Peristaltic transport of a second-grade dusty fluid in a tube

These article intents to analyze the behavior of second-grade dusty fluid flowing through a flexible tube whose walls are induced by the peristaltic movement. Coupled equations for the fluid and solid particles have been modelled by considering streamline conversions. Regular perturbation technique has been implemented to get the solutions and the outcomes are demonstrated through graphs. The impact of diverse parameters on the contour presentations and on the velocity of the solid grains and fluid has been illustrated. The velocity for both fluid and solid grains is increased with the enhancement in wave number and Reynolds number. In retrograde region, pumping rate increases with the increase in and it decreases for increased values of


INTRODUCTION
The flow in a flexible tube is induced by progressive wave along its walls. This progressive wave phenomenon is known as peristalsis. The particular application of peristalsis in an elastic tube is found in many biological systems such as cardiovascular system, fertile egg in the female fallopian tube, trachea and duodenum. The mechanism of peristalsis was first modelled by assuming long wavelength analysis along with low Reynolds number. Morgan and Kiely (1957) investigated the propagation of a harmonic wave in a flexible tube. Womersley (1957) considered the axisymmetric flow in a tube. Singh and Singh (2014) considered the Rabinowitsch fluid model in a tube. Hameed et al. (2015) considered the fractional second-grade fluid in a tube. Shaheen and Asjad (2018) presented a study that deals with the convectively heated surface of the tube. The fluid that they assumed in this study was Sisko fluid. There after theories were expanded to non-creeping flow. Yin and Fung (1969) studied the peristaltic movement in a tube. Streeter et al. (1964) investigated the non-creeping flow in an arterial system of a dog. Olsen and Shapiro (1967) studied the unsteady movement in a tube which was elastic. In a tube, the second order fluid was studied by Siddiqui and Schwarz (1994). Pandey and Claube (2010) considered the creeping and non-creeping movement of the fluids which were viscoelastic in nature and flowing in a tube which was non-uniform. A recent study that deals with the non-creeping flow is presented by (Pantokratoras, 2018;Hayat et al., 2016aHayat et al., , b, 2017aHayat et al., , b, c, 2018Hayat et al., , b, 2017bKhan et al., 2017Khan et al., , 2018aKhan et al., , b, 2019Rashid et al., 2019;Qayyum et al., 2018a, b;Asghar et al.,2018Asghar et al., , 2019aAsghar et al., , b, 2019cAsghar et al., , 2020a Javid et al., 2019a, b;Ali et al., 2019b;Dogonchi et al., 2020) are notable researches in the same context.
It is pertinent to mention here that in the above works, the pure form of the fluids is considered. However, water and air in natural form contains impurities like foreign bodies and dust particles. The fluid that has dust particles mixed in it is labelled as dusty fluid. The dusty fluids are of great importance. The unrefined crude oil and petroleum, fruit juices with pulp fibers, paints with particles like glitter suspended in it are few examples of dusty fluids. In Saffman (1962) studied the gas flow containing the dust particles in it. This research derived the interest of many researchers towards the dusty fluids and their applications. Kumar (1980) assumed the geometry of the wavy wall and studied two-dimensional dusty fluid past through it. The effects of wall properties on the dusty fluid were presented by Rathod and Kulkarni (2011). The passage was considered porous as well. Hayat et al. (2014) investigated the dusty fluid under the influence of Joule and radiative heating effects along with the wall properties. Khan and Tariq (2018).assumed the Walter's B dusty fluid in a passage under the impact of wall properties. Bhatti et al. (2017) used the Jeffrey fluid with particle suspended in it to analyze the thermal radiation effects in a duct.
The ambition of this study is to analyze the dusty fluid passing through a tube. No attempt has been made to investigate the dusty fluid in a cylinder or tube that is exhibiting the peristaltic transport. Dusty second-grade fluid flowing past the asymmetric passage has been considered. This research can be useful in petroleum industry, paint industry and papermaking industry. Motion equations for dust and fluid have been modeled. DSolver has been exploited to get the solution. Results are exhibited by streamline and pressure graphs.

