1 Introduction

There is no doubt that non-Newtonian fluids exist widely in physiological and engineering processes. Explicit examples of such processes include bubble columns,biofluids,hematology, polymer solutions,food industries,micro/nano fluidic,chemical engineering,mineral industry, building,confectionery industries,etc. The equations governing the flow of viscous fluids are known as the Navier-Stokes equations. These equations are capable of describing the viscosity effects successfully. However,there are materials,especially in industry and engineering,where fluids are not viscous but non-Newtonian. Several non-Newtonian fluids are viscoelastic. Such fluids have elastic properties in addition to viscous effects. The inadequacy of the classical Navier-Stokes model describing viscoelastic fluids has led to the development of several con- stitutive equations. Generally,the non-Newtonian equations with diverse characteristics are classified into three categories,namely,the differential,rate,and integral types. The prob- lems arising in the flows of non-Newtonian fluids pose a challenge to applied mathematicians, modelers,engineers,and computer simulationists. These stem from the fact that the viscoelas- ticity of the fluids introduces some extra terms in the momentum equations which include a particular term that has a derivative whose order is greater than the number of available bound- ary/initial conditions. In fact,the orders of the differential equations in such problems increase because of the fluid viscoelasticity. However,there is no corresponding increase in the number of the boundary/initial conditions. The detailed discussion of such issues is given in Refs.^{[1, 2, 3, 4, 5]}. Hence,one requires additional boundary/initial conditions for the computation of the unique solution in the flows of non-Newtonian fluids. This topic is of current interest,and many recent researchers are engaged in the analysis of the flows of non-Newtonian fluids through abrupt changes in geometry and different non-Newtonian fluid models. We can mention some useful studies in this direction through Refs.^{[6, 7, 8, 9, 10, 11, 12, 13, 14, 15]}.

The peristaltic transport of fluids widely occurs in physiologicaland engineering applications. Its involvement is quite obvious in roller and finger pumps,passage of urine from kidney to bladder,chyme movement in gastrointestinal tract,swallowing food through esophagus,sanitary fluid transport,egg movement in the female fallopian tube,blood circulation in vessels,etc. Latham ^[16] firstly conducted an experimental investigation for the peristaltic motion of a viscous fluid. Shapiro et al. ^[17] initially carried out the analysis of the peristaltic flow of a viscous fluid in an asymmetric channel and tube when the long wavelength and low Reynolds number approximations held. Since then,extensive studies have been made to discuss the peristalsis in different geometries and assumptions of long wavelength,low Reynolds number,small wave number,small amplitude ratio,etc. However,it is noted that much attention in such studies is given to the peristaltic motion of fluids in symmetric channels and tubes though these studies are further narrowed down when non-Newtonian fluids are considered. The importance of the peristaltic motion in an asymmetric channel was firstly pointed out by Eytan and Elad ^[18] . Such consideration is of great value regarding an application in the intra uterine fluid flow in a non-pregnant uterus. Afterwards,some attempts have been made for the peristaltic transport in an asymmetric channel. One may look at the relevant investigations in this direction in Refs.^{[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]}.

The mechanism of peristalsis in the presence of heat transfer has relevance in various in- dustrial processes. The thermodynamic aspect of blood is significant in the processes like oxygenation and hemodialysis when the blood is drawn out of the body. Moreover,the inter- action of peristalsis is important in the metabolic processes involved in food digestion,heat conduction in tissues,and heat transfer due to the perfusion of arterialvenus blood. Moti- vated by such facts,the object of the present study is to examine the peristaltic flow of the Johnson-Segalman fluid in an asymmetric channel with convective boundary conditions. The Johnson-Segalman fluid is considered in the sense that it can explain the “spurt” (slip at the wall) phenomenon. Mathematical modeling is done,and the solution analysis is constructed by using long wavelength and low Reynolds number approximations. The solutions for small Weissenberg number are constructed. Important flow quantities of interest are examined with respect to the involved pertinent parameters.

2 Flow equations

The constitutive equations in the Johnson-Segalman fluid are ^[31]

in which σ is the Cauchy stress tensor,p is the pressure,I is the unit tensor,τ is the extra stress tensor,µ is the dynamic viscosity,η is the elastic shear modulus,m is the relaxation time, and e is the slip parameter. Here,D and W are the symmetric and skew-symmetric parts of the velocity gradient (L = gradV ),respectively,which are given by where

the model (2) reduces to the Maxwell model. Further,when m = 0,the model recovers viscous fluids.

