic lubrication applicable to thin films~\cite{33}, the
equation of motion are reduced to:
{\begin{equation}\label{eq6}
\left\{\begin{array}{l} 
\dfrac{\partial p}{\partial x} = \mu 
\dfrac{\partial ^{2}u}{\partial y^{2}} - \eta 
\dfrac{\partial ^{4}u}{\partial y^{4}} \\
\dfrac{\partial p}{\partial y} = 0 \\
\dfrac{\partial p}{\partial z} = \mu 
\dfrac{\partial ^{2}w}{\partial y^{2}} - \eta 
\dfrac{\partial ^{4}w}{\partial y^{4}},
\end{array}\right.
\end{equation}}\unskip
where $u$ and $w$ are the velocity components in the $x$ and $z$
direction.

Then, as descripted in~\cite{26}, a modified Reynolds equation based on
a couple stress models can be obtained by introducing the following
dimensionless variables:
{$$
\theta = \dfrac{x}{R}; \quad
\tilde{z} = \dfrac{z}{L}; \quad
\tilde{h} = \dfrac{h}{C}; \quad
\tilde{l} = \dfrac{l}{C}; \quad
\tilde{p} = \dfrac{PC^{2}}{\mu \omega _{a}R^{2}}; \quad
\tilde{G}(\tilde{h},\tilde{l}) = \dfrac{G(h,l)}{C^{3}}.
$$}\unskip
The dimensionless form of modified Reynolds equation can be given as:
{\begin{eqnarray}
&& \dfrac{\partial}{\partial \theta}
\left[\tilde{G}(\tilde{h},\tilde{l})
\dfrac{\partial \tilde{p}}{\partial \theta}\right] + 
\left[\dfrac{R}{L}\right]^{2}
\dfrac{\partial}{\partial \tilde{z}}
\left[\tilde{G}(\tilde{h},\tilde{l})
\dfrac{\partial \tilde{p}}{\partial \tilde{z}}\right] = 6
\dfrac{\partial \tilde{h}}{\partial \theta}
\label{eq7} \Seqnsplit
&& \tilde{G} (\tilde{h},\tilde{l}) = \tilde{h}^{3} - 12\tilde{l}^{2}
\tilde{h} + 24\tilde{l}^{3}\tanh \left[\dfrac{\tilde{h}}{2\tilde{l}}\right]. 
\label{eq8}
\end{eqnarray}}\unskip
In the previous equation $\omega_{a}$ is the velocity of journal and $l
= (\eta/\mu)^{1/2}$ is the couple stress parameter. It can vary from 
0 to 1.

This variation illustrates the transition from Newtonian behavior
($\tilde{l}=0$) to that of non-Newtonian lubricants.

Physically, it represents the relative length of the molecular chains
in additives that are added to the base oil to enhance lubricant
performance, such as improving viscosity.

The deterministic approach is employed to solve the Reynolds equation.
Unlike the stochastic method, which introduces empirical flow factors
to account for surface texture effects, the deterministic approach
models these effects directly by incorporating the surface texture into
the fluid film thickness equation through explicit geometric
representation. 

\subsection{Boundary conditions}\label{ssec23}

The Reynolds boundary conditions are used. They respect flow continuity
and film rupture. They assume that for an unknown abscissa
$\theta_{c}$, the pressure and pressure gradient become zero.
{\begin{eqnarray}
&& \tilde{p} [\theta, \tilde{z} = \pm \textstyle\frac{1}{2}] = 0 
\label{eq9} \Seqnsplit
&& \tilde{p} = \dfrac{\partial \tilde{p}}{\partial \theta}=0; \quad
\mbox{for } \theta = \theta _{c}. \label{eq10}
\end{eqnarray}}\unskip
However, these conditions do not satisfy film reformation. Their use
requires the determination of an additional unknown, $\theta_{c}$,
which is the cavitation angle. Thus, an unknown boundary between the
active and inactive zones of the lubricant film is defined by $\theta =
\theta_c$. It is assumed that the abscissa of the film reformation is
known and coincides with ($\theta = 0$).

The respect of these conditions is made by the application of the
Cristopherson algorithm~\cite{34}. This algorithm allows the suppression of
negative pressure terms during the computations and the study will be
limited to the active zone of the bearing.

Thus, the determination of the static pressure distribution in the
lubricant film for a given position of the shaft in the bearing allows
the calculation of the static performance of the bearing such as:
load-carrying capacity, attitude angle, Sommerfeld number, friction
factor, and side leakage flow.

