1 Introduction

The peristaltic flow is a transport of fluid generated by waves traveling along the wall in a channel or a tube, which is commonly seen in organisms, such as the vasomotion, gastrointestinal tract, and urethra. Latham^[1] first experimentally studied the peristaltic flow. Then, Burns and Parkes^[2] established a mathematical model to describe peristalsis. Ensuing year, Shapiro et al.^[3] developed a long wavelength approximation method to solve the mathematical equations and discovered reflux and trapping phenomena. Brown and Hung^[4] employed an implicit finite-difference technique to solve the Navier-Stokes equations for peristaltic flows. All these investigations have been carried out by assuming that physiological fluids behaved as Newtonian fluids.

The hemolymph was discovered to be non-Newtonian fluids^[5-7], which caused more attention to the peristaltic transport of non-Newtonian fluids. Radhakrishnamacharya^[8] studied the peristaltic flow of power-law fluids by the perturbation method, while Srivastava and Srivastava^[9] investigated the cases for the Casson fluids. Siddiqui and Schwarz^[10] analyzed the motion of a third-order fluid in a channel and found that the pumping rate of a shear-thinning fluid was less than that of a Newtonian fluid. El Hakeem et al.^[11] studied the separated flow through peristaltic motion of Carreau fluid and concluded that the radial velocity of fluid at the separated flow points for the Newtonian fluid was larger than that for the non-Newtonian fluid. Haroun^[12] investigated the peristaltic transport of a fourth grade fluid in an inclined asymmetric channel and found that the trapped bolus volume decreased with the increase in the Deborah number, the phase difference, and the Froud number. Hayat et al.^[13] studied the peristaltic flow of a micropolar fluid in a channel and obtained the exact solutions of the flow quantities under low Reynolds number assumptions. Ali et al.^[14] explored the peristaltic motion of Oldroyd 4-constant fluid in a planar channel and discussed the effect of material parameters on the pumping. Soon afterwards, Ali et al.^[15] studied the peristaltic flow of a non-Newtonian fluid in a curved channel, and found that an increase in the curvature of the channel helped in reducing the pressure rise and the trapping phenomenon was altered by the presence of curvature for a non-Newtonian third grade fluid. Nadeem et al.^[16] examined the effects of heat transfer on a peristaltic flow of a micropolar fluid in a vertical annulus and discussed the trapping phenomena for different parameters. Hina et al.^[17] analyzed 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. The size and circulation of the trapped boluses increased with an increase in the slip parameter. Akbar and Nadeem^[18] studied the Rabinowitsch fluid model for the peristaltic flow, and calculated the exact solutions for the velocity and pressure gradient. Ali et al.^[19] analyzed the peristaltic motion of a non-Newtonian Carreau fluid in a curved channel and found that an increase in the Weissenberg number generated a small eddy in the vicinity of the lower wall of the channel. Tripathi et al.^[20] developed a mathematical model to simulate the transport phenomena via peristaltic transport, and tested the effects of the curvature parameter, apparent viscosity coefficient, and volumetric flow rate on the flow characteristics. Srinivas et al.^[21] investigated the peristaltic pumping of Carreau fluid in an elastic tube and showed that the flux of Carreau fluid with peristalsis was more when the tension relation was a fifth degree polynomial as compared with the exponential curve.

Blood circulation, a multiphase flow including plasma, red blood cells (RBCs), and white blood cells (WBCs), is a limitless subject in the research of peristaltic flow. Hung and Brown^[22] experimentally studied the behavior of one particle in a peristaltic pump. Since the RBCs constitute up to 45% of the blood volume (hematocrit) in humans and due to the lack of calculation power, the continuum model for the multiphase flow of blood is commonly seen in theoretical research. For example, Srivastava and Srivastava^[23] chose a planar two-dimensional symmetric geometry for the peristalsis of particle-fluid suspension, and Misra and Pandey^[24] adopted an axisymmetric case. In addition, Srivastava and Saxena^[25] studied the peristalsis of blood considering erythrocytes as a Casson fluid and plasma as a Newtonian fluid. Later, Nagarani and Sarojamma^[26] analyzed the peristaltic motion of a non-Newtonian, power-law fluid with suspension of small particles in a two-dimensional channel, and explained the formation and growth of trapping zone for variations in the amplitude ratio and volume flow rate. Hayat et al.^[27] modeled the peristaltic transport of viscous nanofluid in an asymmetric channel by taking the effects of Brownian motion and thermophoresis into account. Nadeem et al.^[28] checked the effect of lateral walls on nanoparticle phenomenon in peristalsis with a non-Newtonian fluid, and discussed the features of walls, flow rate, Jeffrey fluid parameter, Brownian motion, and thermophoresis. Maiti and Pandey^[29] presented a theoretical study of a nonlinear rheological fluid transport in an axisymmetric tube by cilia. Eldesoky et al.^[30] theoretically investigated the peristaltic pumping of a viscous compressible liquid mixed with rigid spherical particles of the same size in a channel.

