1 Introduction

The ductus efferentes are a series of microvessels which provide a vital link between the testis and the epididymis^[1-2]. It is known that the wall structure of these vessels is a single layer of columnar epithelium which is supported by a thin layer of smooth muscle and connective tissue^[3]. The functions of the vessels are sperm transport from the rete testis to the epididymis and reabsorbing high amount of fluid which comes from the rete testis. Due to the second function, spermatozoa concentration rises manifold. Epithelium of the ductus efferentes is composed by ciliated and non-ciliated cells. The well-formed tufts of cilia are projected by the ciliated cells from the lumen of the duct, and closely-packed, long and regular microvilli are born by the non-ciliated cells. However, microvilli, which are scattered between the cilia, are also projected by the ciliated cells. Moreover, they are fewer, shorter and thinner than those of the non-ciliated cells. Since ductus efferentes are lined by ciliated epithelium, they are unique of the male reproductive tract^[2-3].

A cilium is a hairlike slender appendage/protuberance that projects from the free surfaces of certain cells (e.g., eukaryotic cells). Its presence has been observed in almost all animals. Due to their motility, it engages a significant role in various physiological processes like locomotion, alimentation, circulation, respiration, and reproduction^[1].

Cilia are classified into motile and non-motile cilia. The latter is also known as primary cilia. In this study, we have considered motile cilia which do not beat randomly, but rather in coordinated manner. The nature of the cilia possesses some important aspects of ciliated epithelium. A great number of general observations and inferences on the cilia of the gill of several aquatic species were presented by Rivera^[4] as follows: (i) In any given tissue, a beat rate of all the cilia is quite uniform. (ii) The lashings of a single cilium and of cilia on adjacent cells are very much coordinated. (iii) A definite metachronal rhythm is established. A metachronal rhythm is a movement consisting of a sequence of beat from row to row of cilia in a given row of cells and then from one row of cells to the next one. As a result, any object, which is at rest on the surface of cilia, always moves forward while keeping the direction fixed.

Since the metachronal rhythm gives more fixed passage of water with time over the surface of cilia or perhaps it is impractical to stimulate synchronous beat of large area^[5-6], it is believed that cilia beat in a metachronal rhythm to the contrary of synchronous manner. However, the metachronal rhythm along the surface of cilia may alter their pattern^[5]. The change depends on whether the metachronal rhythms move towards the effective stroke of the ciliary beat (symplectic metachronism), or the metachronal rhythms pass towards the opposite of the effective stroke of the beat and so in the reverse direction of flow (antiplectic metachronism), or the cilia beat at right angles to the line of wave movement (diaplectic metachronism). There are some data available for some other animals such as wave-lengths, metachronal rhythm velocities, and frequencies^[5].

It is a bit debatable to what extent the cilia and to what extent smooth muscles drive the fluid in the efferent ducts of the male reproductive tract. It was reported that the prime contribution of the net fluid transport in efferent ducts was coming from metachronal movement of cilia^[7-8]. However, Winet^[9] created a cilio-peristaltic model which suggested that the greater contribution to the flow was coming from the smooth muscle contraction. He remarked that "we may conclude that if the peristaltic wave has a 33% or more constriction, the spermatozoa concentration in the ductus efferentes is at least 4×10⁸ cells·cm^-3 and that flow rates in the ductus efferentes are the same as in the rete testis". However, he added that "no observations of φ (the occlusion factor) have been made for any of the male tubes". Epithelium of ductus efferentes is supported by only a thin layer of smooth muscle and connective tissue (see Ref.[3]). Therefore, the suggestion, the major contribution to the flow was from smooth muscle contraction, is questionable. Moreover, if the peristaltic wave has a 50% or more constriction, and the cilia length is more than 20 μm, cilia of opposite boundary will clash with each other at the contraction region. In addition, at the contraction region, the driving force (which is opposite direction of the flow) at the central region will be active on the cilia (which obstruct the free space at the central portion). As a result, there may be damage of cilia indicating that smooth muscle contraction (i.e., peristaltic motion) may not be the main driving force of fluid transport at ductus efferentes. The conclusions of this paper will throw some more light on this issue.

