Peristaltic Transport of Power-Law Fluid in an Elastic Tapered Tube with Variable Cross-Section Induced by Dilating Peristaltic Wave

The peristaltic transport of power-law fluid in an elastic tapered tube with variable cross-section induced by dilating peristaltic wave is studied. The exact solution of the expression for axial velocity, radial velocity, stream function, local shear stress, volume of flow rate and pressure gradient are obtained under the assumption of long wavelength and low Reynolds number. The effects of all parameters that appear in the problem are analyzed through graphs. The results showed that the flux is sinusoidal in nature and it is an increasing function with the increase of whereas it is a decreasing function with the increase of . An opposite behavior for shear strain is noticed compared to pressure gradient. Finally, trapping phenomenon is presented to explain the physical behavior of various parameters. It is noted that the size of the trapping bolus increases with increasing whereas it decreases as increases. MATHEMATICA software is used to plot all figures.


Introduction
The need to transport fluids through peristaltic channels/tubes has dramatically increased in recent days due partly to its application in engineering and biological system. The peristaltic flow is the transport of fluid generated by wave traveling with invariable speed along the elastic wall in a channel or a tube.
The flow geometry plays an important role to understand the characteristics of different fluid flows. Most of the earlier research works were concentrated on rigid channels. The biological systems deal with elastic boundaries, so the non-Newtonian fluid flow through elastic walls gives some influential applications like swallowing of food though esophagus, blood flow in a small blood vessel, movement of chyme through intestine and colonic vessel, to understand the evolution of pathogen due to vessel deformation. Some of the non-Newtonian fluid models which are accepted by researcher are Jeffery fluid, Herschel-Bulkley fluid, Bingham fluid, power-law fluid etc. The power-law fluid is the simplest and is accepted as a model to understand the rheological properties of physiological fluids in living organisms [1][2][3].
Srivastava and Srivastava [4] studied the peristaltic transport of a power-law fluid in a uniform and non-uniform channel. The helical flow of a power-law fluid in a thin annulus with permeable walls is discussed by Vajravelu et al. [5]. Pandey and Chaube [6] investigated the peristaltic transport of a visco-elastic fluid in a tube of non-uniform cross section. Sadeghi and Jalali Talab [7] studied the analytical investigation of peristaltic transport of power law fluid through a tube. Hina et al. [8] examined the slip effects on the peristaltic flow of non-Newtonian fluid in a curved channel and found that an intensification in the slip effect resulted in a larger axial velocity. Vajravelu et al. [9] studied the peristaltic transport of a Casson fluid in an elastic tube. Maiti and Pandey [10] analyzed a theoretical study of a nonlinear rheological fluid transport in an axisymmetric tube by cilia. Selvi et al. [11] considered a mathematical model to study the influence of elasticity on the peristaltic flow of a power-law fluid in a tube. Pandey and Singh [12] discussed the peristaltic transport of Herschel-Bulkley fluids in tubes of variable cross sections induced by dilating peristaltic waves (application to sliding hiatus hernia). The effect of elasticity on peristaltic transport of Herschel-Bulkley fluids in tube of non-uniform cross-section was studied by Selvi and Srinivas [13]. Adnan and Abdulhadi [14] investigated the effect of an inclined magnetic field on peristaltic flow of Bingham plastic fluid in an inclined symmetric channel with slip conditions. Kareem and Abdulhadi [15] studied the impacts of heat and mass transfer on magneto hydrodynamic peristaltic flow having temperature-dependent properties in an inclined channel through porous media. Hasen and Abdulhadi [16] discussed the MHD effect on peristaltic transport for rabinowitsch fluid through a porous medium in cilia channel.
In view of the above studies, a mathematical model is considered to analyze the peristaltic transport of power-law fluid in an elastic tapered tube with variable cross-sections induced by dilating peristaltic wave. The exact expressions for axial velocity, radial velocity, stream function, pressure gradient, local shear stress and volume of flow rate are obtained. The effects of all pertinent parameters on flow are explained through graphs.

Mathematical Formulation
Consider an axisymmetric peristaltic flow of an incompressible power-law fluid through a cylinder, with variable cross sections, of an elastic tapered tube of length and radius . The tube walls are subjected to infinite sinusoidal wave with dilating amplitude movement and constant speed , as shown in Figure-1.
In a cylindrical coordinate system ̅ ̅ , where ̅ lies along the center line of the tube and ̅ is the radius of the tube, at axial station ̅ , the instantaneous radius of the tube is given by [12] where ̅ ̅ ̅ ̅ ̅ ̅ respectively, denote radial displacement of the wall from the center line, axial coordinate, dilation parameter, time parameter, radius of the tube at the inlet ( in the absence of elasticity ), slope of the tube wall (non-uniform parameter), amplitude of the wave, and the wavelength . Because of the axisymmetric condition, the angle in the cylindrical coordinate is eliminated.