\subsection{Static characteristics}\label{ssec24}
%\mbox{}
\begin{itemize}
\item The components of the dimensionless load carrying capacity are
expressed by:
{\begin{equation}\label{eq11}
\left\{\begin{array}{rcl} 
\tilde{F}_{1} &=& \displaystyle \int _{-\frac{1}{2}}^{\frac{1}{2}} \int _{0}^{2\uppi }\tilde{p} (\theta ,\tilde{z})\cos \theta\,\mathrm{d}\theta\, \mathrm{d}\tilde{z} \vspace*{3pt} \\
\tilde{F}_{2} &=& \displaystyle \int _{-\frac{1}{2}}^{\frac{1}{2}} \int _{0}^{2\uppi }\tilde{p} (\theta ,\tilde{z})\sin \theta\,\mathrm{d}\theta\, \mathrm{d}\tilde{z}.
\end{array}\right.
\end{equation}}\unskip
\end{itemize}
It is from these two relationships that we can determine the following
expressions: 
\begin{itemize}
\item Static attitude angle: 
{\begin{equation}\label{eq12}
\varphi = \tan ^{-1} \left[-\dfrac{\tilde{F}_{2}}{\tilde{F}_{1}}\right].
\end{equation}}\unskip
\item The resultant of load carrying capacity: 
{\begin{equation}\label{eq13}
\tilde{F} = \sqrt{{\tilde{F}}_{1}^{2}+{\tilde{F}}_{2}^{2}}.
\end{equation}}\unskip
\item The Sommerfeld number: 
{\begin{equation}\label{eq14}
S = \dfrac{1}{\uppi \tilde{F}}.
\end{equation}}\unskip
\item The friction couple is obtained by integrating the shear stresses at 
the shaft surface ($y=h$):
{\begin{equation}\label{eq15}
T_{a} = R \displaystyle \int _{s} [\sigma _{xy}]_{y=0,h}\,\mathrm{d}s.
\end{equation}}\unskip
\end{itemize}
In laminar flow, the expression for shear stress in a couple stress
fluids is given by:
{\begin{equation}\label{eq16}
[\sigma _{xy}]_{y=0,h} = \mu \left.\dfrac{\partial u}{\partial
y}\right| _{y=0,h} + \underset{=0}{\underbrace{\eta \left.\dfrac{\partial
^{3}u}{\partial y^{3}}\right| _{y=0,h}}} = \mu \left(\dfrac{\omega
_{a}R}{h}\mp \dfrac{1}{2\mu R}\dfrac{\partial P}{\partial \theta}
\left(h-2l \tanh \left(\dfrac{h}{2l}\right)\right)\right).
\end{equation}}\unskip
The non dimensionless friction couple on the journal is written as:
{\begin{equation}\label{eq17}
\tilde{T}_{a} = \displaystyle \int _{-\frac{1}{2}}^{\frac{1}{2}} \int
_{0}^{2\uppi}\left[\dfrac{1}{\tilde{h}} + \dfrac{1}{2}  \dfrac{\partial
\tilde{p}}{\partial \theta }\left(\tilde{h}-2\tilde{l}\tanh
\left(\dfrac{\tilde{h}}{2\tilde{l}}\right)\right)\right]\mathrm{d}\theta
\,\mathrm{d}\tilde{z}.
\end{equation}}\unskip
\begin{itemize}
\item The friction factor is defined as:
{\begin{equation}\label{eq18}
\tilde{f} = \dfrac{\tilde{T}_{a}}{\tilde{F}}.
\end{equation}}\unskip
\item Side leakage flow:
{\begin{equation}\label{eq19}
\tilde{Q} = -\dfrac{1}{12} \displaystyle \int _{0}^{2\uppi
}\tilde{h}^{3} \dfrac{\partial \tilde{p}}{\partial
\tilde{z}}\,\mathrm{d}\theta + \tilde{l}^{2}\int _{0}^{2\uppi}
\dfrac{\partial \tilde{p}}{\partial
\tilde{z}}\left[\tilde{h}-2\tilde{l}\tanh
\left(\dfrac{\tilde{h}}{2\tilde{l}}\right)\right]\mathrm{d}\theta.
\end{equation}}\unskip
\end{itemize}

\section{Numerical analysis}\label{sec3}

The governing equations are solved numerically using the centered
finite-difference method. The field of solution is divided in to grid
spacing ($N~X~M$) each of a mesh size is $\Delta \theta X \Delta
\tilde{z}$.