Ilio and Hess^[3] stated that the ratio of ciliated and non-ciliated cells in the epithelium in different animals generally varies in the range between 1:3 and 1:8^[10]. Therefore, the vast majority of the inner surface is non-ciliated. However, Aire and Josling^[11] reported (based on their experimental observations) that the cilia of the ciliated cells usually over-shadowed the luminal surfaces of non-ciliated cells even though the non-ciliated cells were greater than the ciliated cells in number. Moreover, in histological photographs^[11], it is clear that ciliated and non-ciliated cells are not distinctly separated into two parts, and they are rather adequately mixed to justify the model considered here.

It is to be noted that data are not easily obtainable for bigger animal species, especially mammals. However, it was measured that the beat frequency for cilia lining in the rabbit oviduct was approximately 20-30 beats/s^[12]. It is worth mentioning that we can compare this order of magnitude of the frequency of cilia beat with the lower animals^[5].

It is to be further noted that several investigators (see Refs.[13] and [14]) studied the hydrodynamics of protozoa which utilized cilia for locomotion. Blake^[15-16] took a spherical envelope model in order to study the swimming of the protozoan opalina and the swimming motion considering the ciliated body either a two-dimensional or cylindrical shape. Miller^[17-19] investigated mucus transport with the help of mechanical simulation of the cilia in the trachea, while Barton and Raynor^[20] studied mucus transport analytically in the trachea without considering the metachronal rhythm. Those investigations on protozoology and mucus transport in the respiratory tract gave conception in terms of locomotion in protozoa as well as movement of particles in the respiratory tract. However, only a little has been done finding the relation between the properties of the cilia and the nature of the metachronal rhythm for fluid movement in efferent ducts. Our motivation came up from the questions related to the interpretation of the fluid movement through the ductuli efferentes of male reproductive tract of human^[21] and the effect of cilia on ovum transport and sperm movement in fallopian tubes^[22-23]. Particularly, this paper will deal with a comparison of the results for the flow rate of this model to the corresponding estimated flow rate in the ductuli efferentes of male reproductive tract.

It is worth mentioning that there are limited data available on the flow rate due to ciliary activity. Based on the experimental observations (see Refs.[8], [24], and [25]), Lardner and Shack^[1] estimated a flow rate for human testes as 6×10^-3 ml/h with approximate values, i.e., a=50 μm and frequency of beat of the cilia as 20/s. But the theoretical model of Lardner and Shack^[1] obtained a flow rate of 0.12×10^-3 ml/h. Hence, further investigations are indeed required.

The previous experimental observations indicated that most of the biological fluids possessed non-Newtonian behaviour^[26-37]. Analyses on the basis of simple Newtonian fluid yield non-realistic results. The power-law model is one of the widely used models for rheological fluid transport^{[30-33, 38-40]}. The rheological nature of this model is strongly dependent on the rheological fluid index n. The model approximates both shear-thinning (n < 1) and shear-thickening (n > 1) fluids behaviour over a large range of rheological conditions^{[26-27, 30-31]}. Viscous properties of human semen are experimentally found to exhibit power-law behaviour (see Refs.[27] and [28]). It has been reported experimentally that semen proves to fit in a power-law model with pseudoplastic behavior^[27-28]. Thus, studies on fluid transport of the power-law model by ciliary activity are expected to yield some important inferences. The constitutive equation of a Newtonian fluid (i.e., relationship between shear stress and strain rate) is linear, whereas it is nonlinear for a rheological power-law fluid.

There are two different approaches for investigating the periodic motion of cilia: the sub-layer model and the envelope model^[41]. A sub-layer model approach helps us by computing the forces and bending moments generated by each cilium to neighbouring liquid and it obtains the mean flow generated above the cilia layer. However, the hydrodynamic representation is wearisome and normally obtained numerically^{[15, 42-44]}. The envelope model approach has advantage for the consideration of metachronal rhythms above the cilia layer ignoring the details of the sub-layer dynamics. Moreover, the envelope model can be utilized for comparing, even quantitatively, e.g., for comparing the swimming velocities evaluated mathematically with those reported in water for a number of microorganisms^{[16, 44-45]}. In addition, this approach is accountable to perturbation analysis and has been used to combine some non-Newtonian effects by a systematic way^{[44, 46-47]}.

