Channel flow of Ellis fluid due to cilia motion

Abstract In this report, a theoretical study is directed at the cilia motion of the Ellis fluid model due to the propagation of an infinite metachronal wave train traveling along the walls of the channel due to the continuous beating of the cilia. The governing equations of continuity and equation of motion for the well-known Ellis fluid model are carried out by taking long-wavelength approximations. The equations were solved with the help of the computational software Mathematica 13.0, and the results were obtained. The influence of physical parameters on pressure rise ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>∆</m:mo> <m:mi>p</m:mi> </m:math> \triangle p ), axial velocity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>)</m:mo> </m:math> u(x\left,y) , and stream function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ψ</m:mi> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \psi (x,y) are exhibited graphically and discussed briefly. It is observed from graphical results that the velocity field shows a parabolic nature, achieves its highest magnitudes at the mid of the channel, and then reduces drastically at the walls of the channel. Velocity reduces with increasing material constant α and rises with increasing material constant β . It is also noted that pressure increases with rising <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>β</m:mi> </m:math> \beta . The present inquiry is also valid in the treatment of different symptomatic problems and different medicament delivery methods in pharmacological and biomedical engineering.