The discrete system of equations obtained is solved using the
Gauss--Seidel method with the over relaxation coefficient ${\Omega}$
varying from 1 to 2. Applying this algorithm, we can write:
{\begin{equation}\label{eq20}
{\tilde{p}}_{ij}^{(k+1)}=(1-\Omega) {\tilde{p}}_{ij}^{(k)}+\Omega
\left[a_{ij}{\tilde{p}}_{i+1,j}^{(k)}+b_{ij}{\tilde{p}}_{i-1,j}^{(k+1)}+
C_{ij}{\tilde{p}}_{i,j+1}^{(k')}+d_{ij}{\tilde{p}}_{i,j-1}^{(k+1)}
+e_{ij}\right]
\end{equation}}\unskip
with
{\begin{eqnarray*}
a_{ij} &=& \left[\dfrac{\tilde{G}}{\Delta \theta
^{2}}+\frac{1}{2\Delta \theta }\left(\frac{\partial
\tilde{G}}{\partial \theta }\right)_{ij} \right]\frac{1}{F_{ij}}; \quad
b_{ij}=\left[\frac{\tilde{G}}{\Delta \theta ^{2}}-\frac{1}{2\Delta
\theta }\left(\frac{\partial \tilde{G}}{\partial \theta }\right)_{ij}
\right]\frac{1}{F_{ij}}; \\
c_{ij} &=& \left[\frac{\tilde{G}}{\Delta
\tilde{z}^{2}}+\frac{1}{2\Delta z}\left(\frac{\partial
\tilde{G}}{\partial \tilde{z}}\right)_{ij}
\right]\frac{\lambda}{F_{ij}}; \quad
d_{ij} = \left[\dfrac{\tilde{G}}{\Delta
\tilde{z}^{2}}-\frac{1}{2\Delta z}\left(\frac{\partial
\tilde{G}}{\partial \tilde{z}}\right)_{ij} \right]\frac{\lambda
}{F_{ij}}; \quad
e_{ij}=6\left[\frac{\partial \tilde{h}}{\partial \theta
}\right]_{ij}\frac{1}{F_{ij}}; \\
F_{ij} &=& 2\tilde{G}\left[\frac{1}{\Delta \theta ^{2}}+\frac{\lambda
}{\Delta \tilde{z}^{2}}\right];\quad 
\lambda =\left[\frac{R}{L}\right]^{2},
\end{eqnarray*}}\unskip
where $k$ is the iteration number, $\tilde{P}_{ij}$ is the
dimensionless pressure in node ($i$, $j$), and $a_{ij}$, $b_{ij}$,
$c_{ij}$, $d_{ij}$, $e_{ij}$, $f_{ij}$ are the discretization
coefficients.

Several mesh densities were tested to ensure grid-independent results,
and a $600\times 100$ grid was finally selected based on the
convergence analysis presented in the Appendix~\ref{appA}. The
iterative process was considered converged when, at every computational
point, the relative error between two successive iterations dropped
below a tolerance of $10^{-6}$.

\section{Validation}\label{sec4}

To validate the numerical method developed for the study of the
influence of surface texturing on the static characteristics of a
bearing lubricated with a polar fluid, we compared our results with
those obtained by Constantinescu et~al.~\cite{35}. The compared
parameters are: the attitude angle and the Sommerfeld number. The
calculations were performed for a Newtonian fluid ($\tilde{l} = 0$) for
three values of relative eccentricity ($\tilde{\varepsilon} = 0.0962$;
0.5374; 0.8349). The results obtained are reported in Table~\ref{tab1}.

%tab1
\begin{table}[b!]
\caption{\label{tab1}Comparison of theoretical results obtained for a
journal bearing of finite length ($R/L=0.5$)}
\begin{tabular}{ccccc}
\thead
$\tilde{\varepsilon}$ & 0.0962 & 0.5374 & 0.8349 & \\ 
\endthead
\xmorerows{1}{$S$} & 1.03540 & 0.1549 & 0.0345 & Reference results \\ 
& 1.35970 & 0.1555 & 0.0352 & Present work \\ 
\xmorerows{1}{$\phi$ (deg)} & 84.03\0\0\0\0 & 56.07\0\0\0 & 33.03\0\0\0 & Reference results \\
& 84.09\0\0\0\0 & 56.18\0\0\0 & 33.52\0\0\0 & Present work
\botline
\end{tabular}
\end{table}

The 30\% deviation observed at low eccentricity required further
mesh-sensitivity analyses (Appendix~\ref{appB}), which confirmed that
grid refinement has a negligible effect on the results. The deviation
is attributed to the low-load operating regime, which is characterized
by a thicker lubricant film and lower pressure, making the
load-carrying capacity sensitive to small variations. It is also
attributed to the higher experimental uncertainty typically associated
with such conditions.

However, the numerical model shows good agreement with the results of
Constantinescu et~al.\ for medium and high eccentricity ratios.

\section{Results and discussion}\label{sec5}

A parametric study was conducted to evaluate the combined effects of
surface texturing and rheology flow on the static performance of a
hydrodynamic bearing with finite length ($R/L=0.5$), lubricated with a
polar fluid where the dimensionless couple stress parameter vary from
0.0 to 0.50 ($\tilde{l}=0.0$ case of Newtonian fluid).

The solutions have been obtained for some combinations of relative
amplitude of texture (0.0; 0.10; 0.20; 0.35) and number of asperity
(10; 30; 60; 100). Noting that the relative amplitude of texture
($\tilde{a}_p=0.0$) correspond to the smooth bearing case.

The circumferential distributions of the static pressure are determined
for two values of the static relative eccentricity
($\tilde{\varepsilon}=0.5$ and 0.9) along the median section of the
bearing. In addition, the static performances were determined for a
relative eccentricity ratio varying from 0.05 to 0.9. 