Recently, Siddiqui et al.^[48] studied the flow of a power-law fluid due to the ciliary motion in an infinite channel. They pointed out that the power-law fluid gave results closer to the estimated one as 6×10^-3 ml/h. However, we believe that in real physiology, the wall contraction/length related to the cilia and favourable pressure gradient are less than that they have considered. They took ε=0.9, giving reference to Agarwal and Anawaruddin^[49] who reported an application of their model for fluid transport in vas deferens. It is worth mentioning that ductus efferentes and vas deferens are quite different ducts, and the latter one may undergo possibly that much wall contraction. ε, a non-dimensional measure with respect to a mean radius a of the tube and the cilia length, would be much less than ε=0.9^{[1, 50]}. To the authors' knowledge, the other unacceptable considerations in Ref.[48] are as follows: (i) They considered a large constant favourable pressure gradient. (ii) They studied dependence of flow characteristics on the higher value of δ, wave number of the metachronal wave.

It is well known that the pressure gradient in small biological vessels (at least when there is a wave-like peristaltic wave^{[5, 31, 51-53]}, metachronal wave etc. in the vessels) varies with the length of the vessels, and favourable pressure gradients are physiologically insignificant, in general, and, at least, in the ductus efferentes^[9]. Moreover, study of flow characteristics is not true for higher values of δ as they have done analysis for small values of δ. However, their power-law model is somehow different and complex, compared with the widely used power-law model.

In view of all the above, a theoretical model is being considered for more realistic consequences for the flow in an axisymmetric tube under the influence of metachronal rhythm of cilia movement. We study here a non-linear problem of Ostwald-de Waele power-law fluid transport induced by means of a sequence of cilia beat from row-to-row of cilia in a given row of cells and from one row of cells to the next (metachronal wave movement). The conditions considered in this study are the small Reynolds number and the small wave number.

Based upon the analytical investigation, the velocity, pressure difference, flow rate have been calculated. Thereafter, we have carried out extensive numerical calculations. Keeping in view a specific situation of fluid movement in micro-vessels by ciliary motion, the numerical results are exhibited in graphs. The results discussed for shear thinning and shear-thickening fluids are closely connected in the circumstances of various situations of fluid transport due to ciliary activity. For example, semen behaviour is normally of shear-thinning kind when it contains a number of spermatozoa up to a certain limit. However, in the case of heavily concentrated spermatozoa suspensions, the nature of semen may be considered as a shear-thickening type^[27-29]. Moulik et al.^[54] reported that high viscosity of semen might indicate antibodies in the plasma and/or genital tract infection.

Some novel features are discussed to have a better insight of fluid motion by ciliary activity. We can utilize ciliary pumping mechanism in the study of the hydrodynamics of protozoa which use cilia for locomotion. The results may find useful applications in cilia-based actuators as micro-mixers for flow control in tiny bio-sensors and also as micro-pumps for drug-delivery systems.

In this investigation, the term cilia will not consider flagella but ciliated epithelium. Comparative physiology of flagella including sperm tails may be available in Refs.[5] and [55].

2 Formulation

A non-linear problem, concerning the fluid movement characteristics in an axisymmetric tube under the action of ciliary beat that generates a metachronal rhythm, is to be studied here by considering the fluid as an incompressible non-Newtonian viscous fluid. We consider an axisymmetric tube with ciliated walls of large length compared with its radius, and a symplectic metachronal wave is moving on the right hand side with the velocity c. The non-Newtonian viscous fluid behaviour within the tube is considered to be incompressible Ostwald-de Waele power-law rheological fluid.

We treat (R, θ, Z) as the cylindrical coordinates for the position of any fluid particle. Here, R is measured along the radius of the tube, the coordinate Z is along the wave propagation direction, and θ is the rotational coordinate. The schematic illustration of fluid motion by metachronal waves of cilia has been given in Fig. 1. When τ and Δ denote the stress tensor and the symmetric rate of deformation tensor, respectively, the governing law of shear stress for the Ostwald-de Waele power-law fluid may be written as^[56]

(1)

in which

where γ and n are the flow consistency index and the power-law index, respectively. We introduce U and V as the velocity components in the Z-and R-directions, respectively. We know that the shear thinning fluid has been classified for n < 1, and n > 1 corresponds to the shear thickening fluid.

