Transportation of micro-polar fluid by dilating peristaltic waves

1 Introduction

Combination of periodically alternating contractions and relaxations of muscles responsible for most of the physiological flows is peristalsis. This combination forms a transverse and progressive wave. This mechanism has several engineering applications. Several innovators, across various fields, are developing peristaltic machines that can move in cylindrical tubes to locate ruptures at the joints of gas and water pipelines and those caused by cracks. Many such machines are being designed to serve industries for sanitary fluid transport, blood pumps in heart/lung machines, transport of corrosive fluids without contacting machinery components. Studies are also focused on realization of machines that can pass through intestines and blood vessels. Peristalsis, observed in earthworms and nematodes, induces shape variation and a shift in the center of gravity. This causes extensional waves to propagate and thus progress without injury to the vulnerable inner walls of blood vessels. This moving mechanism together with catheters can reach a diseased site by itself (Nakazato et al., 2010).

Eringen (1966) formulated the effects of individual particles such as micro-rotation in flow, which are concentrated suspension of non-deformable neutrally buoyant rigid particles in a viscous medium. Micro polar fluids contain micro constituents which can undergo rotation. Rotation of micro constituents can affect the hydrodynamics of flow and make the fluid distinctly non-Newtonian. Physically, micro polar fluids represent fluids consisting of rigid, spherical or randomly oriented particles with ignored deformation suspended in a viscous medium (Lukaszewicz, 1999). Liquid crystals, blood, some edible solutions resemble micro-polar property. Engineering applications using polymer solutions, colloidal solutions, drilling fluids in oil industries etc. may be better understood by this investigation (Pandey and Tripathi, 2011a).

Devi and Devanathan (1975) studied peristaltic transport of micro-polar fluids in a cylindrical tube with a sinusoidal wave of small amplitude. Philip and Chandra (1995) explored peristaltic transport of a simple micro-polar fluid which accounts for micro-rotation and micro-stretching of the particles contained in a small volume element using long wave length approximations. Srinivasacharaya et al. (2003) examined different micro-polar properties on pressure across one wavelength and also on trapping; Hayat et al. (2007) investigated the effects of different wave forms; Muthu et al. (2005, 2008a, b) studied wall properties in channels and tubes respectively whereas Hayat and Ali (2008) investigated effects of an endoscope. The same authors (Ali and Hayat, 2008) studied the effects of asymmetricity of wave propagation of channel while Mekheimer and Elmaboud (2008) studied the flow in an annulus. Asghar et al. (2018) used micro-polar fluid to characterize the rheology of a thin layer of slime and its dominant micro-rotation effects. Recently some interesting papers dealing with the flow of micro-organisms such as bacteria and cilia driven flows under different conditions have been published (Ali et al., 2016; Asghar et al., 2017, 2018, 2019a; Asghar and Ali, 2019; Asghar et al., 2019b; Ali et al., 2019a; Asghar et al., 2019c; Ali et al., 2019b; Javid et al., 2019; Asghar et al., 2020a, b, c).

Unlike aforementioned authors, Pandey and Tripathi (2011a) investigated flow of a micro-polar fluid in a finite tube with the consideration that peristaltic waves do not move beyond the stationary boundary of the tube to match such a flow in oesophagus. Such a wave propagation was designed by Misra and Pandey (2001).

Pandey et al. (2017) concluded in their investigation that the wave amplitude does not remain constant during the wave propagation in a peristaltic motion when anything swallows in the oesophagus. The conclusion was derived in order to model the experimental reports of Kahrilas et al. (1995) who had located a higher pressure zone in the distal part of the oesophagus in normal as well as pathological state. Pandey and Tiwari (2017) further investigated swallowing of fluids that match the properties of Casson fluid, due to peristaltic waves of dilating amplitude.

In light of the observation of dilating peristaltic waves by Pandey et al. (2017) in oesophagus which validates experimental investigation, peristaltic swallowing of micro-polar fluids (Pandey and Tripathi, 2011a) requires a revisit of investigation. Particularly the impact of dilation of wave amplitude on the non-dimensional parameters such as coupling number and micro-polar parameter may be worth reporting.

2 Formulation of the problem

We consider the flow of micro-polar fluid in a tube of length of $\stackrel{\sim }{l}$ caused by continuous contraction waves that propagate along the walls of the tube (cf. Fig. 1), which are given by

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Fig. 1

Schematic diagram of oesophagus (based on Eq. (16)) represents the propagation of a progressive wave, where $\delta =\frac{a}{\lambda }$, $\mathit{Re}$, $Q$ denote respectively the wave number, the Reynolds number and the volume flow rate.

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The governing equations of the flow of micro-polar fluid in the absence of body forces and body couple are given by

For details of the equations for micro-polar fluids, please see Asghar et al. (2018).

