1 Introduction

Peristalsis is a form of fluid transport induced by a progressive wave of area contraction or expansion along the walls of a distensible duct containing some material. In physiology, a peristaltic transport is a radially symmetrical contraction and relaxation of muscles propagating in a wave down a tube in an anterograde and sometimes retrograde direction. In many digestive canals, e.g., human gastrointestinal tracts, the smooth muscle tissue contracts in sequence to produce a peristaltic wave that propels/mixes a bolus of food along the canal. Likewise, peristaltic pumping occurs in many practical applications involving the biomechanical systems such as rollers, finger pumps, and heart-lung machines. The problem has been introduced, and a literature survey of relevant works on peristaltic motion has been provided. Latham^[1] was the first to introduce the mechanism of peristaltic motion. Theoretical and experimental studies have been conducted to understand the different peristaltic models in various situations. A summary of the investigations has been presented with details by Rath^[2], Srivastava and Srivastava^[3-6], and Srivastava and Saxena^[7].

The modelling of two-phase flows has a tremendous variety of engineering and scientific applications, e.g., pollution dispersion in the atmosphere, fluidization processes, and aerosol deposition in spray medication. The sedimentation of particles in a liquid is of interest in many chemical engineering processes, where erythrocyte sedimentation has become a standard clinical test. A number of research works on this topic, with and without peristalsis, have been investigated^[5-40]. The first attempt of studying particulate suspension through peristalsis was conducted by Hung and Brown^[8]. They initiated experimental studies on the particle transport in the two-dimensional (2D) vertical channels with various geometries through peristalsis. Srivastava and Srivastava^[5] studied the peristaltic pumping of a particle-fluid composition in a 2D channel, and obtained a perturbation solution. Mekheimer et al.^[9] studied the peristaltic pumping of the particle-fluid suspension in a planar channel. El-Misery et al.^[10] studied the peristaltic motion of an incompressible generalized Newtonian fluid in a planar channel.

Most liquids are usually considered to be incompressible. However, Aarts and Ooms^[11] showed that compressibility had a great impact on liquid flows. They investigated the influence of ultrasonic radiation on the flow of a compressible liquid through a porous medium in an axisymmetric cylindrical pore, and found that the effect of the ultrasonic radiation on the liquid was like peristaltic pumping. Antanovskii and Ramkissoon^[12] studied the peristaltic motion of a compressible fluid in a finite pipe subjected to a time-dependent pressure drop with the lubrication theory approach, and showed that such flow was applicable in the peristalsis of blood due to the heart pumping along with the synchronic expansion to the constricted vessels. Tsiklauri and Beresenev^[13] studied the non-Newtonian effects in the peristaltic flow of a compressible Maxwell fluid. Castulik^[14] studied the Boltzmann equation for a compressible boundary layer. Kawashita^[15] investigated the global solutions of Cauchy problems for compressible Navier-Stokes equations. Eldesoky and Mousa^[16] studied the peristaltic transport of a Newtonian and non-Newtonian compressible fluid in a cylindrical tube with the perturbation technique. Eldesoky^[17] studied the influence of the slip condition on the peristaltic transport of a compressible Maxwell fluid through porous media in a tube, and constructed the analytical solutions of the mean axial velocity and net flow rate of the liquid in a closed form with a perturbation analysis in the latter investigation.

A short review of the pertinent literature, in which the momentum equations, along with the characteristic response of a liquid to compression, are solved by means of the perturbation technique, is further presented. Mekheimer and Abdel-Wahab^[18] studied the net annulus flow of a compressible fluid with peristalsis. Mekheimer et al.^[19] investigated the effects of magnetic field and space porosity on the peristaltic motion of a compressible non-Newtonian fluid in a microchannel. Felderhof^[20] analyzed the dissipation in a compressible fluid caused by peristaltic pumping, and showed that it had a similar resonance. The calculations were performed in terms of the amplitude of the surface wave. Elshehawey et al.^[21] studied the axisymmetric peristaltic flow of a viscous compressible liquid through a tapered pore, and showed that the compressibility of the liquid has a strong effect on the induced net flow. Hayat et al.^[22] investigated the peristaltic mechanism of a Jeffrey fluid in a circular tube under the effects of the rheological properties and compressibility of the fluid. El-Shehawy et al.^[23] investigated the effects of the slip boundary conditions and the compressibility on the dynamics of fluids in porous media by studying the flow of Newtonian and non-Newtonian fluids in a cylindrical pore.