To illustrate the combined effect of the textured surfaces and the
couple stress parameter on the static hydrodynamic pressure field, the
results obtained for a hydrodynamic textured bearing lubricated with a
polar fluid are compared to those of a hydrodynamic journal bearing
lubricated by a Newtonian fluid.

\subsection{Static hydrodynamic pressure}\label{ssec51}

In order to illustrate the combined effect of the textured surfaces and
the couple stress parameter on the static hydrodynamic pressure field,
the results obtained for a hydrodynamic textured bearing lubricated
with a polar fluid are compared to those of a hydrodynamic journal
bearing lubricated by a Newtonian fluid.

\subsubsection{Texture amplitude effects}\label{sssec511}

In order to analyze the effect of the texture amplitude on the static
pressure in the lubricant film, the number of periods was fixed at 60,
varying the value of the texture amplitude between 0.10 and 0.35.

Figure~\ref{fig3} shows the two-dimensional circumferential variations
of dimensionless static pressures in the median section of the
hydrodynamic bearing for different values of $\tilde{a}_{p}$ and
$\tilde{l}$, with $n_p=60$. It clearly shows that an increase in
texturing amplitude leads to a rise in maximum pressure, for both
Newtonian and non-Newtonian lubricants.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Circumferential static pressure in journal bearing
for different value of $\tilde{a}_{p}$ and $\tilde{l}$.}
\end{figure}

The effect of texturing amplitude becomes more pronounced as the couple
stress parameter increases. An analysis of the variations reveals that
the sensitivity of maximum pressure to texturing amplitude is
significantly enhanced for high values of the couple stress parameter
and relative eccentricity ratio ($\tilde{l}=0.50$ and
$\tilde{\varepsilon}=0.9$). Therefore, the effect of texturing amplitude
is more pronounced at high values of the couple stress.

The analysis of the static pressure distributions indicates that
increasing the texture amplitude generates local pressure peaks in the
convergence zones formed by the surface asperities, in agreement with
previous studies~\cite{5,6}. The regular and continuous arrangement of
textures enables the superposition of these local effects, leading to a
globally increasing pressure field that reflects the cumulative
influence of the surface texture. Moreover, the use of a polar
lubricant enhances pressure development.

\subsubsection{Texture number periods effects}\label{sssec512}

In order to show the effect of texture number on the static pressure in
the lubricant film, the texture amplitude was set to 0.1, varying the
texture number value between 10 and 100.

Figure~\ref{fig4} presents the circumferential variations of
dimensionless static pressure in the median section of the bearing.
Analysis of these profiles shows that: the effect of the number of
texture periods on the maximum pressure is positive. It also increases
with increasing of couple stress parameter and relative eccentricity
ratio.

\begin{figure}
\vspace*{4pt}
\includegraphics{fig04}
\vspace*{4pt}
\caption{\label{fig4}Circumferential static pressure in journal bearing
for different value of $n_{p}$ and $\tilde{l}$.} 
\vspace*{4pt}
\end{figure}

An increase in texture density, characterized by a higher number of
textures, creates additional converging zones, which induce local
pressure peaks. The superposition of these localized effects enhances
the global pressure field. This effect is further amplified by the use
of a couple stress lubricants.

\subsection{Static characteristics of journal bearing}\label{ssec52}

Figure~\ref{fig5}(a--d) presents static characteristics variation of
the hydrodynamic bearing with relative eccentricity for different
values of amplitude texture. The obtained results show:

\begin{figure}
\vspace*{4pt}
\includegraphics{fig05}
\vspace*{4pt}
\caption{\label{fig5}Dimensionless static characteristics for various
values of $\tilde{a}_{p}$ and $\tilde{l}$.}
\vspace*{4pt}
\end{figure}

For a given value of relative eccentricity ratio, the load-carrying
capacity increases with increasing of the amplitude of textures 
(Figure~\ref{fig4}a). This increase is more relevant for bearings
lubricated with non-Newtonian fluids and highly loaded. While the
attitude angle and friction number show a diminution with the increase
of the amplitude (Figure~\ref{fig5}b,c), which becomes more important
for the high values of couple stress parameter and those of the
relative eccentricity ratio. It can be concluded that the combination
of surface texturing and polar fluids improves the bearing load
capacity and reduces frictional losses in the hydrodynamic contacts.

At low relative eccentricity ratio the side leakage flow has a
significant increase with the increase of the amplitude
(Figure~\ref{fig5}d). This effect is more evident for high values of
the couple stresses parameter. Therefore, the association of textured
surfaces and polar fluids has a positive effect on the side leakage
flow at low relative eccentricity ratio, which assures a good lubricant
supply. This constant regeneration of lubricant in the contact area
helps to maintain a stable and continuous lubricating film.

