Because of the diverse applications of cilia propulsions and viscoelastic fluids in biochemical engineering, we consider the flow of a non-Newtonian fluid through both convergent and divergent channels. The transportation of fluid through the flow geometry takes place due to metachronal waves that exist at both walls. Expressions of the Jeffrey fluid in the presence of magnetic and electric fields are used in the development of rheological equations. The mathematical analysis is carried out under the valid physiological assumption of creeping phenomena and long-wavelength approximation in the wave frames. The solutions of rheological equations are obtained analytically with an integration scheme. The transport features are discussed for numerous sundry variables via the Mathematica Software 11.0. Comparative studies among viscous and viscoelastic liquids also argued for both complex convergent and divergent channels. The special nature of metachronal waves (i.e. wavy pattern) is used in both boundary walls. The viscoelastic and magnetic parameters have played an important role to boost of the velocity profile magnitude. The impacts of the cilia length and Helmholtz–Smoluchowski velocity parameters have a significant act in the augmentation of pressure gradient and flow rate under the influence of magnetic and electric fields (EMHD phenomena) for the divergent channel. The current mathematical formulation gives information to the medical students about how to handle the transportation of viscoelastic fluids in both convergent and divergent channels via ciliated walls and electro-magnetic devices. The consequences of the present analysis have remarkable applications in micro-ciliated pumps, blood pumping in uniform and non-uniform arteries, and the rheology of viscoelastic liquids in different chemical processes.
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