The following dimensionless parameters are introduced for the subsequent analysis:

Introduction of the dimensionless parameters gets Eqs. (2)–(5) transformed to

Employing the long wavelength and low Reynolds number approximations, the dimensionless Eqs. (8)–(11) reduce to

Similarly the wall Eq. (1), under non-dimensionalisation reduce to

The following are the boundary conditions imposed on the governing equations:$$u\left(x,r,t\right){|}_{r=H}=0,v\left(x,r,t\right){|}_{r=H}=\frac{\partial H}{\partial t},$$

3 Solution of the problem

Integration of Eq. (13), once with respect to$r$, yields

Further, integrating Eq. (15) twice with respect to $r$ and also using Eq. (19), we obtain non-homogeneous Bessel equation in the cylindrical coordinates as$$\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}-\left(m^2+\frac{1}{r^2}\right)w=\frac{m^2}{2-N}\left\{\left(1-N\right)\frac{r}{2}\frac{\partial p}{\partial x}+\frac{C_1}{r}\right\}.$$

The general solution of above equation is as follow:

Then applying the fourth boundary condition of Eq. (17), and the boundary conditions (18), Eqs. (19) and (20) become

And further integrating Eq. (21) and applying the no-slip condition of Eq. (17), the axial velocity is found as

The radial velocity is derived from Eq. (15), by substituting $u$ from Eq. (23) and integrating it once with respect to$r$. The regularity condition, given in Eq. (17), determines the constant term and gives the radial velocity as

In order to get pressure gradient, we apply the radial velocity of the wall, given in Eq. (17), on Eq. (24). This gives

Integrating this, with respect to $x$, yields the pressure gradient as

Integrating it once again from $0$ to $x$, the pressure is obtained as

Substituting $x=l$ in Eq. (27), the pressure between the inlet and the outlet of the tube, is obtained as

The volume flow rate is defined as$Q\left(x,t\right)={\int }_0^{H}2rudr$, yields, on performing the integration, the following:

The time-averaged volume flow rate is obtained by averaging the volume flow rate for one time period. This gives

The time-averaged volume flow rate may be given in terms of the flow rate in the wave frame, and also in the laboratory frame, as$$\stackrel{\sim }{Q}=q+1-\varphi e^{\omega x}+\frac{3}{8}{\varphi }^2{e}^{2\omega x},$$

This helps us express the pressure gradient in terms of the time-averaged volume flow rate. With some manipulations Eqs. (30) and (32) give

which yields, on integration, pressure difference in terms of the time-averaged volume flow rate as

for $x=l$, which gives

Finally, the local wall shear stress is defined as$${\tau }_w=\frac{\partial u}{\partial r}{|}_{r=H},$$

Which, by virtue of Eq. (21), takes the form$${\tau }_w=\frac{\left(1-N\right)H}{2}\frac{\partial p}{\partial x},$$

and further reduces, in view Eq. (26), to

4 Reflux limit

Reflux is an important phenomenon of peristaltic movement and refers to the presence of fluid particles that move, on the average, in a direction opposite to the net flow in the close vicinity of the wall (Shapiro et al., 1969).

For the axi-symmetric case, the dimensional form of the stream function in the wave frame is defined as

Using the following transformations between the wave and the laboratory frames, defined as

Stream function at the wall, ${\psi }_w$ is solved from Eq. (39) by substituting$r=H$. A simplification yields

Reflux flow rate, $Q_{\psi }\left(x\right)$ associated with a particle at the position $x$ is given by

which, on averaging over one cycle, gives

Moreover, in order to evaluate the reflux limit, ${\stackrel{\sim }{Q}}_{\psi }$ is expanded in a power series, in terms of a small parameter $\epsilon$ about the wall, where $\epsilon \left({=\psi -\psi }_w\right)$ is subjected to the reflux condition

The coefficient of the first two terms in the expansion of $r$ is obtained only for small values of $m$. Substituting the expansion $r^2\left(\psi ,x\right)=H^2+a_1\epsilon +a_2{\epsilon }^2+a_3{\epsilon }^3+\cdots$ into equations (39), and using equations (40), we get

Then integrating Eq. (41) with respect to $x$ and using Eqs. (42)–(45), we obtain the reflux limit (i.e. The occurrence of the reflux) as

5 Results and discussion

In order to explore the effects of various parameters such as coupling number, micro-polar parameter, wave amplitude dilation parameter and the wall shear stress on swallowing of micro-polar fluid, we plot graphs for local pressure distribution along the axis. In order to examine the contribution exclusively of peristalsis, pressure of zero magnitude is prescribed at the two ends of the oesophagus, which makes $\mathrm{\Delta }{p}_l\left(t\right)=0$. At a particular time, only one bolus moves in the oesophagus, which is easily experienced when a non-Newtonian fluid swallows. Therefore, for analysis, we consider a single bolus swallowing in the tube which has the capacity to accommodate three boluses at a time, so far our discussion is concerned.