The equations of continuity,momentum,and energy are

where the vector with an over bar refers to a dimensional quantity,ρ is the fluid density,t is the time,V is the velocity,T is the fluid temperature,c p is the specific heat,k is the thermal conductivity of the material,and

3 Problem statement

We investigate the flow of peristaltic transport of an incompressible Johnson-Segalman fluid in an asymmetric channel (see Fig. 1). The X- and Y -axes are selected along and perpendicular to the channel walls,respectively.

The flow is caused by the waves propagating along the channel walls. The channel walls for heat transfer satisfy the convective conditions. The geometry of the walls is put into the forms as follows:

where c is the wave speed,a₁ and a₂ are the wave amplitudes,λ is the wavelength,d₁ + d₂ is the width of the asymmetric channel,the phase difference φ varies in the range of 0 ≤ φ ≤ π (φ = 0 corresponds to the symmetric channel with waves out of the phase and φ = π corresponds to the waves in the phase). Moreover,a₁ ,a₂ ,d₁ ,d₂ ,and φ satisfy the condition

The exchanges of heat with ambient at the walls through Newton’s cooling law are given by

where η₁ /η₀ and T₁ /T₀ are the heat transfer coefficient and the temperature at the upper/lower channel wall,respectively.

The velocity V for two-dimensional flow is

If (x,y) and (u,v) are the coordinates and velocity components in the wave frame (x,y),then we have the following transformations:

Employ the above transformations. Introduce the following dimensionless variables:

Introduce the stream function ψ(x,y) by

Then,Eq.(4) is satisfied identically. From Eqs.(5) and (6),we can obtain the equations in terms of the stream function ψ as follows:

In the above equations,the dimensionless wave number δ,the Reynolds number Re,the Weis- senberg number We,the Prandtl number Pr,and the Eckert number Ec are defined as follows:

Equations (9) and (10) give where Bi₁ and Bi₂ are the Biot numbers expressed by

Let δ ＜＜ 1 and Re → 0 ^[17] . Then,Eqs.(15) and (16) can be reduced to

The above equation indicates that p is independent of y. Eliminating the pressure p from Eqs.(24) and (25),we arrive at From Eq.(17),we can obtain where the Brinkman number Br is Then,Eqs.(18)−(20) become From Eqs.(29)−(31),we can write Then,Eq.(24) takes the form where

In the fixed frame,the instantaneous volume flow rate is given by The volume flow rate in the wave frame is From Eqs.(12),(34),and (35),we have The time-mean flow over a period τ is defined as Substituting Eq.(36) into Eq.(37) and then integrating the resulting expression yield Define Θ and F as the dimensionless time-mean flows in the laboratory and wave frames, respectively,by Then,Eq.(38) becomes with The dimensionless forms of h i (i = 1,2) are where The conditions for the dimensionless stream function in the wave frame are

4 Solution scheme

The resulting equation (26) is highly non-linear. It seems difficult to find the exact solution of this equation. Therefore,our interest now is to obtain the series solution for small We 2 . Hence,we expand ψ,θ,S_xy ,p,and F in the following forms:

Substituting Eq.(45) into Eqs.(22),(23),(26),(27),(32),(33),(43),and (44),and then col- lecting the terms of like powers of We 2 ,we can write the following systems.

4.1 Zeroth-order system 4.2 First-order system

The next two subsections will develop the solutions of the zeroth- and first-order systems.

4.3 Zeroth-order solution

The solutions of Eqs.(46) and (47) satisfying the boundary conditions (49) and (50) are

where the quantities appearing in the above equations are given in Appendix A.

The expressions of the longitudinal velocity and pressure gradient are

The non-dimensional pressure rise per wavelength (∆P λ 0 ) can be put into the form It is worth mentioning that the solution expressions at this order correspond to the Newtonian fluid.

4.4 First-order solution

Substituting Eq.(56) into Eqs.(51)−(53),solving the resulting equations,and then applying the corresponding boundary conditions,we can get the solutions for ψ₁ ,u₁ ,dp₁ /dx,and θ 1 in the forms presented as follows:

where the involved quantities are shown in Appendix A.