To the best of the authors' knowledge, there has not been any attempt to study the peristaltic transport of a compressible liquid with suspended particles in a planar channel. Therefore, the aim of this article is to investigate the peristaltic model of a mixture of small, spherical, and relatively identical rigid particles in a compressible Newtonian viscous fluid between the infinite parallel walls on which traveling waves of sinusoidal form are imposed. A perturbation technique is employed to analyze the problem in terms of a small amplitude ratio. The current model can be reduced to present the application of a 2D peristaltic transport of the solid-fluid composition investigated by Srivastava and Srivastava^[5]. The theoretical study of the two-phase mixture is very useful in understanding a number of diverse physical problems concerned with the biofluid transports through peristaltic contractions in human organs.

2 Formulation of problem

Consider a 2D infinite channel with the mean width 2d filled with a mixture of small spherical rigid particles in a compressible Newtonian viscous liquid (see Fig. 1). The walls of the channel are flexible, on which the non-conducting travelling waves of the sinusoidal form with small amplitude are imposed.

Bearing in mind that the liquid is compressible, we can express the equations governing the conservation of mass and the linear momentum for both the fluid and the particle phase as follows^{[5, 25-26]}.

(i) Fluid phase

(1)

(2)

(3)

where C is the volume fraction density of the particles which is considered as a constant (see Ref. [5]). Thus, Eq. (3) becomes

(4)

(ii) Particulate phase

(5)

(6)

(7)

where x and y are the Cartesian coordinates with x along the direction of the wave propagation and y along the direction normal to the mean position of the channel walls. u_f and v_f denote the liquid phase velocities. u_p and v_p denote the particulate phase velocities. ρ_f and ρ_p are the actual densities of the materials constituting the liquid and the particulate phase, respectively. (1-C)ρ_f and Cρ_p are, respectively, the densities of the liquid phase and the particulate phase. p denotes the pressure. _μI is the particle liquid mixture viscosity, i.e., the effective viscosity of the suspension. S is the drag coefficient of the interaction for the force exerted by one phase on the other.

The characteristic response of the liquid to a compression is described by the following constitutive equation^[27]:

(8)

where k^* is the compressibility of the liquid. The solution of Eq. (8) for the density as a function of pressure is given by

(9)

where ρ₀ is a constant density at the reference pressure.

The concentration of the particles is considered so small that the field interaction between the particles may be neglected. Thus, the diffusivity terms, which can model the particle-particle effects due to the Brownian motion, are neglected. It is worth mentioning here that the effect of the Brownian motion has been considered by others including Batchelor^[28-29].

The expression for the drag coefficient for the present problem is selected as follows:

(10)

where μ₀ is the fluid viscosity, and a₀ is the radius of the particle. Relation (10) represents the classical Stokes' drag for the small-particle Reynolds number modified to account for the finite particulate fractional volume through the function λ'(C)^[30].

For the present problem, the following empirical relation for the viscosity of the suspension, suggested by Charm and Kurland^[31], is going to be used in the foregoing analysis:

(11a)

(11b)

where T is the absolute temperature. The viscosity of the suspension expressed by this formula is found to be reasonably accurate up to C=0.6. Charm and Kurland^[31] tested Eq. (11) with a cone and plate viscometer, and found it to be in agreement within ten percent in the case of blood suspension.

The nonslip and impermeability conditions constitute the boundary that must be satisfied by the liquid on the walls. The walls of the channel are assumed to be flexible but extensible with a travelling sinusoidal wave, and the displacement in the channel walls is in the transverse direction only. Hence, the boundary conditions are

(12)

The transverse displacement η of the wall is given by

(13)

where a is the amplitude, λ is the wavelength, and c is the wave speed.

We introduce the following dimensionless variables based on c and d:

(14a)

and the dimensionless parameters as follows:

(14b)

where Re is the suspension Reynolds number, χ is the Compressibility number, α is the wave number, ε is the amplitude ratio, and M and N are the suspension parameters.

