1 Introduction

Fluid passage and propulsion owing to motile cilia are universal occurrences in different tissues and several uni- and multi-cellular creatures. Motile cilia on the exterior of a cell generate a dynamic whip-like signal, which pushes the fluid along the exterior of the tissues and cells. In motile cilia, the beating comprises of a fast powerful blow in which the cilium has a stretched form and a gentler recovery stroke in which the cilium is bent and is nearer to the cell surface^[1].

Nowadays, cilia transport phenomena have drawn the attention of many researchers for its physiological as well as industrial uses. In humans, the flow resulting from the movement of the cilia is involved in the motion of several biological fluids, such as the tracheobronchial mucus in the respiratory tract^[2-3]. Agrawal and Anawaruddin^[4] discussed cilia flows of bio-fluids assuming variable viscosity. They made a theoretical analysis of spermatic fluid transport through the vas deferens. Blake^[5] discussed flows in tubules due to ciliary activity and found that back flows (reflux) can occur near the walls for cilia that exhibit antiplectic metachronism. For a detailed study of various current investigations, we refer the reader to Refs. [6]-[12]. Nowadays, much attention is given to the investigation of heat transfer analysis in different flow geometries with different assumptions owing to its applicability to essential processes such as oxygenation and hemodialysis. Mills et al.^[13] described the periodical fluctuations of flexible cilia that blend the animated fluid and produce secondary drifts in the micro channel that smoothens heat passage inside the channel walls. Metachronal whipping of cilia under the influence of heat transfer and the Hartmann layer was debated by Akbar et al.^[14]. More recently, various problems related to this phenomenon are cited in Refs. [15]-[17].

In most of the works that deal with ciliary and peristaltic motion, the fluid viscosity is assumed to be constant. This postulation does not hold in all conditions. In general, the coefficients of viscosity for real fluids are functions of pressure, temperature, and position. In fluids such as oils, blood, and water, the variation in viscosity is influenced mainly by the temperature and position. Consequently, it is preferable to combine the effect of variable viscosity in its place of being constant viscosity fluid. Some significant works connected to the theme of the variable viscosity are quoted in Refs. [18]-[21].

However, to the best of our knowledge, no mechanism has been elucidated that describes the fluid flow related to cilia considering the effects of temperature and viscosity. Currently, we study the ciliary motion phenomenon of Newtonian fluids considering the temperature-reliant viscosity as well as heat transfer. An exponential viscosity-temperature relation known as the Reynolds model of viscosity is applied. The equations are simplified by assuming small values of the Reynolds number and large wavelength. The exact analytical form solutions are provided for the temperature and velocity profiles. Results of the temperature, pressure gradient , pressure rise Δp, and stream plots have been designed for various values of the corporal parameters.

2 Problem statement

We suppose the incompressible viscous fluid flow inside a curved channel with the centre O and length or radius R^*, velocity modules in the axial and radial directions are Ũ and Ṽ, respectively (see Fig. 1). The flow is demonstrated in both static and wave frames of position. The conservation of mass and momentum equations for viscous fluid flows can be inscribed as^[22-23]

(1)

(2)

(3)

(4)

where Ũ, Ṽ, , ρ, μ, κ, (ρc_p), and Q₀ are the components of velocity profiles in the axial and radial directions, pressure, density, viscosity of the constant fluid, thermal conductivity, specific heat, and constant of heat source or sink parameter, respectively. According to the metachronal waveform geometry, it is supposed that the exterior of cilia tip beating can be inscribed in the subsequent form,

(5)

where a represents the mean half length of the bending channel, ϵ represents the non-dimensional quantity with respect to the cilia length, c stands for the wave speed, and λ represents the wave length of the metachronal wave. The axial location of the cilia tips is in an implicit form,

(6)

where and α describe a reference point of the tip and a measure of the eccentricity of the elliptical wave, respectively. The transverse and longitudinal velocities on the cilia limit are

(7)

Invoking Eqs. (5) and (6) into Eq. (7), we can obtain

(8)

The limitations of the assumed problem can be described as

(9)

The conversions among the fixed and moving frames of reference can be written as^[24]

(10)

Dimensionless quantities for the considered problem are assumed as

(11)

where μ₀ is the reference viscosity at the reference temperature , - is the difference of temperature due to buoyancy forces, and α is the dimensional viscosity parameter.

Substituting Eqs. (10) and (11) into Eqs. (1)-(4), (8), and (9), assuming a large wavelength and a small Reynolds number, simplifying the expressions, and excluding β², Re and higher, we obtain the following equations:

(12)

(13)

(14)

Equation (12) indicates that p≠p(r), and a no-slip boundary condition is assumed at the ciliated wavy surface. Therefore, boundary conditions are expressed in the dimensionless form as follows:

(15)

The expression μ(θ) in Eq. (13) is computed as^[25-26]

(16)

where s is known as the coefficient of the Reynolds model.

3 Problem methodology

The solution to the considered boundary value problem (13) and (14) after substituting Eq. (16) into Eqs. (13) and (14) along with the boundary limits can be written as

(17)

(18)

where

The constants a₁ and a₂ are computed by using the boundary assumptions defined in Eq. (15).