Recently, the development of computer technology and the growing computer resource make the implementation of particle-trajectory model possible. Connington et al.^[31] analyzed the motion of a finite micro-particle in peristaltic transport using the lattice Boltzmann method. Jiménez-Lozano et al.^[32] simulated the peristaltic transport of 1 280 particles utilizing the Basset-Boussinesq-Oseen (BBO) equation. Later, Jiménez-Lozano and Sen^[33] extended the method to the case beyond ureter possible condition, and a complete prospect of the particle trapping phenomenon at different amplitude ratios and mean flow rates was obtained. It shows that the particle-trajectory model is convenient to discover the behavior of one particle or a group of particles in the peristaltic transport, especially for the trajection of bacteria in the biological flow. Moreover, the solution of the particle-trajectory model is based on the solution of the Eulerian velocity field of the multiphase flow, where the analytical solution is better than the numerical one^[32].

The present work aims to investigate the mixture flow of particles and power-law fluid in a round peristaltic tube with flexible walls using the continuum model. The two-way coupling approach for the fluid phase and particle phase is used, treating the particles as a continuum phase rather than resolving each particle. The phenomena of reflux and trapping are analyzed, and the influence of different governing parameters on the size of trapping zone is clarified.

2 Problem description 2.1 Mathematical model

Consider an axisymmetric flow of a mixture of small particles uniformly distributed in an incompressible power-law fluid through a circular cylindrical tube. The tube has a radius a, and the wall is subject to sinusoidal waves with a speed of c, as shown in Fig. 1. In the cylindrical coordinate system (r, φ, z), the flexible walls are given by

(1)

where a is the mean radius of the tube, b is the amplitude, t is the time, and λ is the wavelength. Because of the axisymmetric condition, the angle φ in the cylindrical coordinate is eliminated.

The particles are treated as a continuum, following the model of Saffman^[34]. Assume that the particle number density N is a constant. Then, the equations governing the conservation of mass and momentum for the fluid and particles can be written as

(2)

(3)

(4)

(5)

(6)

where (v_z, v_r) and (v_pz, v_pr) denote the axial and radial velocity components of the fluid phase and the particulate phase, respectively, m, p, and ρ are, respectively, the particle mass, the fluid pressure, and the density of the mixture, and K is the Stokes resistance coefficient.

The Ostwald-De Waelepower-law model is chosen, and the stress-strain relation is given by^[35]

(7)

where σ is the modulus of the rate-of-strain tensor, and η is the power-law viscosity coefficient. n is the power-law index. n<1, n=1, and n>1 correspond to the dilatant fluid, the Newtonian fluid, and the pseudoplastic fluid, respectively.

In the cylindrical coordinate system, the rate of strain is denoted as

The boundary conditions for the peristaltic tube are

(8)

(9)

(10)

where (8) guarantees the axisymmetry of the flow near the axis, (9) represents the flexible wall without slip, and (10) is the flux across the cross section of the tube.

