1 Introduction

The transportation of the bio-fluids from one place to another place caused by the propulsive movement of biological systems having smooth muscle tubes is known as peristalsis. The mechanism can be seen in the flow of urine from kidney to bladder through ureter,passage of chyme through the gastrointestinal tract,movement of spermatozoa in the cervical canal,embryo motion in non-pregnant uterus,bile movement in a bile duct,transport of lymph in lymphatic vessels,circulation of blood in small blood vessels,arterioles,and venules in capillaries. It also works in many important technological applications such as blood pumps in hurt-lung machine and finger and roller pumps for pumping of corrosive and toxic materials. Peristaltic flow has attracted the attention of many researchers and scientists due to its occurrence in natural phenomena and applications in biomechanical systems. Latham ^[1] was the first who explored the mechanism of peristalsis through experiments. The theoretical results obtained by Shapiro ^[2] were found to be in good agreement with the experimental results given by Latham ^[1] . Later on,many studies on this topic have been added in the literature under different situations and configurations ^{[3, 4, 5, 6, 7, 8, 9, 10, 11]} .

The heat transfer analysis is important to obtain the information about the blood flow rate through the initial thermal conditions and the thermal clearance rate. The blood flow can be estimated through the dilation technique. The thermal clearance in this process is watched when heat is either injected or generated locally. The role of bioheat transfer is frequently used in cryosurgery,laser therapy,and cancer tumor treatment causing hypothermia and destroying undesirable tissues. The study of heat transfer in peristaltic flows is important in some biomedical processes like oxygenation and hemodialysis. Moreover,this phenomenon helps in the transportation of the sanitary fluids,blood pumps in heart lung machine,and the transportation of corrosive materials where the fluid contact with machinery is prohibited. The normal blood temperature of a man is about 37 ℃ . When it increases to 41 ℃ C or above,irretrievable damage will take place in the plasma proteins. F ocusing on such facts,Ogulu ^[12] discussed the effect of heat generation on low Reynolds number fluid and mass transport in a single lymphatic blood vessel with uniform magnetic field. Mekheimer and Abd-Elmaboud ^[13] performed the heat transfer analysis on the peristaltic transport of electrically conducting Newtonian fluid in a vertical annulus. Vajravelu et al. ^[14] discussed the peristalsis with heat transfer by considering the viscous fluid in a vertical porous annulus. Srinivas and Kothandapani ^[15] dealt with peristalsis and heat transfer. Nadeem and Akbar ^[16] considered the effects of variable viscosity on heat transfer analysis of peristaltic flow of magnetohydrodynamic (MHD) Newtonian fluid. Radhakrishnamacharya and Srinivasulu ^[17] studied the influence of wall properties on peristaltic transport with heat transfer. Hayat et al. ^{[18, 19, 20]} and Abd-Elmaboud et al. ^[21] analyzed the peristaltic mechanism with heat transfer.

In all the above mentioned works,Hall and ion slip effects were neglected in applying the Ohm’s law as they have no remarkable effect for a weak magnetic field. However,the current trend in the application of magnetohydrodynamics is towards a strong magnetic field so that the effect of the electromagnetic force is noticeable. Therefore,the Hall and ion slip currents are important and they have remarkable effects on the magnitude,on the direction of the current density,and consequently on the magnetic force term ^[22] . The effects due to the Hall current become significant when the Hall parameter (the ratio of electron-cyclotron frequency and the electron-atom-collision frequency) is high. This situation occurs as a result of high magnetic field or low collision frequency. The previous related works in these directions are given by Hayat et al. ^[23] ,Abo-Eldahab et al. ^{[24, 25]} ,Haroun ^[26] ,and Mekheimer et al. ^[27] . The effects of Hall and ion slip currents with heat transfer are likely to be important in many situations as well as in engineering applications in areas like Hall accelerators,power generators,MHD accelerators,refrigeration coils,electric transformers,and heating elements. The study of the effect of a magnetic field and the Hall currents on the blood flow through an artery with a mild stenosis may be helpful in understanding the magnetic resonance angiography,which is one of the radiological investigations done for atherosclerosis ^[28] .