The pressure gradient expression can be computed from the succeeding expressions,

(19)

Finally, the flow rate, pressure rise, and stream function in the dimensionless forms are determined from the following equations:

(20)

The dimensionless expression for the Nusselt number^[27-28] is given by

(21)

4 Results and discussion

This part is written to inspect the presentation of velocity, pressure gradient , pressure rise Δp, shear stress, Nusselt number, temperature profile, and stream plots of the problem. The effects of the viscosity parameter, curvature of the channel, Grashof number, and cilia length parameter on the velocity profiles are demonstrated in Figs. 2-5. Figure 2 shows that, with an increase in the viscosity parameter, the velocity increases near the left ciliated tip wall (r=-h). However, the contradictory behavior is observed near the right region due to the ciliated effects. Figure 2 also shows that, as the viscosity parameter s is taken to be zero, the velocity for the constant viscosity fluids is less compared with that for variable viscosity fluids (see the region (-0.1≤ r≤ 1.002 51)), but in the region (-1.002 51≤ r≤ -0.1), the opposite behavior can be gotten. Figure 3 is plotted to see the behavior of the curvature effects on the velocity profile. It is found that, close to the right ciliated wall (r=h) of the channel, the increase in the curvature parameter contributes to the decrease in the fluid velocity. However, near the right ciliated wall of the channel, the larger curvature parameter allows fluids to flow more freely, i.e., the fluid velocity increases. Effects of increasing heat input or absorption parameter on the velocity profile are depicted in Fig. 4. One can easily see from Fig. 4 that as heat is introduced into the fluid, the velocity profile widens in the center of the channel (the metabolic scheme is one of the finest heat source utensils to deliver heat or energy), whereas reflux is observed near the walls, which agrees with that in Ref. [5]. Figure 5 is plotted to visualize the influence of the Grashof number on the velocity profile. As the value of the Grashof number increases, the velocity curve near the right wall increases due to the dominant effects of the buoyancy forces, whereas reflux is depicted near the ciliated tips. We can also observe from Figs. 2-5 that the symmetry of the channel is disturbed due to the curvature. In Figs. 6-9, we reveal the pressure rise versus the flow rate for different values of Grashof number Gr, viscosity parameter s, curvedness of the channel k, and cilia length parameter ϵ. Figures 6 and 8 show that the pressure rise increases when Gr and k increase. Figure 7 confirms that an increase in the viscosity parameter results in a decrease in the pressure rise distribution in the section (-1.5≤ Q≤ 0.2) whereas the opposite behavior can be seen in the remaining portion. The profile of the pressure rise against Q for different values of the cilia length parameter is shown in Fig. 9. It is portrayed that the pressure rise is a swelling function of the cilia length parameter in the section (-1.5≤ Q≤ 1.45). However, a reverse variation is shown in the rest of the section. The pressure gradient expresses the mode and expanse at which the pressure deviations are fast. The pressure gradient for various values of the parameters is demonstrated in Figs. 10-13. It is supposed from these figures that the pressure gradient is enormous for the curved channel exposed by the intervals (0.28≤ x≤ 0.73 and 1.34≤ x≤1.68). However, in the remaining section of the bending channel, (pressure gradient) in the sections 0≤ x≤ 0.27 and 0.74≤ x≤ 1.33 gets the smaller values. It is also evident from Fig. 13 that for constant viscosity fluids is reduced when associated with variable viscosity fluids, and variations get closer as the viscosity parameter increases. Temperature profiles are shown in Figs. 14 and 15. It is observed from Fig. 14 that the temperature profile widens as more energies are supplied to the fluid due to the heat source. It is clear from Fig. 15 that the larger curvature parameter increases the fluid temperature. In Figs. 16 and 17, the Nusselt number Nu is designed against the heat absorption parameter for various values of k and ϵ. It is identified that the Nusselt number is directly proportional to the curvature parameter, but inversely proportional to the cilia length parameter. Trapping is a spectacular display of ciliary wave, which is an extension of an esoteric mixing bolus of fluids by the dwindling streamlines. This confined bolus helps to push forward with metachronal wave. Figures 18 and 19 give the streamlines for the parameters k and ϵ. These plots display that the values of the confined boluses increase when the curvature parameter and the cilia length parameter increase. The close evaluation of the profiles reveals that there are more circulations of fluids that clue more sealed boluses. It is clear from Table 1 that the results found in the present case are consistent with those of previous studies^[10].

5 Conclusions

The exact form solutions are computed for velocity and temperature profiles. The pressure gradient for constant viscosity fluids is smaller than that for the variable viscosity fluids. Pressure rise displays the swelling performance for the growing values of the Grashof number Gr due to the increase in buoyancy forces throughout the region. The temperature profile increases for increasing values of heat source or sink parameter B and dimensionless channel radius k. The variation of the Nusselt number against the heat absorption parameter rises with growing values of the curvature parameter, but decreases with increasing values of the cilia length parameter. The symmetric property of the trapped bolus is demolished in a ciliated bending channel.