\section{Conclusion}\label{sec6}

The parametric analysis demonstrated that both surface texturing with
continuous geometry and lubrication with polar fluids have a
significant influence on the static performance of a highly loaded
hydrodynamic journal bearing. Increasing the texture amplitude and
density creates converging zones where local pressure peaks are
generated, which results in a net increase in the static pressure. The
influence on the side leakage flow is limited; however, a noticeable
positive effect is observed at low eccentricity ratios. 

Compared with the plain bearing lubricated by Newtonian oils, the
textured bearing lubricated by couple stress fluids allows:
\begin{itemize}
\item High increase in load-carrying capacity and static pressure peak
in the lubricating film, especially for high values of the couple
stress parameter and relative eccentricity.
\item Reduction in the attitude angle and friction factor in the
contact improving energy efficiency and tribological system stability.
\item Provide a good lubricant supply, particularly beneficial for low
load bearings. This provides more reliable operation and reduces the
risk of failing due to insufficient lubrication.
\end{itemize}
These results confirm the importance of optimizing surface textures and
selecting the appropriate lubricating fluids to improve the performance
of hydrodynamic bearings under extreme operating conditions. They also
open up interesting prospects for the design of more efficient and
energy-saving tribological systems, particularly in industrial
applications where performance and durability requirements are becoming
increasingly stringent.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Nomenclature}

\begin{inftab}
\fontsize{10}{11.8}\selectfont
\begin{tabular}{ll}
\tbody
$\overset{\rightarrow}{A}$ & Body force per unit mass, (N${\cdot}$kg$^{-1}$) \\
$a_{p}$ & Amplitude of asperity, (m) \\
$\tilde{a}_{p}$ & Dimensionless amplitude of asperity \\
$C$ & Radial clearance, (m) \\
$\overset{\rightarrow}{E}$ & Body couple per unit mass, (N${\cdot}$m${\cdot}$kg$^{-1}$) \\
$e$ & Eccentricity, $e = |\overset{\rightarrow}{o_{a}o_{c}}|$, (m) \\
$F$ & Load carrying capacity, (N) \\
$\tilde{F}$ & Dimensionless load carrying capacity, $\tilde{F} = \sqrt{{\tilde{F}}_{1}^{2} + {\tilde{F}}_{2}^{2}}$ \\
$F_{1},F_{2}$ & Load carrying capacity components, (N) \\
$\tilde{F}_{1},\tilde{F}_{2}$ & Dimensionless load carrying capacity components, $(\tilde{F}_{1},\tilde{F}_{2}) = (F_{1},F_{2})/(\mu \omega _{a}RL(R/C)^{2})$ \\
$\tilde{f}$ & Friction number, $\tilde{f} = \tilde{C}_{a}/\tilde{F}$ \\
$h_{0}$ & Film thickness without texture, (m) \\
$h$ & Film thickness with texture, (m)  \\
$\tilde{h}$ & Dimensionless film thickness with texture, $\tilde{h} = h/C$ \\
$L$ & Bearing length, (m) \\
$l$ & Characteristics length of the additives, $l = \sqrt{\eta/\mu}$, (m) \\
$\tilde{l}$ & Dimensionless couple stress parameter, $\tilde{l} = l/C$ \\ 
$n_{p}$ & Number of asperity \\
$o_{a}$ & Center of the journal \\
$o_{c}$ & Center of the bearing \\
$p$ & Lubricant pressure, (Pa) \\
$\tilde{p}$ & Dimensionless lubricant pressure, $\tilde{p} = p/(\mu\omega _{a}(R/C)^{2})$ \\
$Q$ & Side leakage flow, (m\tsup{3}${\cdot}$s$^{-1}$) \\\inftabbreak
$\tilde{Q}$ & Dimensionless side leakage flow, $\tilde{Q} = LQ/(C\omega _{a}R^{3})$ \\
$R_{a}$ & Journal radius, (m) \\    
$R_{c}$ & Bearing radius, (m) \\
$S$ & Sommerfeld number, $S = (\mu \omega _{a}RL(R/C)^{2})/\uppi F$ \\
$T_{a}$ & Friction couple, (N/m) \\    
$\tilde{T}_{a}$ & Dimensionless friction couple, $(\tilde{T}_{a}) = (C/(\mu \omega _{a}R^{3}L))(T_{a})$ \\
$t$ & Temps, (s) \\
$\overset{\rightharpoonup}{V}$ & Velocity field, (m/s) \\
$u$ & The velocity components in the $x$ direction, (m/s) \\
$w$ & The velocity components in the $z$ direction, (m/s) \\
$x,y,z$ & Axial coordinates of bearing, (m) \\
$\tilde{z}$ & Dimensionless axial coordinate of bearing, $\tilde{z} = z/L$ \\
$\delta$ & Film thickness variation caused by the presence of texture, (m) \\
$\tilde{\varepsilon}$ & Relative eccentricity, $\tilde{\varepsilon} = e/C$ \\
$\eta$ & Material constant responsible for the couple stress fluid property, (N${\cdot}$s) \\
$\theta$ & Circumferential coordinate starting from the line of the centers, $\theta = x/R$, (rad) \\
$\theta _{c}$ & Cavitation angle, (rad) \\
$\mu$ & Lubricant dynamic viscosity, (Pa${\cdot}$s) \\
$\rho$ & Lubricant density, (kg/m\tsup{3}) \\
$\phi$ & Attitude angle, (rad) \\
$\Omega$ & Over relaxation coefficient, $1 < \Omega < 2$ \\
$\lambda$ & Wavelength of asperity, (m) \\
$\omega_{a}$ & Journal angular velocity, (rad/s) \\
$\nabla$ & Gradient operator \\
$\sim$ & On top of a variable denotes dimensionless quantity \\
$X$ & Vector (cross) product
\end{tabular}
\end{inftab}