The above literature review indicates that no attempt has been made to explain the combined effects of Hall and ion slip currents and Ohmic heating on the peristaltic transport of electrically conducting viscous fluid with heat transfer. In fact,the Joule heating effect arises from the applied electric field and fluid electrical resistivity. Studying the heat transfer with Joule heating effects is very important for many applications in food industries and biomedical engineering. One of the important applications of electroosmosis is the fluid delivery in lab-on-a-chip devices, where one deals with thermally labile samples. The temperature rise due to the Joule heating can result in low column separation efficiency,reduction of analysis resolution,and even loss of injected samples. In addition,the viscous dissipation effects in the energy equation cannot be neglected. Such effects have an important role in the dynamics of fluids with strongly temperature-dependent viscosity because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local temperature increase near the tube walls with a consequent decrease in the viscosity which may dramatically change the temperature and velocity profiles. The basic theme behind this theoretical work is to fill the gap in these directions. Therefore,the present attempt deals with such facts for peristaltic flow in an asymmetric channel. The modeled problems are first solved,and then the obtained solutions are analyzed carefully. 2 Problem description

Consider the peristaltic flow of an incompressible and electrically conducting viscous fluid in a two-dimensional asymmetric channel with the uniform width d ₁ + d ₂ . The channel length is assumed to be infinite along the X-axis,whereas the width of the channel bounded by two walls H₁ and H₂ is parallel to the Y -axis and hence perpendicular to the length. Two sinusoidal waves propagate along the upper and lower walls with a constant speed c,which cause the fluid to propel in the direction of wave propagation. It is pertinent to mention that there is no flow in the channel prior and the imposition of sinusoidal waves along the channel walls can introduce the disturbance in the fluid which consequently drives the flow inside the channel. The shapes of the channel walls in a laboratory reference frame is defined through the following expressions:

where H₁ and H₂ are the shapes of the upper and lower walls,respectively. The wall shapes are functions of the wave amplitudes b ₁ and b ₂ ,the wave length λ,the time t,the channel widths associated with the upper and lower halves of the channel d ₁ and d ₂ and the phase difference of two waves φ which varies in the interval 0 φ π. The channel’s asymmetry is produced due to the different amplitudes and phases of the waves on the upper and lower walls. The values of φ = 0 and φ = π retrieve the symmetric shape of the channel. When φ = 0,two waves become out of phase,and when φ = π,the two waves are inphase along the channel walls. Moreover, the quantities b ₁ ,b ₂ ,d ₁ ,d ₂ ,and φ satisfy the following inequality: Furthermore,the channel walls are assumed to be non-conducting,and two constant but different temperatures T₁ and T₂ are maintained at the upper and lower walls,respectively. It is further assumed that the temperature at the upper wall is less than that at the lower wall,i.e., T₁ < T₂ . In addition,the fluid is conducted under the application of a uniform magnetic field with the magnetic flux density vector B=(0,0,B₀ ). Since we are also interested to analyze the effects of Hall and ion slip on the peristaltic flow,the values of necessary magnetic Reynolds number for the case of partially ionized fluid is considered ^[29] . To maintain the mathematical simplicity for this study,we ignore the effects through the change in the magnetic field due to the change in the electric field. 2.1 Governing equations

The conservation of mass and the balance of momentum and energy are given by

In the above equations,d/dt is the sum of local and convective time derivatives,P is the pressure,V is the fluid velocity vector,ρ is the constant density,µ is the dynamic viscosity, J × B is the Lorentz force vector,T is the fluid temperature,c _p is the specific heat,κ is the thermal conductivity,τ is the Cauchy stress tensor,τ ·L is the viscous dissipation term,and (J · J ) is the Joule heating term.

The generalized Ohm’s law with Hall and ion slip effects is given by ^[24]

where J is the current density vector,σ is the fluid electrical conductivity,ω_e is the cyclotron frequency,t_e is the electron collision time,and β _i is the ion slip parameter. Using the Maxwell equations and solving E_q. (7) in J yield where U and V are the axial and transverse components of the fluid velocity ,respectively, i and j denote the unit vectors in the X- and Y -directions,respectively,β_e is the Hall parameter expressed by β _e = ω_et_e ,and α_e = 1 + β _i β _e. 2.2 Transformation and non-dimensionalization

To transform our flow analysis from the laboratory reference frame to the wave reference frame,we use the standard Galilean transformations given by

where p and u,v are the pressure and the velocity components in the wave reference frame, respectively. Substituting these transformations in E_qs. (4)-(6) yields The dimensionless parameters for our problem are given by where δ,Re,P r,Ec,Br,and Ha represent the wave number,the Reynolds number,the Prandtl number,the Eckert number,the Brinkman number,and the Hartman number,respectively.