2.2 Fixed and moving frames

The above equations are derived in the fixed frame, i.e., the laboratory frame as shown in Fig. 2, where the problem is non-steady, and the flow field changes with time. For simplification, the moving frame, i.e., the wave frame in Fig. 3 which moves at the speed of c in the direction of wave propagation, is thus used. The corresponding transformations between fixed and moving frames are as follows:

(11)

Then, the govering equations in the moving frame are easily obtained. The instantaneous wall fluxes across the tube between the axis and boundary in fixed and moving frames are, respectively,

(12)

(13)

The relation of the two fluxes is q=Q+cπs², and the time mean flow at each cross section is obtained by integrating over the period T,

(14)

2.3 Non-dimensional parameters

Burns and Parkes^[2] assumed that the amplitude ratio ϕ ≡b/a was very small, but Shapiro et al.^[3] suggested that the wavenumber a/λ tends to zero. Tao^[36] also concluded that the infinite wavelength assumption was more pertinent to the pulsating flow in the blood vessel. Hence, we assume that the wavenumber is very small and choose the expansion quantities for the perturbation method, i.e., ε =a/λ ≪ 1.

Other non-dimensional quantities are as follows:

(15)

(16)

As a result, the non-dimensional forms for the flexible wall, governing equations, and boundary conditions are listed in the following. Note that the asterisk is dropped for conciseness.

For the flexible wall,

(17)

For the governing equations,

(18)

(19)

(20)

(21)

(22)

For the boundary conditions,

(23)

(24)

(25)

where denotes the mass concentration of the suspended particles, is the relaxation time, and is the Reynolds number for the power-law fluid flows^[37].

3 Method of perturbation 3.1 Expansion of perturbation

By assuming that the wavelength is infinite, the velocities and the pressure gradient are expanded in a power series of the wavenumber ε,

(26)

(27)

(28)

Substituting (26)-(28) into (18)-(22) and collecting terms of equal powers of ε up to O(ε), we obtain the zeroth-order equations as follows:

(29)

(30)

(31)

(32)

(33)

and the corresponding boundary conditions as follows:

(34)

(35)

(36)

In addition, the first-order equations are also documented,

(37)

(38)

(39)

(40)

(41)

as well as the corresponding boundary conditions

(42)

(43)

(44)

3.2 Solution of zeroth-order equations

Substituting (32) into (30) yields

(45)

With the boundary conditions (34)-(36), (45) is solved to obtain the zeroth-order axial velocity,

(46)

Applying the continuity equation (29) and boundary conditions (34) and (35), the zeroth-order transverse velocity is also obtained,

(47)

In the cylindrical coordinate system, the stream function is defined as dψ=rv_zdr-rv_rdz. Thus, we get the stream function for the zeroth order,

(48)

Solving (45) again with the result of (46), the expression for the pressure gradient is acquired,

(49)

The results are similar to the solution for the Poiseuille flow of power-law fluid^[35]. Furthermore, the results also agree with the expansions obtained by Shapiro et al.^[3] when n=1, which corresponds to the Newtonian fluid.

According to (32) and (33), the velocities of particles can be obtained from the following expressions:

(50)

(51)

3.3 Solution of first-order equations

From (40), we have

(52)

Substituting (52) and applying (50) and (51) into (38), the moment equation can be rearranged to

(53)

Using the boundary conditions (42)-(44), we solve (53). For convenience, the first-order solution for the axial velocity is expressed as

(54)

where

Using the continuity equation (37), we can get the transverse velocity for the first order,

(55)

Then, the stream function for the first order can be obtained,

(56)

The pressure gradient at the z-axis is

(57)

Now, we have the zeroth and first orders of the fluid velocities. Using (40) and (41), we can obtain the velocities of the particles for the first order. Therefore, the perturbation solution to the problem is found.

3.4 Mathematical analysis of the solution

The zeroth-order equation simplifies the case as a laminar flow omitting the inertia term. The first-order equation is a modification and superposition to the zeroth-order result, which considers the influence of the Reynolds number, the particle concentration, and the momentum coupling in the radial direction.

However, the first-order equation (38) contains a viscosity term which is not smooth (n<1) in the whole range of definitions (0≤z<1). Consequently, the result of (38) introduces the discrete results into the continuity equation (37). As (55) shows, there exist discrete points of v_r1 near the zone of z=z_c, where z_c satisfies

(58)

The solution of z_c exists as θ satisfies

(59)

The reason behind the phenomenon is that the viscous effect is very sensitive to the velocity gradient especially for the low time-mean flow in the case of dilatant fluid. For the condition of θ≤1, the velocity gradient at some certain parts of the flow field will eventually reduce to zero, causing a great decrease in the viscous effect of the fluids and producing a pretty large velocity in the r-direction. With the combination of dilatant fluid and peristaltic effect, the axial velocity appears very small, while the radial velocity can be quite large.