The pressure rise per wavelength at the order ∆P_λ1 is

The perturbation expressions of ψ,θ,∆P_λ ,and dp/dx upto O(We²)¹ are

5 Results and discussion 5.1 Pumping characteristics

In this subsection,the developed series solutions in the flow of the Johnson-Segalman fluid model are illustrated in Fig. 2,Fig. 3,Fig. 4,Fig. 5,Fig. 6,Fig. 7,Fig. 8,Fig. 9,Fig. 10,Fig. 11,Fig. 12,Fig. 13,Fig. 14,Fig. 15.

In order to see the effects of the emerging parameters We,γ,e,φ,a,b,and d on the axial pressure gradient dp/dx,we plot Fig. 2,Fig. 3,Fig. 4,Fig. 5,Fig. 6,Fig. 7,Fig. 8. From these figures,we can see that the axial pressure gradient decreases with the increase in the Weissenberg number We,the viscosity ratio parameter γ,the phase difference φ,and the channel width d whereas increases with the increases in the slip parameter e,the upper wave amplitude a,and the lower wave amplitude b. Figure 2 indicates that the pressure gradient is larger for the Newtonian fluid (We = 0) in comparison with that of the Johnson-Segalman fluid (0 ＜ We ＜ 1).

Fig. 9,Fig. 10,Fig. 11,Fig. 12,Fig. 13,Fig. 14,Fig. 15 are prepared for the pressure rise ∆p λ per wavelength against the volume flow rate Θ for different values of We,γ,e,φ,d,b,and a,respectively. It is observed from these figures that the pressure rise per wavelength ∆p λ increases with the increases in We, γ,φ,and d while decreases with the increases in e,b,and a. Moreover,the pressure rise within one wavelength ∆p λ gives larger values for small Θ (peristaltic pumping occurs in the region −1 ＜ Θ ＜ 1) and smaller values for large Θ (augmented pumping occurs in the region 1 ＜ Θ ＜ 3) (see Figs.9,10,12,and 13),whereas it has an opposite behavior in Figs.11,14, and 15.

5.2 Velocity behavior

This subsection addresses the behavior of the velocity profile through Fig. 16,Fig. 17,Fig. 18. It is observed that with the increases in We and γ,the velocity profile decreases in the narrow part of the channel while increases in the wider part of the channel (see Fig. 16,Fig. 17). An opposite behavior is noted in Fig. 18 for different e.

5.3 Temperature profile

Here,the effects of various emerging parameters on the temperature profile θ are discussed. Fig. 19 and Fig. 23 show a decrease in the fluid temperature due to the increase in We or Bi₁ (due to the convective condition at the upper wall). It is worth pointing out that the fluid temperature θ generally increases when the viscosity ratio parameter γ,the slip parameter e, the Brinkman number Br,and the Biot number Bi₂ (due to the convective condition at the lower wall) increase (see Figs.20−22 and Fig. 24).

5.4 Trapping

In general,the shape of streamlines is similar to that of a boundary wall in the wave frame as the walls are stationary. However,some of the streamlines split and enclose a bolus under certain conditions,and this bolus moves as a whole with the wave. This phenomenon is known as trapping. Fig. 25 and Fig. 26 show the streamlines for three different values of We and γ. The streamlines near the channel walls nearly strictly follow the wall waves,which are mainly generated by the relative movement of the walls. In the center region,a bolus is formed. These figures reveal that with the increases in We and γ,the size of the trapping bolus decreases. Figure 27 gives the sketch of the streamlines for various e. It is noticed that the size of the trapping bolus increases with an increase in the slip parameter e.

6 Closing remarks

In this paper,the peristaltic motion of the Johnson-Segalman fluid in an asymmetric channel with convective conditions is investigated. Based on the behaviors of the velocity and temper- ature profiles,pumping and trapping phenomena are discussed. It is noticed that a combined increase in the Biot number at the lower wall Bi₂ ,the viscosity ratio parameter γ,the slip parameter e,and the Brinkman number Br enhances the thermal stability of the flow. Further, an increase in the Weissenberg number We or the Biot number at the upper channel wall Bi₁ decreases the temperature profile.

Acknowledgements We are grateful to the reviewers for their constructive suggestions.

Appendix A

Here,we provide the quantities appearing in the performed analysis.