Thus, Eqs. (l), (2), (4)-(7), (9), (12), and (13) now become (after dropping the primes)

(15)

(16)

(17)

(18)

(19)

where

(20)

(21)

The boundary conditions become

(22)

3 Method of solution

To illustrate the nature of the solution, we shall consider the important case of no flow in absence of the peristaltic wave. Assuming the amplitude ratio ε of the wave to be small, we obtain the solution for the governing equations as a power series in terms of ε by expanding p, u_f, v_f, u_p, v_p, and ρ_f as follows:

(23)

(24)

(25)

(26)

(27)

(28)

Substituting Eqs. (23)-(28) into Eqs. (15)-(20) and (22), and collecting the terms of like powers of ε, we obtain two sets, for the first two powers of ε, of coupled linear differential equations with their corresponding boundary conditions in u_f1, u_p1, v_f1, v_p1, ρ_f1, p1, u_f2, u_p2, v_f2, v_p2, ρ_f2, and p₂.

Thus, the effect of the particles on the fluid velocity profile is to cause an increase in the viscosity, i.e., the fluid viscosity μ₀ is replaced by the suspension viscosity (see Eq. (11a)). Therefore, for a given pressure difference, less fluid will flow through the channel.

We seek the solution of the governing equations as follows^{[11, 13]}:

(29a)

(29b)

(29c)

(29d)

(29e)

(29f)

and

(30a)

(30b)

(30c)

(30d)

(30e)

(30f)

where the bar denotes a complex conjugate for the given variable.

The latter choice of the solution is motivated by the fact that the peristaltic flow is essentially a nonlinear (second-order) effect^[11] and adding a non-oscillatory term in the first-order gives only a trivial solution. Thus, we can add some non-oscillatory terms, such as U_f20(y), V_f20(y), U_p20(y), V_p20(y), P₂₀(y), and D₂₀(y), which cannot be cancelled in the solution after the time averaging over the period, only in the second and higher orders.

Substituting Eqs. (29) and (30) into the differential equations and their corresponding boundary conditions leads to two sets of differential equations, i.e., the first-order and the second-order. The first-order terms are

(31)

(32)

(33)

(34)

(35)

(36)

The second-order terms are

(37)

(38)

(39)

(40)

(41)

(42)

(43)

Thus, we obtain a set of differential equations together with the corresponding boundary conditions, which are sufficient to determine the solution of the problem up to the second-order in ε. Now, our main concern is to find out the solution of the differential equations for U_{(f, p)1}, V_{(f, p)1}, and P₁.

Following the solution procedure introduced by Mekheimer and Abd-Elmaboud^[38], we repeat the same analysis, and obtain the master equations for the velocity and pressure. We omit the lengthy calculations here. The obtained first-order solution to Eqs. (31)-(35) subject to the boundary condition (36) is

(44)

(45)

(46)

(47)

(48)

where

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

We notice that ν=α when χ=0, i.e., for incompressible liquids.

Next, in the expansion of U_f2, U_p2, V_f2, V_p2, P₂, and D₂, we are only concerned with the terms U_f20(y), U_p20(y), V_f20(y), V_p20(y), P₂₀(y), and D₂₀(y) since our aim is to determine the mean flow only. Thus, the solution of the coupled differential Eqs. (37)-(42) subjected to the boundary conditions (43) is given by

(57)

(58)

(59)

(60)

(61)

where

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

The mean time average velocities may now be written as

(70)

(71)

We notice that the results of Srivastava and Srivastava^[5] can be recovered if we put χ=0 in the present problem, i.e., for incompressible liquids. In addition, if the fluid is both incompressible and particle free (C=0), the results of the present problems reduce to those found by Fung and Yih^[24].

4 Results and discussion

A close look at Eq. (70) reveals that the net axial velocity of the fluid phase is dominated by the constant W and the perturbation term Re(E(y) -E(1)), which varies across the channel. The constant W, which initially arises from the nonslip condition of the net axial velocity on the wall, is due to the value of U_f20 at the boundary, and is related to the net axial velocity at the boundaries of the channel (at y=±1) by u_f=ε²W. This shows that the nonslip boundary condition applies to the wavy wall, not to the mean position of the wall. It may be reminded that the corresponding W does not appear in the particulate phase net axial velocity when the particulate phase velocity at the walls is unspecified.