4 Results and discussion

The nonlinear system of the peristaltic transport for the mixture of power-law fluid and particles is studied by the above perturbation technique. For a small wavenumber ε, the results of zeroth order and first order can be linearly superimposed (see (26)-(28)). Setting ε=0.1, Re=5, and χ=0.2, the influence of θ, ϕ, α, and n on the velocities is explored.

4.1 Axial velocity

Figure 4 shows the axial velocities of fluid v_z and particles v_pz along the radial direction at z=0. In general, the axial velocity of fluid reaches its maximum at the axis and gradually decreases as the position moves far away from the axis. Finally, the value attains -1 because of the no-slip condition (in the non-dimensional moving frame). Accordingly, the axial velocity of particles behaves like the fluid but reaches -0.2 at the boundary on account of the relaxation time χ. Turning the values into the dimensional form finds that the speed of the particles is generally smaller than that of the fluid.

Note that, for n=0.5, 1.0, and 1.5, the velocity profiles of the fluid are blunt, parabolic, and penetrating, respectively. It means that as n increases, the axial velocity increases in the core region and decreases near the tube wall, as shown in Fig. 4(a). Due to the cylindrical geometry, the decrease in the velocity beyond the core region is less distinct than the results of Nagarani and Sarojamma^[26]. It is also seen that as the time-mean flow θ and the amplitude ratio ϕ increase, the magnitude of the axial velocity rises, as shown in Figs. 4(b) and 4(c). In addition, Fig. 4(d) reveals that the axial velocity slightly enhances with α.

The results of maximum and minimum axial velocities varying from ϕ=0 to ϕ=1 at the crest and trough at the axis are compared with the theoretical results of Shapiro et al.^[3] and numerical results of Xiao and Damodaran^[38] for Newtonian fluids (see Fig. 5). Since the present results are in good agreement with the previous works, the validation of the mathematical model is confirmed.

4.2 Transverse velocity

Peristalsis over the flexible wall of the tube is the major reason for studying the transverse velocity, meaning that the transverse velocity profile differs at different cross sections. In order to explore the influence of peristaltic transport, Fig. 6 presents the transverse velocities of fluid and particles along the radial direction. Note that the sections at z=0.25 and z=0.230 1 (i.e., at the crest and near the crest) are chosen to show the fluid and particle velocities, respectively. As shown in Fig. 6(a), the power-law index greatly affects the transverse velocity. The direction of transverse velocity in the case of shear-thickening fluid (n=1.5) is completely different from those of the Newtonian (n=1.0) and shear-thinning fluids (n=0.5). For n=0.5, the transverse velocity is always positive from the axis to the wall. This result refers to the effect of trapping flow, where the center of trapping bolus zone is not precisely located at the z=0.25 section for the non-Newtonian fluid. The results also imply that there is a deflection of the flow field when the power-law index n=0.5 or 1.5 compared with the Newtonian fluid, where the trapping zone is symmetric to a certain cross section, shown in Fig. 7 and confirmed in Subsection 5.2. Also shown in Figs. 6(b)-6(d), the amplitude ratio ϕ exerts the greatest influence on the transverse velocity, followed by the time-mean flow θ and the particle concentration α. The higher amplitude ratio means a more powerful peristaltic effect, which causes faster movement in the r-direction for both the fluid and particles.

5 Reflux and trapping phenomena 5.1 Reflux phenomenon

To analyze the reflux phenomenon in the Euler coordinate, Shapiro et al.^[3] defined the instantaneous flow q_ψ which is the flux of all the materials between the axis and the coordinate ψ. In the cylindrical coordinate system, it is given by

(60)

Substituting the stream function into the equation, we get

(61)

Taking the time-mean over the period T, we obtain

(62)

Transforming the fixed frame to the moving frame and introducing the dimensionless variables, it becomes

(63)

After dropping asterisks, we have

(64)

The integration result is difficult to obtain. However, according to the analysis in Subsection 5.1, the reflux phenomenon occurs near the wall. Therefore, we just focus on the integration results at the wall ψ=ψ_w and near the wall |ψ₀-ψ_0w|=δ→0, and compare their values. indicates that there is reflux between ψ₀ and ψ_0w.