The Basic and Constitutive Equations
The basic governing equations of non-Newtonian power-law fluid flow are given by: [17] the equation of mass conservation ⃑ ⃑ and the equations of momentum conservation (Navier-Stokes equations) ⃑ ⃑ ̅ in which ⃑ is the velocity, is the density, is the material time derivative and ̅ is the Cauchy stress tensor.
Let ⃑ ̅ ̅ ̅ be the velocity components in the fixed frame cylindrical coordinate system ̅ ̅ . According to the assumption of the problem, the velocity components can be written as The power-law fluid model of Ostwald-De Waele is chosen and the constitutive equations can be expressed as: [11,17] ̅ where ̅ is the extra stress tensor, is the identity tensor, ̅ is the pressure, ̅ is the rate of deformation (the rate of strain), is the consistency parameter (the consistency index of non-Newtonian viscosity), { [ ] ⃑ } is the symmetric part of the velocity gradient (first Rivin-Ericksen tensor), and ̅ is defined as: where is the second invariant of strain-rate tensor ̅ ), where the numbers 1,2 and 3 are the coordinates ̅ ̅ respectively. Now, from equations (2) and (3) the continuity and Navier-Stokes equations in the fixed frame are given by The continuity The ̅ component of Navier-Stokes equations (7), the rate of strain tensor ̅ in the fixed frame has the following components [18] ̅ The flow is unsteady in the laboratory frame (fixed frame) and it becomes steady in the wave frame (move frame). The transformation between stationary coordinates ̅ ̅ and move coordinates ̅ ̅ is given by where ̅ and ̅ are the radial and axial velocity components in the moving coordinates.
We introduce the dimensionless quantities that are used to find out the non-dimensional analysis as where is the wave number and is the Reynolds number ( . p, F, , and are the pressure, the flux, the extra stress tensor, the rate of deformation, and the stream function in nondimensional form, respectively. Equations (1) and (9)- (13) in the move frame of non-dimensional form are given by: By applying low Reynolds number and long wavelength approximation (i.e. neglecting wave number), an assumption which is based on the fact that the radius of the tube is small compared to the wavelength of the peristaltic wave, then equations (17)-(19) reduces to

( )
The local shear stress can be calculated from equation (6) as follows: The dimensionless of equation (6), in view of equation (21), is given by As mentioned in the previous that implies , otherwise substituting equation (21) into equations (26), in view of , to get and the shear rate has been assumed to be negative throughout the moving frame, thus: [3] Now, by substituting equation (28)

Result and Discussion
This section deals with the computational results of the peristaltic transport of non-Newtonian power-law fluid in an elastic tapered tube with variable cross sections induced by dilating peristaltic wave. The effect of various parameters that appear in the problem, such as power-law index , slope of the tube wall (non-uniform parameter) , dilation parameter and amplitude ratio on the axial velocity , radial velocity , pressure gradient , volume flux , local shear stress and trapping of the flow, are discussed graphically. MATHEMATICA software is used to plot all figures.

Axial Velocity
The variation of axial velocity along the radial direction at for different values of are analyzed in Figures-2 -6, respectively. Note that the axial velocity exhibits an oscillating behavior with the increase of , as shown in Figures-2 -4. It is observed that an increase in leads to an increase in the axial velocity at the center of the tube along with a decrease near the boundary. Also, as shown in Figure-5, the non-uniform parameter exerts the greatest impact on the axial velocity. It is found that, at the center of the tube, the axial velocity decreased with increasing while it is increasing at the boundary. The axial velocity increased with increasing , as explained in Figure-6.

Radial Velocity
Figures 7 -10 present the behavior of various parameters involved in the radial velocity along the radial direction. Figure -7 shows the effect of power law index on radial velocity. It is clear that the radial velocity reduces with an increase in . Figure -8 elucidates the effect of on the radial velocity. It is noted that an increase in causes the greatest decreases in the radial velocity. The increase in amplitude ratio means a powerful peristaltic movement effect, which causes faster movement in the radial direction for the radial velocity. The impact of the non-uniform parameter on radial velocity is shown in Figure -9. It is observed that radial velocity increases with increasing . An opposite trend is observed in the case of , as shown in Figure -10.