Invoke these dimensionless parameters and introduce the stream function ψ ^* by writing the velocity components as

such that the continuity equation can be identically satisfied. Then,E_qs. (12)-(14) can be reduced to The above equations are subject to long wavelength assumption,i.e.,λ d ₁,which means that δ → 0. Moreover,the asterisks are omitted for simplicity.

The boundary conditions in the wave reference frame are expressed as follows:

where

φ also satisfies The dimensionless flux F in the wave reference frame and averaged over time is given by which is related to the time averaged flow rate Q in the laboratory reference frame through the following equation: 3 Solution of problem

Equation (18) indicates that p is a function of x only. Operating on E_q. (17) leads to

Equations (19) and (25) will be satisfied if where The constants of integration A_s (s = 1,2,· · · ,6) involved in E_qs. (26)-(28) can be obtained by E_qs. (20) and (21). They are given as follows:

The pressure rise per wavelength and the frictional forces at h _s (s = 1,2) can be computed as where dp/dx is the axial pressure gradient,which is given by The heat transfer coefficient at the upper wall is 4 Graphical results and discussion

In this section,we will discuss the salient features of the peristaltic flow and the heat transfer under the influence of the Hall parameter β _e,the ion slip parameter β _i ,the Hartman number Ha,and the Brinkman number B_r. The complicated integrals in the expressions of pressure rise and frictional forces are computed numerically by using built-in routine NIntegrate in MATHEMATICA-8.

The variation of pressure rise ∆P_λ over one cycle of the wave is shown in Fig. 1. It can be clearly seen that the pumping rate in the retrograde pumping region (∆P_λ > 0,Q < 0) decreases when β _e and β _i increase while increases when Ha increases. This situation remains the same in the peristaltic pumping region (∆P_λ > 0,Q > 0) up to the flow rate Q = 0.21. The situation is reverse when Q exceeds 0.21. In the copumping region (∆P_λ < 0,Q > 0), when β _e and Ha increase,∆P_λ is disturbed in an opposite direction to that observed in the retrograde pumping region (∆P_λ > 0,Q < 0). Further,the maximum pressure,∆P_λ when Q = 0,is less in magnitude in the presence of Hall current and ion slip. The value of the flow rate Q when ∆P_λ = 0 is known as free pumping flux and denoted by Q₀.

Table 1 is prepared to have a better look on Q₀ for various values of β _e ,β _i ,and Ha. From Table 1 ,we can observe that Q₀ increases when β _e and β _i increase whereas decreases when Ha increases. The effects of β _e and β _ion pumping in all pumping regions become more apparent for large fixed Ha. However,this observation for Q₀ becomes true when either β _e or β _i varies and Ha is fixed and small.

The variations in the frictional forces F_λs (s = 1,2) at the upper and lower walls,respectively,can be seen in Fig. 2. The left-hand panels explain the variation of F_λ1 at the upper wall y = h ₁(x),while the right-hand panels illustrate the variation of F_λ2 at the lower wall y = h ₂(x). Figure 2(a) describes that F_λ1 increases when β _e increases whilst decreases when Ha increases. This situation becomes reverse after the flow rate is appropriately chosen,i.e.,Q = −0.24. The variations of F_λ2 with different β _e and Ha are given in Fig. 2(b). It can be seen from Fig. 2(b) that increasing β _e increases F_λ2 for Q < −0.11 whereas decreases F_λ2 for Q > −0.11. Comparing Fig. 2(a) with Fig. 2(b),one can see that F_λ1 is less in magnitude when compared with Fλ2. From Figs. 2(c) and 2(d),it is noted that the impact of β _i on F_λs is similar to that of β _e on F_λs . Moreover,the effects of β _e and β _i on both F_λ1 and F_λ2 are more significant for higher values of Ha. Figure 2 also illustrates that there exist critical values,i.e.,Q_h1 and Q_h2 ,corresponding to F_λ1 = 0 and F_λ2 = 0,respectively. The critical values define that F_λs (s = 1,2) resist the flow when Q < Q_hs (s = 1,2) and assist the flow when Q > Q_hs (s = 1,2).