Under the above consideration, the flow of an incompressible viscous Ostwald-de Waele power-law fluid in an axisymmetric tube together with the equation of continuity may be considered to be governed by

(2)

(3)

(4)

where ρ and P are the density and pressure of the fluid, respectively. The geometry of the metachronal wave pattern helps us to assume the envelope of the cilia tips mathematically by

(5)

Therefore, this can be assumed as the equation of the extensible vessel wall. Here, a is the radius of the tube, and ε is a non-dimensional parameter which together with a in the form of aε makes the amplitude of the metachronal wave. λ and c are the metachronal wave length and velocity, respectively. Observations of the various patterns of cilia movement in Ref.[6] motivate us to assume the cilia tips to move in elliptical paths so that the horizontal position of a cilia tip is in an implicit form, i.e.,

(6)

in which Z₀ represents a reference location of the particle, and α stands for a measure of the eccentricity of the elliptical motion. Under the no-slip condition, the velocities imparted to fluid particles are just those of the cilia tips. Thus, the axial and vertical velocities of the cilia are evaluated as

(7)

(8)

If (5) and (6) are applied to (7) and (8), we obtain

(9)

(10)

These boundary conditions enable us to distinguish between the effective stroke of the cilia and the slow less effective recovery stroke by approximately accounting for the shortening of the cilia. Hence, the tube is narrower when U is positive at the boundary.

For a wave frame (z, r) moving with a velocity c away from a fixed frame (Z, R), apply the transformations,

(11)

where (u, v) and (U, V) denote the velocity components, and p and P are pressures in the wave frame and the fixed frame of reference, respectively. Afterward, we will introduce the following non-dimensional variables:

(12)

where Q₁(Z, t) is the instantaneous volume flow rate, and Re is the Reynolds number of the fluid flow. Dropping the bars over the symbols, the equations governing the flow can be rewritten as

(13)

(14)

(15)

(16)

whereas the boundary conditions are

(17)

(18)

Since in most of the cases of flow in small diameter tubules, Reynolds numbers are very small (Re«1), the analysis can be carried out by the approximation of the inertia-free flow. Moreover, in the wave frame of reference, if the tube length is finite and equal to an integral number of wavelengths together with constant pressure difference across the ends of the tube, the flow may be steady^[53]. In this case, the governing equations under the consideration of the long-wavelength approximation (which makes δ«1) can be simplified as follows:

(19)

(20)

and also the simplified forms of the boundary conditions may be given by

(21)

(22)

By solving (19) subject to the boundary conditions (21), we find the axial velocity in the form,

(23)

where and . If we integrate the continuity (13) across the cross section of the tube, we get

(24)

(25)

(26)

where

(27)

which implies that .

This shows the fixed volume flow rate q in the wave frame of reference. By integrating (23) across the cross section of the tube, the pressure gradient is obtained in terms of the volume flow rate in the following form:

(28)

Substituting , which is obtained from (28) into (23), we calculate

(29)

By integrating the above equation, the instantaneous non-dimensional volume flow rate, Q₁(Z, t), in the fixed frame of reference can be obtained as

(30)

Thus, the time-mean volume flow rate over a wave period is given by

(31)

which, on integration for the sinusoidal wall of (5), gives

(32)

By solving from (28) and using (32), the pressure rise per wavelength may be given by

(33)

It may be noted that if we set n=1 and α=0 in (23), (28), and (33), the expressions reduce to those reported earlier in Ref.[53]. If α=0, the results also match well those of Ref.[30] when the peristaltic wave form in Ref.[30] is replaced by the metachronal wave in this study. Similarly, if the yield stress is set equal to zero, and the fluid flow is considered in an axisymmetric uniform vessel in Ref.[52], the expressions said above reduce to those obtained in Ref.[52].