Volume of Flow Rate (Flux)
In this subsection, the variation of flux for different values of parameters, such as power-law index , non-uniform parameter , dilation parameter and amplitude ratio are discussed. Figures  11 -15 show that the flux along non-uniform tube radius for different values of pertinent parameters are sinusoidal in nature. Figure -11 illustrates the effect of power-law index on the flux. It is noticed that the flux in an elastic tube increases with the increase in . This result is in conformity with the result in Sadeghi and Jalali Talab [7]. The influence of non-uniform parameter on flux variation is explained in Figure -12. It is found that the flow rate decreases with the increase in . The opposite behavior is observed in the case of dilation parameter , as shown in Figure -13. The effect of amplitude ratio on flux variation is described in Figure -14. It is clear that the flow rate enhances for increasing values of due to an increase in maximum displacement of the fluid particles. Hence, the increases in fluid velocity yields an enhanced flow rate. This result is in agreement with the result obtained by Selvi et al. [9]. Figure -15 shows that the flux increases with the increase in the pressure gradient.

Pressure Gradient
The influence of physical parameters on pressure gradient is represented in Figures 16 -20. As a result of peristaltic property of the tube wall, it may expand or contract. Due to this property of the tube wall, there exist changes in the shape of cross-section of the tube. It can be noticed that, in the wider part of the tube [ ] and [ ], the pressure is relatively small, that is, the flow is easily passing throughout the tube. On the other hand, in a narrow part of the tube [ ] a much higher pressure is required to maintain the same flux to pass it, especially at approximately

Murad and Abdulhadi
Iraqi Journal of Science, 2021, Vol. 62, No. 4, pp: 1293-1306 1302 .The impact of fluid behavior index on pressure gradient is illustrated in Figure -16 and it is clear that the pressure gradient is reduced with increasing . The effect of the non-uniform parameter on is shown in Figure -17. It is observed that increase with the increase in . Figure -18 displays the impact of the dilation parameter on pressure gradient, where it is noted that the pressure gradient decreases with the increase of . Figures -19 and 20 show that pressure gradient is inversely proportional to the increase of the amplitude of the wave and rate of flux .

Shear Stress
The behavior of shear stress at the wall of the tube (i.e. at ) for the variation of different parameters is illustrated in Figures 21 -25. It can be observed from Figures 21 -24 that the shear stress increases with the increase of . Figure -25 shows that the shear stress decreases with the increase of the non-uniform parameter .

Trapping Phenomenon
The formation of an internally circulation bolus of fluid by the closed streamlines is known as trapping. Trapping is an interesting phenomenon of peristalsis mechanism in the wave frame which is pushed ahead with the peristaltic wave. The influences of different parameters on the size of trapped bolus are shown in Figures 26 -30. It is noted from Figure -26 that the size of the trapped bolus increases as the power-law index increases. This result is in agreement with the result obtained by Selvi et al. [9]. The impact of the non-uniform parameter is explained in Figure -27. It is observed that the trapped bolus increases with increasing It is clear from Figure -28 that the bolus size is reduced with increasing values of dilation parameter . The bolus size enhances with increasing values of amplitude ratio as illustrated in Figure -29. This result is in agreement with the result obtained by Selvi et al. [9] and Selvi and Srinivas [11]. Figure -

Conclusions
The present study deals with the peristaltic transport of power-law fluid in an elastic tapered tube of variable cross sections induced by dilating peristaltic wave . The biological systems deal with elastic boundaries, so that the non-Newtonian fluid flow through elastic walls gives some influential applications such as the swallowing of food through the esophagus. The presented investigation can be applicable for the patients suffering from sliding hiatus hernia, which happens when the esophagus is divided into two parts. First we consider that the esophagus gets diverged near the distal end, while the second consideration is that, at the distal end, some part of the esophagus gets diverged and then converged. The exact solutions of the problem are obtained under the assumptions of low Reynolds number and long wavelength . The influence of various parameters in the problem is explained graphically. From the above results, we concluded that the increase in leads to an increase in the axial velocity at the center of the tube, which is decreased near the boundary. Whereas the opposite behavior is noted for which is increased with increasing F in all cross sections of the tube. Also, we concluded that the radial velocity distribution is parabolic in nature and it is reduced with increasing it is increased with increasing and it is very sensitive to the variation of . The volume flow rate is sinusoidal in nature and it is an increasing function with the increase of whereas it is a decreasing function with the increase of . It is noted that the pressure gradient decreases with the increase of while it increases with the increase of . An opposite behavior for shear strain is noticed compared to pressure gradient. The size of the trapping bolus increases with increasing whereas it decreases as increases.