Abstract
This paper proposes a mathematical analysis of the inertial flow of an MHD second-grade non-Newtonian fluid in a ciliated channel. The two-dimensional flow is modelled under the effect of inertial forces, magnetic field and Darcy’s resistance, which make the system of partial differential equations highly non-linear. To solve the complex system of partial differential equations, the Homotopy Perturbation Method (HPM) is preferred. The HPM solutions for the velocity profile, stream function and pressure gradient are obtained using the software MATHEMATICA. The significances of the Reynolds number (due to inertial forces), Hartmann number (due to magnetic field), porosity parameter (due to Darcy’s resistance) and fluid parameters (related to the second-grade fluid) on the pressure gradient, stream function and velocity profile are discussed in detail. The pertinent parameters show that the horizontal velocity decays in the presence of a magnetic field, whereas it rises under the effect of inertial forces, Darcy’s resistance and fluid viscosity in the centre of the channel. This research indicates that, for the ciliary flow of a second-grade fluid, a favourable pressure gradient (negative pressure gradient) in the horizontal direction increases when applying a magnetic field, whereas it decreases due to the porous medium. This mathematical model can be helpful to observe ciliary activity under magnetic resonance imaging, when ciliary activity is abnormal.
Highlights
Cilia are small hair-like structures found in most microorganisms, human and animals.Cilia play a key role in the respiratory system, and in the reproductive and digestive systems of humans and for the locomotion of small animals in biofluids
It is observed that the horizontal flow of the Newtonian fluid is larger than that of the second-grade fluid because the second-grade fluid has high viscosity compared with the Newtonian fluid
The distributions of velocity, pressure and shear stress are smooth. They satisfy the boundary conditions defined for the velocity profile with decreasing coefficients of higher-order solutions, which guarantees that the series solution obtained with the Homotopy Perturbation Method (HPM) converges in the region −h < y < h and provides the best results of the present study
Summary
Cilia are small hair-like structures found in most microorganisms, human and animals. Cilia play a key role in the respiratory system, and in the reproductive and digestive systems of humans and for the locomotion of small animals in biofluids. Motile cilia responsible for the transport of biofluids exhibit a rhythmic waving motion that forms metachronal waves. The key role of metachronal waves is to regulate the continuity of the flow. Ciliated surfaces can exhibit various beating patterns and generate multiple types of metachronal waves, of which, the two most frequent ones are symplectic and antiplectic metachronal waves, considered for the first time by Knight [1]. Few authors explored the motion of cilia forming symplectic wave patterns with numerous biotic fluids using analytical and numerical techniques, including
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