\back{}

\appendix{}

\def\appendixlabel{Appendix}

\section{Numerical experiment and mesh convergence analysis}
\label{appA}

A mesh convergence study was carried out to ensure the reliability and
accuracy of the numerical results. This step is an essential part of
validating the numerical model and aims to verify that the computed
quantities no longer depend significantly on mesh refinement.

Several meshes with increasing densities were tested and the results
were compared using a relative variation criterion. The objective of
this numerical experiment is to determine the optimal mesh that
balances computational cost with solution accuracy and ensures
numerical stability.

Table~\ref{tab2} presents the results obtained from the computational
code, illustrating the variation of the journal bearing's
characteristic quantities with mesh refinement.

%tab2
\begin{table}
\caption{\label{tab2}Influence of mesh refinement on the computed
bearing performance characteristics}
\tabcolsep=0pt
\begin{tabular}{cccccc}
\thead
Parameters & $N_{\theta}\times N_z$ & 
\parbox[t]{5pc}{\centering Calculation time CPU (s)} &
\parbox[t]{6pc}{\centering Maximum pressure} & 
\parbox[t]{6pc}{\centering Load-carrying capacity} & 
\parbox[t]{3pc}{\centering Attitude angle}\vspace*{2pt} \\ 
\endthead
\morerows{3}{\parbox[t]{6pc}{\centering $\varepsilon = 0.1$, $\tilde{l} = 0.0$, $\tilde{a}_{p} = 0.0$, $n_p = 0.0$}} 
& $M1 = 150 \times 25$  & 7.08       & 0.22 & 0.24 & 83.84 \\
& $M2 = 300 \times 50$  & 7.80       & 0.22 & 0.24 & 83.84 \\
& $M3 = 600 \times 100$ & 30.88\0    & 0.22 & 0.24 & 83.84 \\
& $M4 = 1200\times 200$ & 117.39\0\0 & 0.22 & 0.24 & 83.84 \vspace*{6pt}\\ 

\morerows{3}{\parbox[t]{6pc}{\centering $\varepsilon = 0.9$, $\tilde{l} = 0.0$, $\tilde{a}_{p} = 0.0$, $n_p = 0.0$}}  
& $M1 = 150 \times 25 $ & 4.84       & 34.03 & 16.77 & 26.66 \\
& $M2 = 300 \times 50 $ & 9.91       & 34.10 & 16.85 & 26.69 \\
& $M3 = 600 \times 100$ & 24.97\0    & 34.10 & 16.87 & 26.70 \\
& $M4 = 1200\times 200$ & 133.14\0\0 & 34.10 & 16.88 & 26.70 \vspace*{6pt}\\

\morerows{3}{\parbox[t]{6pc}{\centering $\varepsilon = 0.1$, $\tilde{l} = 0.5$, $\tilde{a}_{p} = 0.0$, $n_p = 0.0$}}  
& $M1 = 150 \times 25 $ & 4.96       & 0.81 & 0.87 & 80.94 \\
& $M2 = 300 \times 50 $ & 6.88       & 0.81 & 0.87 & 80.93 \\
& $M3 = 600 \times 100$ & 21.83\0    & 0.81 & 0.87 & 80.93 \\
& $M4 = 1200\times 200$ & 126.40\0\0 & 0.81 & 0.87 & 80.93 \vspace*{6pt}\\ 

\morerows{3}{\parbox[t]{6pc}{\centering $\varepsilon = 0.9$, $\tilde{l} = 0.5$, $\tilde{a}_{p} = 0.0$, $n_p = 0.0$}}  
& $M1 = 150 \times 25 $ & 6.05       & 2690.84 & 871.34 & 13.60 \\
& $M2 = 300 \times 50 $ & 7.14       & 2735.62 & 891.07 & 13.68 \\
& $M3 = 600 \times 100$ & 29.16\0    & 2739.58 & 895.41 & 13.71 \\
& $M4 = 1200\times 200$ & 131.45\0\0 & 2740.97 & 896.51 & 13.72 \vspace*{6pt}\\ 

