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A mathematical model is developed to study the transport mechanism of a Casson fluid flow inspired by the metachronal coordination between the beating cilia in a cylindrical tube. A two-dimensional system of nonlinear equations governing the flow problem is formulated by using axisymmetric cylindrical coordinates and then simplified by employing the long wavelength and low Reynolds number assumptions. Exact solutions are derived for the velocity components, the axial pressure gradient, and the stream function. However, the expressions for the pressure rise and the volume flow rate are evaluated numerically. The features of the flow characteristics such as pumping and trapping are illustrated and discussed with the help of graphs. It is observed that the volume flow rate is influenced significantly by the width of plug flow region

The study of fluid transport due to systems of beating cilia has attracted the attention of many researchers due to its applications in bioengineering and medical sciences. It is generally believed that cilia are responsible for the transport of biological fluids in several physiological processes such as the removal of tracheobronchial mucus in the respiratory track, the transport of ovulatory mucus and ovum in the oviduct of the female reproductive tract system, and the motion of epididymal fluid in the efferent ductus of the human male reproductive tract [

Cilia are hair-like appendages extending from the surface of many cells and deform in a wave-like fashion to propel either cell itself or the fluid around it. A ciliated organism carries high densities of cilia arranged in rows along and across the body surface. Cilia beat in a whip-like asymmetric manner consisting of an effective stroke and a recovery stroke. Moreover, when many cilia operate together, hydrodynamic interactions cause them to beat out of phase leading to the formation of metachronal waves and an enhanced fluid flow [

A survey of the literature shows that Jahn and Bovee [

The basic motivation of this study is the hope that such a problem will be helpful in many biomedical as well as industrial applications especially in the study of infertility problems in humans and in the manufacturing of micropumps for drug-delivery systems. Ciliary pumping mechanism may be utilized in the manufacturing of swimming microrobots for biomedical applications [

Consider the axially symmetric flow of an incompressible Casson fluid in a uniform cylindrical tube whose inner surface is ciliated (see Figure

Wave motion due to cilia: (a) ciliated tube and (b) metachronal wave pattern.

The constitutive equation (relationship between the shear stress and strain rate) of a Casson fluid model may be defined in a simplified form as [

The fundamental equations governing the axially symmetric flow of an incompressible fluid are given by

The present investigation will be carried out in the coordinate system

We introduce the following nondimensional quantities:

After using the above nondimensional parameters and then employing the assumptions of long wavelength and low Reynolds number, the equations governing the flow of Casson fluid can be reduced to the following forms:

Equation (

Making use of (

The constant flux

In this section, we provide a careful analysis of the pressure rise per wavelength

Variations of

Variations of

Variations of

Variations of

Figures

Variation of axial pressure gradient

Variation of axial pressure gradient

The axial velocity profile

Variation of axial velocity

Variation of axial velocity

Another interesting phenomenon in the cilia transport is trapping. In the wave frame, streamlines under certain conditions split to trap a bolus of fluid which moves as a whole with the speed of metachronal wave. The effect of the plug flow width

Streamlines for (a)

Streamlines for (a)

We have formulated a mathematical model to study the fluid transport characteristics in an axisymmetric tube under the action of ciliary beat that generates a metachronal wave. This type of fluid transport is observed in the ductus efferentes of the human male reproductive tract. The ductuli efferentes in human body are usually 10–15 tubules connecting the rete testis to the epididymis. The cells lining these tubules are ciliated and are responsible for the transport of fluids. As pointed out by Lardner and Shack [

In the present analysis, we have examined the role of cilia motion in terms of metachronal waves in the transport of a Casson fluid through an axially symmetric tube. The implication of long wavelength and low Reynolds number allows us to obtain the flow exactly. The main findings of the above analysis may be summarized as follows.

It is noted that the relation between

The magnitude of the pressure rise increases with an increase in the plug flow width which shows that the pumping rate decreases for the Casson fluid in comparison to the Newtonian fluid.

The pressure gradient required to pass the same amount of a Casson fluid is comparatively larger than that of a Newtonian fluid under the same set of conditions.

It is observed that, with an increase in

The size and the number of circulations of the closed streamlines reduce as we increase the width of plug flow region.

It is found that the calculated value of the volume flow rate by using our model is

The corresponding results for a Newtonian fluid can be recovered as a special case from our results by taking

The authors declare that there is no conflict of interests regarding the publication of this paper.