By solving the continuity equation (13) subject to the boundary condition (22), we get the radial velocity as

(34)

3 Quantitative study

This section deals with a quantitative study of the mathematical model considered in the above sections. It may be noted that, unlike the Newtonian model, it has been not possible to find an explicit expression for the pressure difference ΔP in terms of the time averaged flow rate Q for the Ostwald-de Waele power-law model. This is due to the complexity of the problem of a rheological fluid arising out of arbitrary wave shape. However, a numerical computation is required to express pumping characteristics or pumping performance of the rheological fluid for the sake of clarity. In addition, it is a convention to exhibit the velocity field in term of the pressure difference between the tube ends. Hence, in view of those observations and also in order to fulfill the requirements, necessary computations have been carried out numerically by using the software MATHEMATICA.

It is to be noted that there are only a few available data on the flow rates due to ciliary activity^{[1, 50]}. For the quantitative study, we shall present mathematical estimates of various physical quantities relevant to physical problems of the flows of rheological fluids under ciliary activity. For the present analysis, the following non-dimensional data for the rheological fluids are used^{[1, 50]}:

3.1 Pumping characteristics

It is natural to determine pressure-flow characteristic (i.e., the pumping characteristics) by the variation of the time averaged flow rate Q with the pressure difference ΔP across one wave length. Variations of the volumetric flow rate by cilia motion for different values of ε, the flow index number n, the wave number δ, and the eccentricity α are exhibited in Figs. 2-5. As expected, the variations are linear nature for an inertia-free flow (Re=0) of a Newtonian fluid. The curves of ΔP versus Q are straight lines with the negative slope and the positive intercepts. However, for the aforementioned case of a rheological fluid, the relationship is nonlinear between the pressure difference and the mean flow rate.

The curves have three portions, namely, (i) the free pumping zone, describing the region in which ΔP=0, (ii) the pumping zone, indicating the region where ΔP > 0, (iii) the co-pumping zone, which is regarded for the region where ΔP < 0, the situation favourable for the flow to take place. We can compute the amount of flow pumped by ciliary activity if the mean pressure gradient (ΔP=0) is absent and also for the case of adverse pressure gradient (ΔP > 0) up to a certain limit. If ε, n, δ, and α are fixed at some values, employment of an adverse mean pressure gradient diminishes the mean flow rate obtained when ΔP=0, and by the time it becomes right amount (certain limit), it is equal to the driving force of motion generated by the cilia movement, i.e., the mean flow reduces to 0. Further, if ΔP (>0) exceeds the limit, the fluid will move in the reverse direction.

Figures 2(a)-2(c) show that the area of the pumping zone and the length of the free pumping zone both increase significantly with a rise in ε for Newtonian, shear thinning as well as shear thickening fluids. Also, they enhance at a very high rate with an increase in the rheological parameter n. Moreover, for fixed values of ΔP, the time averaged flow rate Q rises with an increase in ε and n. However, in the co-pumping zone, there is a critical value of ΔP for which Q can be raised for all fixed values of ΔP with an increase in n when ε>0.1. If ΔP exceeds this critical value, the reverse trend occurs. It is important to mention that for fixed values of parameters, taken by Lardner and Shack together with n=1, the flow rate in the axisymmetric tube is twice the values of Ref.[1] for a two-dimensional channel. In the case of free pumping, Fig. 3 clearly reveals that there is a remarkable enhancement of flow rate with the increase in ε. It further explains that n strongly influences the flow rate. The plots, presented in Figs. 4-5, indicate that, for a non-Newtonian fluid, Q increases as the wave number δ and the eccentricity α increase, and these parameters enhance Q at almost the same rate for both the shear thinning and shear thickening fluids.

It is worthwhile to mention that an increase in ε corresponds to a rise in the cilia length and vice versa. Again, in the case of free pumping, when ε=0, i.e., there is no cilia to the inner surface of the tube, then, Q=0.