\morerows{3}{\parbox[t]{6pc}{\centering $\varepsilon = 0.5$, $\tilde{l} = 0.0$, $\tilde{a}_{p} = 0.1$, $n_p = 10$}}  
& $M1 = 150 \times 25 $ & 6.74       & 2.45 & 1.10 & 60.07 \\
& $M2 = 300 \times 50 $ & 11.89\0    & 2.44 & 1.10 & 60.08 \\
& $M3 = 600 \times 100$ & 26.09\0    & 2.44 & 1.10 & 60.08 \\
& $M4 = 1200\times 200$ & 110.24\0\0 & 2.44 & 1.10 & 60.08 \vspace*{6pt}\\ 

\morerows{3}{\parbox[t]{6pc}{\centering $\varepsilon = 0.5$, $\tilde{l} = 0.5$, $\tilde{a}_{p} = 0.1$, $n_p = 10$}}  
& $M1 = 150 \times 25 $ & 5.81       & 16.80 & 11.13 & 48.25 \\
& $M2 = 300 \times 50 $ & 6.61       & 16.71 & 11.12 & 48.30 \\
& $M3 = 600 \times 100$ & 27.75\0    & 16.70 & 11.11 & 48.31 \\
& $M4 = 1200\times 200$ & 146.68\0\0 & 16.69 & 11.11 & 48.31 \vspace*{6pt}\\ 

\morerows{3}{\parbox[t]{8pc}{\centering $\varepsilon = 0.9$, $\tilde{l} = 0.0$, $\tilde{a}_{p} = 0.35$, $n_p = 100$}}  
& $M1 = 150 \times 25$  & 4.41       & 49.46 & 21.39 & 25.20 \\
& $M2 = 300 \times 50$  & 8.49       & 60.66 & 25.65 & 22.48 \\
& $M3 = 600 \times 100$ & 32.15\0    & 58.06 & 25.08 & 22.77 \\
& $M4 = 1200\times 200$ & 158.38\0\0 & 59.15 & 25.49 & 22.65 \vspace*{6pt}\\ 

\morerows{3}{\parbox[t]{8pc}{\centering $\varepsilon = 0.9$, $\tilde{l} = 0.50$, $\tilde{a}_{p} = 0.35$, $n_p = 100$}}
& $M1 = 150 \times 25$   & 5.49       & \07887.24  & 1788.28 & 10.50 \\
& $M2 = 300 \times 50$   & 10.96\0    & 12119.53 & 2987.36 & \09.04 \\
& $M3 = 600 \times 100$  & 37.67\0    & 10528.61 & 2678.27 & \09.79 \\
& $M4 = 1200\times  200$ & 171.81\0\0 & 10830.71 & 2710.44 & \09.94
\botline
\end{tabular}
\end{table}

The influence of mesh refinement on numerical stability was evaluated
using a relative variation criterion defined as
{\begin{equation*}
\Delta _{i}(\%) = \dfrac{|\psi _{{{M}_{i+1}}} - \psi _{{{M}_{i}}}|}{\psi _{{{M}_{i+1}}}} \times 100,
\end{equation*}}\unskip
where $\Delta _{i}(\%)$ is the relative variation in percent of the
physical quantity ${\psi}$ between two successive meshes $M_i$ and
$M_{i+1}$; $\psi_{{{M}_{{i}}}}$ and $\psi_{{{M}_{{i}+1}}}$ denote the
values of the considered quantity obtained with meshes $M_i$ and
$M_{i+1}$ respectively; $M_i$ considered mesh level, defined by a
specific number of axial and circumferential divisions; $M_{i+1}$
represent a progressive refinement of the computational domain used for
the numerical convergence analysis.

This criterion evaluates the numerical stability and convergence of the
results with respect to mesh refinement. The corresponding results are
summarized in Table~\ref{tab3}.

%tab3
\begin{table}
\caption{\label{tab3}Relative variations of maximum pressure,
load-carrying capacity, and attitude angle}
\tabcolsep=2.8pt
\begin{tabular}{cccccccccc}
\thead
Parameters & 
\multicolumn{3}{c}{\parbox[t]{6pc}{\centering Maximum pressure $\tilde{P}_{\max}$}} & 
\multicolumn{3}{c}{\parbox[t]{6pc}{\centering Load-carrying capacity $\tilde{F}$}} & 
\multicolumn{3}{c}{\parbox[t]{3pc}{\centering Attitude angle $\varphi$}}\vspace*{2pt} 
\\\cline{2-4}\cline{5-7}\cline{8-10}
& $\Delta_1$ (\%) & $\Delta_2$ (\%) & $\Delta_3$ (\%) &
  $\Delta_1$ (\%) & $\Delta_2$ (\%) & $\Delta_3$ (\%) &
  $\Delta_1$ (\%) & $\Delta_2$ (\%) & $\Delta_3$ (\%) \\
\endthead
\parbox[c]{6pc}{\centering $\varepsilon = 0.1$, $\tilde{l} = 0.0$, 
$\tilde{a}_{p} = 0.0 $, $n_p = 0.0$} & 0.00  & 0.00      & 0.00 & 0.00 
& 0.00  & 0.00 & 0.00  & 0.00 & 0.00 \vspace*{5pt}\\

