## 2 Problem Statement  
We consider a vertical channel of uniform width and a rectangular coordinate system \(\left(\hat{X},\hat{Y}\right)\), where the \(\hat{Y}-axis\) of the of the system coincides with its centerline and the \(\hat{X}-axis\) lies along its normal direction. The inner surface was carpeted with a highly dense mat of cilia, which was assumed to behave as a Darcian porous medium [35]. The tips of the cilia performed synchronous periodic beatings and formed a sinusoidal wave called the metachronal wave traveling with a uniform speed \(c\) along the surface made by the cilia tips. A magnetized, viscoelastic, Jeffrey-type liquid was placed within the channel under the influence of a magnetic body force. The imposed magnetic field had a constant strength \(B_{0}\) and acted normal to the flow. The geometry of the problem is shown in Figure 1. We neglected the induced magnetic field by taking the magnetic Reynolds number to be very small. The analysis of the heat and mass transfer was also taken into consideration by giving the temperature \(T_{1}\) and concentration \(C_{1}\) on the walls of the channel. The well-known symmetry conditions for both the temperature and concentration fields were employed at the center of the channel. Moreover, the study of thermal effects on biochemical reactions was necessary to examine the various metabolisms taking place within the human body. Therefore, the effects of chemical reactions and heat addition (absorption) were also considered in the present study. By considering the above assumptions, we defined the mass, momentum, energy, and concentration equations with the usual Boussinesq approximation as:  
\[\frac{\partial\hat{U}}{\partial\hat{X}}+\frac{\partial\hat{V}}{\partial\hat{Y}} =0, \tag{1}\]  
\[\rho\left[\frac{\partial\Omega}{\partial\hat{t}}+\hat{U}\frac{\partial\Omega}{ \partial\hat{X}}+\hat{V}\frac{\partial\Omega}{\partial\hat{Y}}\right]=-\frac{ \partial\hat{P}}{\partial\hat{X}}+\frac{\partial S_{\hat{X}\hat{X}}}{\partial \hat{X}}+\frac{\partial S_{\hat{X}\hat{Y}}}{\partial\hat{Y}}+\rho g\alpha_{T}( T-T_{0})+\rho g\alpha_{m}(C-C_{0})-\left(\sigma B_{0}^{2}+\frac{\mu}{K_{p}} \right)\hat{U}, \tag{2}\]  
\[\rho\left[\frac{\partial\hat{V}}{\partial\hat{t}}+\hat{U}\frac{\partial\hat{V }}{\partial\hat{X}}+\hat{V}\frac{\partial\hat{V}}{\partial\hat{Y}}\right]=- \frac{\partial\hat{P}}{\partial\hat{Y}}+\frac{\partial S_{\hat{Y}\hat{X}}}{ \partial\hat{X}}+\frac{\partial S_{\hat{Y}\hat{Y}}}{\partial\hat{Y}}-\frac{\mu }{K_{p}}\hat{V}, \tag{3}\]  
\[\rho C_{p}\left[\frac{\partial T}{\partial\hat{t}}+\hat{U}\frac{\partial T}{ \partial\hat{X}}+\hat{V}\frac{\partial T}{\partial\hat{Y}}\right]=K\left[ \frac{\partial^{2}T}{\partial\hat{X}^{2}}+\frac{\partial^{2}T}{\partial\hat{Y }^{2}}\right]+Q_{0}, \tag{4}\]  
\[\left[\frac{\partial C}{\partial\hat{t}}+\hat{U}\frac{\partial C}{\partial \hat{X}}+\hat{V}\frac{\partial C}{\partial\hat{Y}}\right]=D\left[\frac{ \partial^{2}C}{\partial\hat{X}^{2}}+\frac{\partial^{2}C}{\partial\hat{Y}^{2}} \right]+\frac{DK_{T}}{T_{m}}\left[\frac{\partial^{2}T}{\partial\hat{X}^{2}}+ \frac{\partial^{2}T}{\partial\hat{Y}^{2}}\right]-K_{c}(C-C_{0}). \tag{5}\]  
In the above equations, \(\hat{U}\), \(\hat{V}\), \(\hat{P}\), and \(T\) denote the axial velocity component, the normal velocity component, the fluid pressure, and the fluid temperature, respectively. Moreover, \(\rho\) represents the fluid density, \(\mu\) denotes the viscosity coefficient, \(g\) is the acceleration due to gravity, \(\alpha_{T}\) is the thermal expansion coefficient, \(\alpha_{m}\) is the concentration expansion coefficient, \(\sigma\) is the electrical conductivity, \(K_{p}\) is the permeability of the porous medium, \(C_{p}\) is the specific heat, \(K\) is the thermal conductivity, \(Q_{0}\) stands for the constant heat addition (absorption), \(D\) represents the mass diffusivity coefficient, \(K_{T}\) is the thermal diffusion ration, \(T_{m}\) is the mean medium temperature, and \(K_{c}\) is the chemical reaction parameter.