MATHEMATICAL FORMULATION
Axisymmetric flow of a second-grade fluid contains small spherical particles in a tube was taken. These particles are even in size. Therefore, the number density N of the solid granules are presumed to be constant. The walls of the tube are propagated with constant speed c. The cylindrical coordinate structure ( ̅ ̅ ) In the radial and axial directions, the velocity elements were taken as ̅ , ̅ for fluid and ̅ and ̅̅̅̅ for dust particles.
The walls of the tube are given as: The geometry is given in Figure 1.
For second-grade fluid, extra stress tensor is specified by ̅ .
In the fixed frame ( ̅ ̅ ), the equations expressing the movement of the fluid are [9] and for solid grains [42] Using the transformations mentioned below to associate the moving ( ̅ ̅ )and fixed frames ( ̅ ̅ ) and taking into account the dimensionless quantities Components of extra stress tensor are: The streamline conversions implied are: The compatibility equations for fluid and dust after removing the pressure gradient are: where ( ).
Walls geometry in dimensionless configuration is: The non-dimension boundary conditions are: The dimensionless time flow of the fluid and the solid granules are given as: The expression for the pressure rise is:

Solution methodology
The equations obtained for the dust and the fluid particles are nonlinear in nature. To obtain the solution, perturbation technique has been adopted.

( )
This solution is same as obtained by Siddiqui and Schwarz (1994).

First order system
Where  In the absence of solid particles, this solution is same as obtained by Siddiqui and Schwarz (1994).

Second order system
Where, With ( ) These systems of equations are solved by using DSolver methodology in Mathematica software.

RESULTS AND DISCUSSION
This section is dedicated to discuss the impact of diverse parameters on the velocity, streamline, pressure rise.  Figure 2, with the increase in wave number, the fluid flows more smoothly and efficiently in the desired direction. With the rise in the Reynolds number (Re), we see an increasing manner in the velocity in the center of the tube. As increase in Re reduces the friction force thus causing rise in the fluid velocity. Growth in enhances the velocity of the fluid as demonstrated in Figure 4. The pressure gradient versus z is given in Figures 5 to 7 for the impact of Re, and respectively. It is noted that small pressure gradient has been occurred in [ ] It means that the flow can smoothly go through without imposition of large pressure gradient and large pressure gradient occurs for [ ]. It means greater pressure gradient is required to  As the amplitude is increased, it is concluded that more pressure is needed to pump the fluid throughout the region. The pumping rate falls in the retrograde region ( ) as the wave number increases. While a reverse situation can be observed in co-pumping region ( ). The retrograde region broadens with the increase in . Pumping rate grows with the increase in second grade parameter. Impression of different parameters on wall shear stress at , are demonstrated in Figures 11 to 13. With the increase in the parameters, it can be seen that the walls of the tube are exhibiting to and fro movement thus the peristaltic motion of the wall can be seen clearly. Figure 14 shows that as the values of the wave number ( ) is raised, the bolus in the channel get enhanced. As wave number is increased, the movement of bolus can be seen stretching towards the upward direction. The impact of Reynolds number (Re) on the streamline pattern is shown in Figure 15. As the Reynolds number is increased, viscous force weakens thus it was observed that the motion of the fluid gets smoother and bolus  Figure 16. For no bolus is formed as shown in Figure 16(a). As the values of are increased, a clear change in the formation and volume of the bolus was observed. The motion of the fluid particles is towards the direction of the amplitude of the tube.
The impact of parameters like ( ) Re and ( ) on the velocity of the solid grains is given in Figures 17 to 19. Increase in the velocity for ( ) and for Re, growth in the velocity of the solid grains was observed. Increase in Reynolds number refers that viscous forces are weakening thus particles may flow more smoothly but for the second-grade parameter ( ), we observe no change in the velocity of the solid grains at least to ( ) Figure 20 shows the impact of ( ) on the contour patterns of the solid grains. As the wave number grows, an expansion in the volume of the bolus was seen. This indicates that the movement of particles is expanding. The enlargement in the Reynolds number (Re) enhances the bolus as illustrated in Figure 21. With lower viscous force, the bolus formed shows that the  Figure 23 shows a comparison between the velocity profile obtained by perturbation technique and numerical technique. The velocity of the fluid seems to be following same pattern

Conclusions
In this article, the second-grade fluid containing fine dust grains in a tube was studied. Perturbation technique up to ( ) has been imposed to get the solutions. The results are shown through graphs. Following results are worth mentioning: 1. The bolus expands for solid grains and fluid as the wave number enhances. 2. The trapped bolus expands for dust grains and the