For the sake of comparison, we define the mean-velocity perturbation function Z(y) in accordance with Fung and Yih^[24] and Srivastava and Srivastava^[5], i.e.,

(72)

which gives the net time axial velocity of the fluid phase as follows:

(73)

The calculations disclose that the results of Fung and Yih^[24] can be obtained by taking χ=0 and C=0 in our problem. Moreover, choosing χ=0 in our investigation implies a well-agreed physical situation as obtained by Srivastava and Srivastava^[5].

The effects of the parameters under the consideration on the net axial velocity are shown in Figs. 2-7. The effects of the particle concentration C on the variation of the net axial velocity versus y are depicted in Figs. 2 and 3. We observe from Fig. 2 that, at a low value of Re, increases with increasing C. We further notice from Figs. 2 and 3 that, increasing Re causes to increase with an increase in C until C=0.4, afterwards, decreases with an increase in C for C > 0.4.

The effects of the compressibility parameter χ on the variation of the net axial velocity versus y are depicted in Figs. 4 and 5. It is noticed that χ has a decreasing effect on for Re=1.0. Figure 5 elucidates that increases with increasing χ at a higher value of Re (=10.0). The effects of the Reynolds number Re on the variation of the net axial velocity versus y are depicted in Figs. 6 and 7. It is observed that increases with an increase in Re for different values of χ.

5 Miscellaneous applications

Next, we return to the dimensional flow problem described in Section 2. The dimensional net axial velocity of the fluid phase V_fx is equal to the dimensionless net axial velocity of the fluid phase multiplied by the factor of the wave speed c (see Eq. (73)). The properties of the blood are given by ρ_f=1 066 kg· m^-3 and μ=4×10^-3 N·s·m^-2 with the compressibility k=5 × 10^-10 m²·N^-1 at 1.013×10⁵ Pa^[11]. The constant particle concentration C possesses values accurate up to C=0.6 as suggested by Charm and Kurland^[31].

We calculate the net axial velocity V_fx, according to the relation (73), based on the aforementioned real values. Thus, we have ε=1 × 10^-4, both and increase with increasing c, and α increases with increasing f while decreases with increasing c as stated by the relation , where f is the wave frequency.

We suggest studying some cases of blood flows in human body.

Case Ⅰ: We consider a very small artery, where the dimensional net axial velocity V_fx is calculated and plotted versus y for various values of the compressibility number χ (=40.4×10^-5, 0.6 × 1^0-5, 0.8 × 10^-5, and 1.0 × 10^-5). We take the wavelength λ=2.0 cm, the half mean width d=1.05 mm, the wave amplitude a₀=10^-4 mm, and the concentration C=0.3. We observe from Fig. 8 that, the net axial velocity V_fx does not affected by χ. It is also noticed from Fig. 9 that V_fx increases with an increase in C.

Case Ⅱ: We consider the left main coronary artery (as a relatively small vessel), whose diameter and wavelength range from 2.0 mm to 5.5 mm (mean 4 mm) and from 2.5 cm to 13 cm, respectively^[32]. The dimensional net axial velocity V_fx is mathematically determined and plotted versus y for various values of the compressibility number χ (=2.0^-4, 2.5^-4, 3.0^-4, and 3.5^-4). The other parameters are taken as 12.5 cm, 2.0 mm, 10^-4 mm, and 0.3 for the wavelength λ, the half mean width d, the wave amplitude a₀, and the concentration C, respectively. It is observed from Fig. 10 that χ does not have a significant effect on the net axial velocity V_fx. However, it is noticed from Fig. 11 that V_fx increases with an increase in the particle concentration C.

Case Ⅲ: We consider the right coronary artery, whose diameter and wavelength range from 1.5 mm to 5.5 mm (mean 3.2 mm) and from 2.0 cm to 10.5 cm, respectively^[32]. The dimensional net axial velocity V_fx is calculated and plotted versus y for various values of the compressibility number χ (=1.5 × 10^-4, 2.0 × 10^-4, 2.5 × 10^-4, and 3.0 × 10^-4). The other parameters are chosen as 10.0 cm, 1.6 mm, 10^-4 mm, and 0.3 for λ, d, a₀, and C, respectively. We observe from Fig. 12 that χ does not have a significant impact on V_fx. In addition, we observe from Fig. 13 that V_fx increases with an increase in the particle concentration C.

