This model introduces the volumetric magnetic force $f_m={\mu }_0\left(M\cdot \nabla \right)H$, where $M$ is the magnetization and $H$ is the magnetic field [3] . It does not account for magnetic moment or rotation.

A static isothermal model based on angular momentum [4] . The equation of motion incorporates:

• Magnetic force coefficient ${\lambda }_3$;

• Jenkins viscosity ${\lambda }_4={\eta }_j$;

• Direct influence of the magnetic field on ferromagnetic particles.

This model accounts for rotational viscosity and magnetic moment [5] . The fluid equations include:

• ${\eta }_s=\frac{{\mu }_0{M}_0^2\tau }{4}$

• Volumetric torque $T_v=M\times H$

• Thermal and relaxation effects: ${\tau }_s$, ${\lambda }_1$, ${\lambda }_2$

This study will focus particularly on the Shliomis model. The Shliomis model takes into account the rotational viscosity and the magnetization parameter, which shows that this model is realistic as it fully considers the effects of the magnetic field without excessive assumptions.

We have:

$h=O_{c}M-O_c{M}^{\prime }$

$R_c-O_c{M}^{\prime }=R_a+C-O_c{M}^{\prime }$

By applying the law of sines to triangle $O_a{M}^{\prime }{O}_c$ ( Figure 1 ), we get:

$\frac{O_a{M}^{\prime }}{\sin \left(O_a{O}_c{M}^{\prime }\right)}=\frac{e}{\sin \alpha }=\frac{O_c{M}^{\prime }}{\sin \left(O_c{O}_a{M}^{\prime }\right)}=\frac{R_a}{\sin \left(\pi -\theta \right)}=\frac{R_a}{\sin \theta }$

Thus,

$\{\begin{array}{l}\sin \alpha =\frac{e}{R_a}\sin \theta \\ O_c{M}^{\prime }=\frac{R_a}{\sin \theta }\cdot \sin \left(O_c{O}_a{M}^{\prime }\right)\end{array}$

Since $O_c{O}_a{M}^{\prime }=\theta -\alpha$, we have:

$O_c{M}^{\prime }=\frac{R_a}{\sin \theta }\cdot \sin \left[\theta -\arcsin \left(\frac{e}{R_a}\sin \theta \right)\right]$

After development and simplification, we obtain:

$O_c{M}^{\prime }=R_a\sqrt{1-{\left(\frac{e}{R_a}\sin \theta \right)}^2}-e+\cos \theta$

Since the ratio $\frac{e}{R_a}<\frac{c}{R}$ is very small (on the order of 10⁻³), we can neglect the term ${\left(\frac{e}{R_a}\sin \theta \right)}^2$.

The film thickness can then be expressed as:

$h\left(\theta \right)=c\left(1+\epsilon \cos \theta \right)$

Finite difference methods are used to discretize the modified Reynolds equation:

$\begin{array}{l}\frac{\partial f\left(\theta ,\tilde{z}\right)}{\partial \theta }\frac{\partial \tilde{p}}{\partial \theta }+f\left(\theta ,\tilde{z}\right)\frac{{\partial }^2\tilde{p}}{\partial {\theta }^2}+{\gamma }_1\frac{\partial g\left(\theta ,\tilde{z}\right)}{\partial \tilde{z}}\frac{\partial \tilde{p}}{\partial \tilde{z}}+{\gamma }_{1}g\left(\theta ,\tilde{z}\right)\frac{{\partial }^2\tilde{p}}{\partial {\tilde{z}}^2}\\ =6\frac{\partial \tilde{h}}{\partial \theta }+{\gamma }_2\frac{\partial F\left(\theta ,\tilde{z}\right)}{\partial \theta }\frac{\partial {\tilde{H}}_x}{\partial \theta }+{\gamma }_{2}F\left(\theta ,\tilde{z}\right)\frac{{\partial }^2{\tilde{H}}_x}{\partial {\theta }^2}\end{array}$(6)

where:

$f\left(\theta ,\tilde{z}\right)=\frac{{\tilde{h}}^3}{1+\frac{3}{2}\phi \frac{\xi -\tanh \xi }{\xi +\tanh \xi }},\text{with}\tilde{h}\left(\theta \right)=1+\epsilon \cos \theta$

$\xi ={\lambda }_1\tilde{H},{\lambda }_1=\frac{{\mu }_0\mu }{K_{B}T}{H}_0,{\gamma }_1=\frac{R^2}{L^2}$

$g\left(\theta ,\tilde{z}\right)=\frac{{\tilde{h}}^3}{1+\frac{3}{2}\phi \frac{\xi -\tanh \xi }{\xi +\tanh \xi }{\left(\frac{{\tilde{H}}_y}{\tilde{H}}\right)}^2},{\gamma }_2={\lambda }_2\left(\frac{\phi }{1+2.5\phi }\right)$

${\lambda }_2=\frac{C^2{\mu }_0{H}_0\mu }{R^2{\eta }_0\omega V},F\left(\theta ,z\right)=\left(\coth \xi -\frac{1}{\xi }\right)\frac{{\tilde{H}}_x}{\tilde{H}}$

After discretizing the modified Reynolds equation, the Gauss-Seidel method with relaxation is used:

${\tilde{p}}_{i,j}^{N_{\text{iter}}+1}=\left(1-\sigma \right){\tilde{p}}_{i,j}^{N_{\text{iter}}}+\sigma \left(a_{ij}{\tilde{p}}_{i+1,j}^{N_{\text{iter}}}+b_{ij}{\tilde{p}}_{i-1,j}^{N_{\text{iter}}+1}+c_{ij}{\tilde{p}}_{i,j+1}^{N_{\text{iter}}}+d_{ij}{\tilde{p}}_{i,j-1}^{N_{\text{iter}}+1}+e_{ij}\right)$(7)

where $\sigma$ is the relaxation coefficient, and $N_{\text{iter}}$ is the iteration number.

To ensure convergence, in the MATLAB solution we set the condition: if $N_{{\text{iter}}_{\text{max}}}=N_{\text{iter}}$, then no convergence; otherwise, convergence.