3.2 Distribution of velocity

As shown in the previous section, the mean flow rate is only responsible for the flow through ciliary activity during free pumping, and only the movement of cilia produces the driving force for fluid transport. For various values of ε, n, δ, and α, the distributions of axial velocity of the current investigation are presented in Figs. 6 and 8-11. Since the flow is unsteady in the fixed frame of reference, and the velocity profiles along with the lumen of the ductus efferentes vary with the time, the distribution of velocity is investigated at a time interval which is a quarter of a metachronal wave period of cilia.

One can observe from Fig. 6 that there exists a retrograde flow region at any time, and the maximum retrograde flow occurs at the narrowest portion of the tube, while the maximum flow in the forward direction takes place at the widest portion of the tube. As the time-average flow rate is positive for free pumping (see Subsection 3.1), the forward flow region is predominant here. Moreover, the study further reveals the existence of two stagnation points on the axis. For example, at the time t=0.75, one stagnation point lies between Z=0.00 and Z=0.25, and the other is situated between Z=0.25 and Z=0.50 which separates the central region of a retrograde flow from two forward flow regions. For peristaltic transport, similar observations were reported numerically by Takabatake and Ayukawa^[57] for a Newtonian fluid and analytically by Maiti and Misra^{[33, 52]} and Misra and Maiti^[31] for non-Newtonian fluids. From the standpoint of ciliary pumping, this retrograde flow at the trough region is considered to be a kind of ineffective leakage.

It is to be noted that the beat of a single cilium has two different phases. One of them is the effective stroke of cilia, when the movement of cilium is in the general fluid movement direction. Reflux occurs near the walls in this case, and the portion of the tube is wide. The other phase is the recovery stroke, when the movement of cilium is in the opposite direction to the general fluid movement. There may be forward flow near the walls and the narrow portion of the tube in this case. Figure 7 shows the instantaneous velocity field, and the corresponding envelop of the cilia tips is shown in Fig. 6(b). The flow is along the direction of the cilia motion in the region 0≤Z≤0.5, while the reverse trend occurs in the region 0.5≤Z≤1.

The effects of ε, n, δ, and α on the velocities at the crest and the trough section of the tube are shown in Figs. 8-11 under consideration of free pumping. It may be observed from Fig. 8 that there is a remarkable increase in the magnitude of the axial velocity due to an increase in the value of ε for both sections. This may be interpreted that the rise in the cilium length (at least up to a certain limit) may generate the greater driving force. Figure 9 illustrates that the magnitude of the axial velocity for both sections enhances significantly at the central region while the reverse trend occurs at the boundary region if the flow index number n increases. That reverse trend might have been initiated due to the increase in n which is related to friction at the boundary region of the tube. As shown in Figs. 10-11, the wave number δ and the eccentricity α of the path of the cilia raise the magnitude of the axial velocity at the central region for both sections, while at the boundary region, the trend is reversed indicating enhancement of friction due to increases in δ and α.

