## 1 Introduction
These authors modelled the fractional Burgers fluid through a ciliated channel with the periciliary layer and mucus, also velocity profiles had obtained for both types of wave pattern and concluded that antiplectic waves are more efficient for the transportation of a viscous fluid. Most bionic fluids are viscoelastic in nature [13]. Viscoelasticity is besides the essential rheological property of bronchial mucus [14,15]. Therefore, it is of prime importance to consider the non-Newtonian behaviour of mucus when investigating mucociliary transport. For the first time, the viscoelastic second-grade fluid flow due to the ciliary motion [16,17] is considered here under the effect of a magnetic field and inertial forces. Furthermore, the interest in magnetohydrodynamics flows (MHD) has grown because of their broad industrial and health applications [18, 19, 20, 21]. [PERSON] et al. [22] investigated the application of mucyltics (i.e., the process of reducing elasticity, rigidity and viscosity of mucus by breaking down the chemical bonds in mucus) to enhance particle retention under moderate magnetic fields. For this, they used the frog plate model to compare the two different mucyltics and found that N-acetylcysteine and dextran sulphate are effective for moderate and sufficiently large particle concentrations, respectively. Some other investigators considered the MHD flow of linear and nonlinear fluids due to ciliary motion. [PERSON] et al. [23] studied the influence of magnetohydrodynamics (MHD) for a non-Newtonian Jeffrey fluid flow produced by ciliary motion in a porous medium. The results were obtained by a semi-analytical Adomian decomposition method. [PERSON] et al. [24] considered the Casson fluid model to study the influence of MHD for solid-liquid suspensions on the ciliary motion in a porous medium. For the same configuration, [PERSON] et al. [25] modelled the problem without applying the longwave length and low Reynolds number approximation. A series solution by the Adomian Decomposition Method was obtained, and the results showed that the magnetic field and [PERSON]'s resistance reduced the magnitude of the velocity field. [PERSON] et al. [26] studied the MHD viscous flow for a ciliated planar channel embedded in a porous medium and concluded that mucus congestion in the respiratory tract can be cleaned by drugs containing magnetic nanoparticles. A magnetic field has been widely used together with artificial cilia in microfluidic devices for mixing applications. These cilia can also be used for cell transportation, antifouling surfaces, biochemical targets such as proteins and chemical agent sensing. [PERSON] et al. [27] investigated fluid motion due to self-assembled artificial cilia using chains of super-paramagnetic beads. The use of a magnetic field to actuate beating cilia in a simple non-reciprocal beating manner caused a uniform motion above the cilia tips. [PERSON] et al. [28] developed a numerical model of fluid transport due to the motion of artificial cilia made by super-paramagnetic elastic filaments that are actuated by an applied magnetic field. A better control of artificial cilia was obtained by driving them externally, using either a magnetic or an electric field. [PERSON] et al. [29] studied the inertial flow by considering flagellated micro-organisms swimming in an MHD viscoelastic second-grade fluid. They demonstrated that propulsion in the presence of a magnetic field surges for low inertial forces and drops for higher ones. Fluid flows through a porous medium have numerous real-world applications in nature, e.g., natural flows in porous medium can be observed under pathological conditions of gallstones when they drop into the bile tubes and close them incompletely or fully. [PERSON]'s law has also been used to study the ciliary flow of a biofluid assuming the ciliated carpet as a porous medium. [PERSON] et al.