The fundamental motive is to study the local pressure distribution along the axis when a bolus travels down the oesophagus towards the cardiac sphincter. Since the mathematical model involves expressions that cannot be integrated by classical methods, the only way out is to go for numerical evaluation. Moreover, the values of all the non-dimensional parameters are merely suitable assumptions to facilitate qualitative investigation.

5.1

5.1 Effect of dilating wave amplitude $\mathit{\omega }$ on pressure

The bolus is supposed to be already there in the tube at$t=0.00$. Dashed line symmetric about the axis of the tube are drawn to indicate the position of the bolus in the direction left to right. Fall of pressure from the left towards right paves way for the bolus to move in the tube. Different inclinations of the pressure curve indicate different pressure gradients of the corresponding part of the bolus. Its gradual rise from the head of the bolus to the end of the tube is the revelation that the motion is under control; the bolus is never let move freely.

The effect of dilating wave amplitude $\omega$ on the flow dynamics is plotted in Fig. 2. We fix the various parameters as$l=3,\varphi =0.7,N=0.10,m=1.0$, and let $\omega$ vary. Fall and rise of pressure throughout the length of the tube from $t=0.00$ to $t=2.00$ in the plots is observed to be dependent on the wave amplitude$\omega$, $\omega =0.00$ corresponding to the constant amplitude (Fig. 2). Variation between the maximum and the minimum pressures becomes larger when the wave-amplitude dilates, e.g., when$\omega =0.01$,$0.02$ (Fig. 2). An observation of Fig. 2(a-f) reveals that pressure gradients, corresponding to $\omega =0.01$ and$\omega =0.02$, are larger in magnitude in the lower oesophageal part than that in the upper oesophageal part and also, if we measure the magnitude, the pressure rises more in the lower part of the oesophagus. It conforms the experimental observations of high pressure zone in the lower oesophageal part even when the fluid transported is of micro-polar nature (Kahrilas et al., 1995).

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Fig. 2

Pressure distribution along axial distance at different time instants showing the effect of dilation parameter$\omega$. Other parameters are taken as$l=3,\varphi =0.7,N=0.10,m=1.0$.

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5.2

5.2 Effect of coupling number $\mathit{N}$ on pressure

Plots in Fig. 3 depict the impact of the coupling effect parameter$N$, a measure of particle coupling with its surroundings, on pressure distribution along the axis of the tube. We set the various parameters as$l=3,\varphi =0.7,\omega =0.01,m=1.0$, and let $N$vary in the range 0–0.75.

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Fig. 3

Pressure distribution along axial distance at different time instants showing the effect of coupling number$N$. Other parameters are set as $l=3,\varphi =0.7,\omega =0.01,m=1.0$.

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It is observed that pressure gradient as well as pressure along the length of the oesophagus increases as the coupling effect parameter $N$ increases throughout the length of the tube. This may be physically interpreted as that internal rotation of the fluid particles increases pressure; and finally the fluid reduces to Newtonian, i.e. as $N\to 0$, pressure is minimum. This leads to the conclusion that physically the oesophagus has to make additional efforts to swallow a micro-polar fluid. Similar is the observation for all values of $t$ ranging from $0\to 2$, i.e., throughout swallowing. Temporal effects are similar to those observed for other fluids such as Newtonian, power-law, visco-elastic, visco-plastic and magneto hydrodynamic fluids (Misra and Pandey, 2001; Pandey and Tripathi, 2010). Figures, together with captions, provide the details (cf. Fig. 3). Achalasia causes inadequate lower oesophageal sphincter relaxation; as a consequence of which oesophageal clearance is delayed. A possible treatment of patients with inadequate lower oesophageal sphincter relaxation is through drugs or by operation (Spechler and Castell, 2001). Thus, this problem will be more acute if the fluid is micro-polar.

5.3

5.3 Effect of micro-polar parameter on pressure

Our next analyse the role of the other micro-polar parameter$m$ by setting other parameters$l=3,\varphi =0.7,\omega =0.01,N=0.50$, and vary $m$ in the range 1.0–5.0. It is noticed that the pressure along the entire length of the tube decreases as $m$ increases. Hence, this parameter has an opposite effect vis-à-vis coupling number $N$ (cf. Fig. 3). Since no value of $m$ can lead to Newtonian nature, no comparison can be made with Newtonian fluids. In fact, the micro-polar fluid has a complicated characteristic that is built up by the combined effects of several parameters. This may be recorded that once$N$ vanishes, $m$ no longer exits (cf. Fig. 4 ).

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Fig. 4

Pressure distribution along axial distance at different time instants showing the effect of micro polar parameter$m$. Other parameters are taken as$l=3,\varphi =0.7,\omega =0.01,N=0.50$.

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