## 2 Materials and Methods  
### Geometrical Modelling and Governing Equations  
Let us consider the ciliary flow of an incompressible second-grade fluid [31] in a symmetric conduit under a constant applied magnetic field, as illustrated in Figure 1a. Assume that an infinite number of continuously beating cilia are present along the inner walls of the conduit, causing symplectic or antiplectic metachronal waves, which move towards the positive or negative x-direction with a wave speed denoted by c. The cilia bed for the mucus flow is considered as a porous medium, as already suggested by [PERSON] et al. [32]. In the present research, the envelope model approach [3] is applied to transport the second-grade fluid by ciliary motion. To study the ciliary flow, the two-dimensional momentum and continuity equations are considered under the effect of a constant magnetic field and inertial forces. A single cilium is assumed to follow an elliptical path (Figure 1b), but collectively, cilia produce a metachronal wave in the (\(x\),\(y\)) plane. The mathematical representation of the fluid near the cilia tip is written as follows:  
\[\overline{X}=F\big{(}\overline{X},\overline{t}\big{)}=\overline{X}_{0}+ \varepsilon aasin\left(\frac{2\pi}{\lambda}\right)\big{(}\overline{X}-c \overline{t}\big{)}, \tag{1}\]  
\[Y=H\big{(}\overline{X},\overline{t}\big{)}=\pm\left[a+\varepsilon acos\left( \frac{2\pi}{\lambda}\right)\big{(}\overline{X}-c\overline{t}\big{)}\right]= \pm L, \tag{2}\]  
in which \((\overline{X}_{0}\),\(a)\) is the reference position of the cilium tip, \(\alpha\) is the eccentricity of the elliptical motion, \(2L\) is the length of the vertical region covered by the mucus, \(\varepsilon\) is the cilium length, \(\lambda\) is the wavelength of the metachronal wave, \(t\) is time, and \(c\) is the speed of the metachronal waves. Figure 1: (**a**) Geometry of the channel with the ciliated bed; (**b**) recovery and effective stroke of a single cilium forming an elliptical path. The expressions for the fixed and wave frames are linked by the following relations:  
\[\overline{u}=\overline{U}-c,\ \ \ \overline{v}=\overline{V},\ \ \ \overline{x}= \overline{X}-c\overline{t},\]  
\[\overline{y}=\overline{Y},\ \ \ \overline{p}\big{(}\overline{x},\overline{y}, \overline{t}\big{)}=\overline{P}\big{(}\overline{X},\overline{Y},\overline{t} \big{)}, \tag{3}\]  
where \((\overline{x},\overline{y})\) and \((\overline{u},\overline{v})\) are the independent and dependent variables in the wave frame and \((\overline{X},\overline{Y})\) and \((\overline{U},\overline{V})\) are the independent and dependent variables in the fixed frame. Non-dimensional quantities are introduced by the following expression:  
\[x=\frac{\overline{x}}{\lambda},\ \ \ y=\frac{\overline{y}}{a},\ \ \ u=\frac{ \overline{u}}{c},\ \ \ v=\frac{\lambda\overline{v}}{ac},\ \ \ p=\frac{a^{2}\overline{p}}{c\lambda\mu},\ \ \ t=\frac{c\overline{t}}{\lambda},\ \ \ h=\frac{L}{a},\ \ \ \beta=\frac{a}{\lambda}\]  
\[Re=\frac{\rho ca}{\mu},\ \ \ \frac{1}{K}=\frac{\rho a^{2}}{k},\ \ \ M^{2}=\frac{\sigma B_{0} ^{2}a^{2}}{\mu},\ \ \ \lambda_{1}=\frac{\alpha_{1}c}{\mu a},\ \ \ \lambda_{2}=\frac{\alpha_{2}c}{\mu a}. \tag{4}\]  
where \(p\) is pressure, \(2h\) is the mean width of the region occupied by the mucus, and the rest of the quantities are defined in the nomenclature. The conservative laws of mass and momentum for the second-grade fluid model [31; 32; 33] in a non-dimensional form are as follows:  
\[\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}, \tag{5}\]  
\[