We notice from the previous study through the arteries of the human body that the net axial velocity does not affected by χ since the compressibility number has been taken to be very small. Therefore, we study the flow case in industrial engineering applications^{[33, 11]}. We consider the ultrasonic radiation influence on the flow of oil through a reservoir. In order to analyze the flow field, we assume that the deformation of the channel walls is caused by the excess pressure waves generated by an acoustic source. The source is placed in the well at a distance h from the oil reservoir. The considered oil properties are the compressibility k, the density ρ₀ at the reference pressure ρ₀, and the viscosity μ. By the radiation from a simple source in the radial direction, the excess pressure Δp generated in the oil reads

(74)

where is the sound speed in the oil^[33], and P is the average power output generated at the acoustic source.

The displacement a of the channel wall due to the excess pressure is given by

(75)

where G is the shear modulus of the porous medium.

The amplitude ratio ε calculated by the combination of Eqs. (74) and (75) is given by

(76)

The wave speed c on the channel wall depends on the compressibility k of the liquid as well as on the distensibility D of the channel^[33], i.e.,

(77)

where the distensibility D is given by D=1/G.

Following the example of Aarts and Ooms^[11], let the power output generated at the acoustic source be P=10 kW and the source be placed at a distance h=0.05 m from the reservoir. For the oil properties, we choose ρ_Oil=800 kg·m^-3 and k=0.7×10^-9 m²·N^-1. For the sandstone, the shear modulus G is approximately 0.5 × 10⁹ N·m^-2, and the distensibility is given by D=2 × 10^-9 m²·N^-1. Consequently, c₀=1 336.3 m·s^-1. Substituting these values in Eq. (74), we obtain the excess pressure generated by the acoustic source as Δp=8.25 × 10⁵ N·m^-2. As a result, the amplitude ratio of Eq. (76) is going to be ε=0.825 × 10^-3. Then, from Eq. (77), we can obtain the wave speed of the travelling wave, i.e., c=680.414 m·s^-1. As a result, the dimensionless parameter χ (=kρ₀c²=c₂/c_o²) becomes equal to 0.26. Let the frequency of the acoustic source f be 20 kHz. Then, the wave length is λ=c/f=0.034 m. Finally, let the channels of the reservoir have the mean width d=0.5 × 10^-4 m and the oil have the viscosity μ=5 × 10^-3 N·s·m^-2. Then, the dimensionless parameters α (=2πd/λ) and will possess the values 9.23 × 10^-3 and 5 443, respectively.

The dimensional net axial velocity V_fx is now plotted versus y for various values of the particle concentration C (=0.0, 0.2, 0.4, and 0.59). We observe from Fig. 14 that C has an increasing effect on the net axial velocity V_fx.

Last, the dimensional net axial velocity is plotted versus y for various values of the power output of the generator P (=10.0, 15.0, and 20.0 kW). From Fig. 15, we can see that the net axial velocity V_fx increases with the power output of P.

6 Conclusions

The peristaltic transport of a macroscopic particle-fluid suspension fluid model has been studied and discussed for different cases of blood flows in human body. The study sets forth the engineering applications for the two-phase compressible fluid flows in a planar channel. A perturbation solution is obtained, which satisfies the momentum equations with small amplitude ratios. The study enables one to observe the effects of the particle concentration, compressibility, Reynolds number, and power output of a generator on the flow characteristics. The expression for the dimensional net axial velocity of the fluid phase is obtained and presented graphically. The main findings can be summarized as follows:

(i) In real dimensional biological problems, the net axial velocity is not affected by the compressibility number.

(ii) The net axial velocity increases with the particle concentration C.

(iii) For a considerably large compressibility, the net axial velocity increases with the power output generated at the acoustic source in industrial engineering applications.

(iv) For incompressible liquids, the results of Srivastava and Srivastava^[5] can be recovered.

(v) The results of Fung and Yih^[24] can be recovered if the compressibility number, together with the fluid particles, vanishes.

Acknowledgements S. I. ABDELSALAM thanks the Binational Fulbright Commission in Egypt and the Council for International Exchange of Scholars in U. S. A. for the honor of the Fulbright Egyptian Scholar Award for the year (2015-2016). In addition, the first two authors declare that they contributed to this work equally.