According to the definition of δ, it can be used as the perturbation parameter,

(65)

Consider the following expansions of the parameter r:

(66)

Substituting (66) and (48) into (65) to solve a₁ and a₂, we get

(67)

Substituting (67) and (65) into (64) and retaining the first three items yield

(68)

Reflux occurs under the condition of , i.e.,

(69)

which is independent of the parameter n.

5.2 Trapping phenomenon

The streamline of flow field in the moving frame can be derived from (48) and (56), plotted in Fig. 7. With the power-law index varying in the range from 0.5 to 2 (three examples are given here), the streamline deflects its axis. Thus, the transverse velocity changes direction at different n.

The closed curve of streamlines represents the trapping zone in the moving frame, denoting that the material in this area transports at the speed of peristaltic wave. The result of first-order perturbation equations makes little difference in the longitudinal transport. Thus, the solution to zeroth-order equations can schematically depict the trapping phenomenon. From (53), three solutions can be obtained using ψ₀ =0,

(70)

(71)

where r₁=0 represents the axial streamline of the tube. If there are non-zero real solutions for ψ₀=0, r₂ and r₃ represent the closed cross-sectional curves.

To obtain the positive solution and avoid the value bigger than the tube radius, the following conditions should be satisfied:

(72)

Solving the equations, we have

(73)

This is the condition under which the trapping phenomenon happens, which coincides with those of Shapiro et al.^[3] for n=1. It is observed that the streamline deflects as the parameter n changes in Fig. 7. Thus, the trapping condition depends on n.

The longitudinal length L and the radial width D are two quantities to measure the trapping zone scale, defined as

(74)

(75)

Figure 8 displays the variations of trapping zone for different parameters. Clearly, the trapping zone is related to not only the tube geometry and flow flux but also the fluid properties. Both the length and width of the trapping zone increase with an increase in θ or ϕ. Furthermore, the trapping zone is more difficult to produce in the shear-thinning fluid than the shear-thickening fluid. There is an effect of n on the onset trapping mean flow rate θ according to Figs. 8(c) and 8(d), and as θ grows, the trapping area tends to be equal. At some extreme conditions, the onset trapping mean flow rate θ eventually decreases to 0, for example, when ϕ=0.7 for Newtonian fluids. Figure 9 shows that the dimension of trapping zone starts with a positive number at θ =0, which coincides with the results of numerical work by Xiao and Damodaran^[38] as the comparison shows.

Based on (73), the trapping onset of θ can be summarized by using ϕ as abscissa and θ as ordinate. Figure 10 shows that the corresponding graphical summary describing the onset trapping mean flow θ is influenced by both the power law index n and the amplitude ratio ϕ.

6 Conclusions

The velocity profile perfectly fits the numerical results for the Newtonian fluids. For the power-law fluid, the properties of the velocity profile are similar with the results of same cases in a two-dimensional channel^{[3, 26]}. Although there is a restriction to the geometry of large wavelength, the perturbation method provides a convenient, reliable, and concise solution to the velocity field and streamlines. Moreover, the effects of limited Reynolds number and particle concentration are concerned in the first-order solution.

The results of the velocity field and streamline give general images to the power-law fluids in peristaltic transport, which differs from the Newtonian fluids denoted as deflection. However, there is no numerical result to identify this phenomenon. Expansion to the streamline flux quantities near the wall using the perturbation method provides a quick way to determine whether the reflux exists.

Finally, the onset condition of trapping is obtained from the mathematical results of streamlines. Two quantities are defined to measure the trapping zone, which provides some possibility to think about how the governing parameters influence the trapping phenomenon. The results of trapping zone by the current work are limited to low Reynolds numbers and large wavelengths, and have a little difference from the prediction by numerical methods, but it is easy to access.