\parbox[c]{6pc}{\centering $\varepsilon = 0.9$, $\tilde{l} = 0.0$, 
$\tilde{a}_{p} = 0.0 $, $n_p = 0.0$} & 0.20  & 0.00      & 0.00 & 0.47 
& 0.11  & 0.05 & 0.11  & 0.04 & 0.00 \vspace*{5pt}\\

\parbox[c]{6pc}{\centering $\varepsilon = 0.1$, $\tilde{l} = 0.5$, 
$\tilde{a}_{p} = 0.0 $, $n_p = 0.0$} & 0.00  & 0.00      & 0.00 & 0.00 
& 0.00  & 0.00 & 0.01  & 0.00 & 0.00 \vspace*{5pt}\\

\parbox[c]{6pc}{\centering $\varepsilon = 0.9$, $\tilde{l} = 0.5$, 
$\tilde{a}_{p} = 0.0 $, $n_p = 0.0$} & 1.64  & 0.14      & 0.05 & 2.21 
& 0.48  & 0.12 & 0.58  & 0.22 & 0.07 \vspace*{5pt}\\

\parbox[c]{6pc}{\centering $\varepsilon = 0.5$, $\tilde{l} = 0.0$, 
$\tilde{a}_{p} = 0.1 $, $n_p = 10 $} & 0.41  & 0.00      & 0.00 & 0.00 
& 0.00  & 0.00 & 0.02  & 0.00 & 0.00 \vspace*{5pt}\\

\parbox[c]{6pc}{\centering $\varepsilon = 0.5$, $\tilde{l} = 0.5$, 
$\tilde{a}_{p} = 0.1 $, $n_p = 10 $} & 0.54  & 0.06      & 0.06 & 0.09 
& 0.09  & 0.00 & 0.10  & 0.02 & 0.00 \vspace*{5pt}\\

\parbox[c]{8pc}{\centering $\varepsilon = 0.9$, $\tilde{l} = 0.0$, 
$\tilde{a}_{p} = 0.35$, $n_p = 100$} & 18.46\0 & 4.48    & 1.84 &
16.60\0 & 2.27  & 1.60 & 12.09\0 & 1.27 & 0.52 \vspace*{5pt}\\

\parbox[c]{8pc}{\centering $\varepsilon = 0.9$, $\tilde{l} = 0.50$,
$\tilde{a}_{p} = 0.35$, $n_p = 100$} & 34.92\0 & 15.11\0 & 2.79 &
40.14\0 &11.54\0& 1.19 & 16.15\0 & 7.67 & 1.51\vspace*{2pt}
\botline
\end{tabular}
\end{table}

The comparison between mesh levels shows that the computed results are
practically insensitive to further refinement, with differences between
the ($600\times 100$) and ($1200\times 200$) meshes are under 1\% for
all  cases and not exceeding 3\% for the case with: $\varepsilon=0.9$,
$\tilde{l}=0.50$, $\tilde{a}_{p}=0.35$, $n_p=100$. This consistency
confirms the numerical robustness of the model and the results'
\mbox{validity.}

Therefore, the ($600 \times 100$) mesh was selected as the optimal
configuration, as it provides an appropriate balance between
computational accuracy and cost.

\section{Low-eccentricity case: additional numerical tests}
\label{appB} 

Table~\ref{tab4} presents the results of additional numerical tests for
the low-eccentricity case. These results highlight the influence of
mesh refinement on the computed bearing performance parameters.

\vspace*{8pt}

%tab4
\begin{table}[h!]
\caption{\label{tab4}Numerical results for the low-eccentricity case}
\begin{tabular}{cccccc}
\thead
Parameters & $N_{\theta}\times N_z$ & 
\parbox[t]{3.4pc}{\centering Maximum pressure} & 
\parbox[t]{5pc}{\centering Load-carrying capacity} & 
\parbox[t]{2.6pc}{\centering Attitude angle} & 
\parbox[t]{4.2pc}{\centering Sommerfeld number}\vspace*{2pt} \\ 
\endthead
\morerows{3}{\parbox[t]{7pc}{\centering $\varepsilon = 0.0962$, $\tilde{l} = 0.0$, $\tilde{a}_{p} = 0.0$, $n_p = 0.0$}}
& $M1 = 150 \times 25 $ & 0.22 & 0.23 & 84.07 & 1.36 \\
& $M2 = 300 \times 50 $ & 0.22 & 0.23 & 84.07 & 1.36 \\
& $M3 = 600 \times 100$ & 0.22 & 0.23 & 84.07 & 1.36 \\
& $M4 = 1200\times 200$ & 0.22 & 0.23 & 84.07 & 1.36
\botline
\end{tabular}
\end{table}

\vspace*{-1pc}

\printbibliography
\refinput{crmeca20250230-reference.tex}

\end{document}