4 Application to fluid movement in ductuli efferentes

The ciliated walls of an axisymmetric tube are modelled by the metachronal wave of cilia which are equivalent to the wavy walls of peristaltic transport. There are only a few available data in the existing studies on the flow rates due to the ciliary activity. The relationship between the pressure difference and the time-mean-volume flow rate through an axisymmetric tube is given by (33). We have to verify whether the computations given by (33) reflect the flow rates measured in efferent ducts of the male reproductive tract. In humans, the efferent ducts are 10-15 tubules connecting the rete testis to the epididymis, and the lining cells in the tubules are ciliated. In general, researchers believe that the cilia drive the fluid motion^[21] through the efferent ducts. Lardner and Shack^[1] estimated the approximate flow rate in human rete testis to each efferent ducts as 6×10^-3 ml/h with corresponding values a=50 μm, the frequency of the cilia beat 20/s, and c=(20 beats/s)×10 μ=200 μ/s on the basis of experimental observations^{[8, 24-25]} of the flow rates in the rete testis of rat, ram, and bull. The said quantities validate the action of the long wavelength and the small Reynolds number theory (Re«1) in this study. The Non-dimensional and dimensional flow rates 2.2×10^-2 and 0.12×10^-3 ml/h were evaluated by Lardner and Shack^[1] from their model in the case of free pumping assuming that ε=0.1, δ=0.1, and α=1. However, if the above said values are retained in the present study under the consideration n=1, this model gives the non-dimensional flow rate in an axisymmetric tube as 4.545 14×10^-2, and consequently, the dimensional flow rate is 0.257×10^-3 ml/h (see Fig. 2(a)). This value, for an axisymmetric tube, is about twice the value of Ref.[1] for a two-dimensional channel flow. The reason for this large difference is that they estimated the flow rate initially for a channel and used it for an axisymmetric case. However, ε is linearly proportional to the cilia length on the basis of the assumption, and n is linked to the rheological fluid (widely known as a non-Newtonian power-law fluid) in the ductus efferentes of the male reproductive tract. It has been found from Fig. 2 that the flow rates increase significantly with an increase in ε. The fact that the flow rates increase with the cilia length was also reported in Ref.[50]. Moreover, for higher values of ε, the rheological fluid index n remarkably affects the fluid transport for adverse pressure gradient (i.e., at the pumping zone). If we consider n=4/3, ε=0.4, ΔP=0, δ=0.1, and α=1, then the non-dimensional flow rate and dimensional flow rate are calculated as 0.581 611 and 3.288 9×10^-3 ml/h, respectively, which is close to the estimated value, 6×10^-3 ml/h. However, the authors have no ideas about the actual length of cilia and hence the value of ε, due to non-unavailability of real physiological data of the concerned variables and parameters used in the analysis. The length of cilia in ductus efferentes is measured about 5 μm in the length in the domestic fowl, 7 μm in the guinea-fowl and quail, and 8 μm in the turkey (meleagris gallopavo).

5 Summary and conclusions

Here, we present an analysis for the rheological fluid transport by means of a sequence of beat of cilia which are in an array and coordinated in such a way to represent the metachronal rhythm. The objective of the present study is related to the questions connecting the understanding of the fluid movement through the efferent ducts of male reproductive tract of human (see Refs.[3] and [21]) and the influence of cilia on ovum and sperm movement in fallopian tubes, etc. Moreover, an application of our results for the flow rates in efferent ducts of male reproductive tract is discussed here. On the basis of the derived analytical expressions, extensive numerical computations are carried out. The effects of various parameters, such as ε, n, δ, and α on the pumping characteristic and velocity distribution are investigated in detail in the case of an axisymmetric tube flow.

The study reveals that, for a particular set of values, say, ε=0.1, δ=1, α=1, n=1, and ΔP=0 (as considered in Ref.[1]), the flow rate in an axisymmetric tube is about twice the value of Ref.[1] for a two-dimensional channel. It further reveals that the flow rate changes remarkably with ε and n. When ε is near about 0.4, the present result for the flow rate in human ductus efferentes is close to the estimated value 6×10^-3 ml/h as suggested by Lardner and Shack^[1] based on the experimental observations for the flow rates in efferent ducts in other animals, e.g., rat, ram, and bull.

The respectable variation between the theoretical and the measured quantities indicates that the metachronal wave of cilia cannot be responsible for the total flow rate in efferent ducts, and there must be some other important factors accountable for semen movement. These factors are as follows (see Ref.[3]): (a) contraction of smooth muscle; (b) invariable fluid secretion in seminiferous epithelium; (c) contraction of the myoepithelial layer of seminiferous tubule as well as tunica albuginea in testis; (d) the devoid space generated due to ejaculation of sperm from the lower tract and by fluid absorption; (e) augmented pressure because of the design of branching and convergence of ductuli. Another important factor may be the shape of an envelope of the tips of cilia beat which is different from the one considered by researchers including us. This motivates us to study the flow through ductus efferentes in the future. Consequently, for adequate understanding the mechanism involved in semen movement in ductus efferentes of male reproductive tract, further theoretical and experimental investigations are required. It may be noted that the forward flow takes place at the wider portion of the tube, while the backward flow occurs at narrower portion of the tube as observed earlier^{[30, 31, 52, 57]}, although the forward flow is dominant for the positive time-averaged flow rate.

Acknowledgements One of the authors, S. MAITI, is grateful to the University Grants Commission (UGC), New Delhi for awarding Dr. D. S. Kothari Post Doctoral Fellowship during this investigation.