The critical values Q_h1 and Q_h2 for different values of β _e,β _i ,and Ha are presented in Table 2. We observe from Table 2 that Q_hs (s = 1,2) increase when β _e and β _i increase. However, when Ha increases,Q_hs (s = 1,2) decrease. Moreover,the critical values of Q_hs (s = 1,2) vary at a sharp rate with different β _e and β _i for small Ha.

The effects of β _e ,β _i ,and Ha on the axial pressure gradient dp/dx are displayed in Fig. 3. The graphs are plotted within one wavelength x ∈ [−0.5,0.5]. The panels on the left-hand side depict the variation of dp/dx corresponding to the symmetric channel,and the right-hand panels represent the variation of dp/dx corresponding to the asymmetric channel. Figure 3(a) illustrates that the pressure gradient dp/dx is symmetric about x = 0 when the channel is symmetric. For all values of Ha,increasing β _e decreases the pressure gradient dp/dx,and the effect becomes more significant when Ha is large. Moreover,the pressure gradient dp/dx increases as a result of increasing Ha. It can be seen from Fig. 3(b) that the effects of β _e and Ha on dp/dx remain the same for the asymmetric channel case. However,the pressure gradient dp/dx is not symmetric when the channel is asymmetric. Increasing β _i disturbs dp/dx in a similar way to that of β _e(see Figs. 3(c) and 3(d)).

The profiles of the dimensionless velocity u for different values of β _e ,β _i ,and Ha are shown in Fig. 4. It is evident that the amplitude of u decreases when Ha increases. This is due to the fact that the magnetic body force (Lorentz force) increases in response to an increase in Ha. Such force offers resistance to the fluid flow. The behavior of β_e on u depending on Ha can be explained through Figs. 4(a) and 4( b). It can be seen from Figs. 4(a) and 4(b) that an increase in β_e gives an increase in the amplitude of u. In fact,an increase in β_e decreases the effective conductivity (σ/((1 + β_i β_e) ² + β_e ² )),and consequently decreases the magnetic damping force. Therefore,u increases. Figures 4(a) and 4(b) also illustrate that the impact of β_e on u is more prominent against the higher values of the magnetic field. This means that when Ha is small,an increase in the value of β_e slightly increases the amplitude of u. This is due to the gradual decrease in the magnetic damping force which is proportional to β_e. Moreover, increasing β_e with larger Ha greatly decreases the effective conductivity which rapidly reduces the magnetic damping force. Therefore,u increases. Figures 4(c) and 4(d) ensure that the effects of the ion slip parameter β_i on u are similar to those of the Hall parameter β_e . The comparison of left-hand panels with right-hand panels reveals that the magnitude of u in the asymmetric channel case is less than that in the symmetric channel case.

The distributions of the temperature profile θ are displayed in Fig. 5. As shown in the figure,Fig. 5(a) indicates the combined effects of β_e and Ha on θ. It is noted that θ increases when Ha increases. In fact,the Lorentz force increases when Ha increases,which opposes the motion of the fluid. Therefore,some energy is converted into the heat due to the frictional forces. Contrarily,an increase in β_e decreases θ for all values of Ha. This is because of the result of decreasing the Joule dissipation which is proportional to (1/((1 + β_i β_e) ² + β_e ² )). The temperature profile θ increases when Br increases (see Fig. 5(b)). When Br increases,the heat energy is stored in the fluid due to the frictional or drag forces,and thus θ increases. Moreover, the variation in θ with β_e is more significant for larger values of Ha and Br,which is due to the change in the viscous and Joule dissipations. Figures 5(c) and 5(d) explain the effects of β_i on θ,which depend on Ha and Br,respectively. Here,one can see that the effects of β_i on θ are similar to the effects of β_e on θ in a qualitative manner.

The effects of β_e ,β_i ,Ha,and Br on the heat transfer coefficient Z at the upper wall y = h ₁(x) are displayed in Fig. 6. In Fig. 6,the plots of Z are sketched for one wavelength x ∈ [−0.5,0.5]. Clearly,for all Ha and Br,Z decreases in magnitude in response to the increases in β_e and β_i . On the other hand,Z increases in magnitude when either Ha or Br increases. Figure 6 also indicates that the variations of Z with varying β_e and β_i become more prominent for large values of Ha and Br.

Figures 7-9 show the effects of β_e ,β_i ,and Ha on trapping. In all these figures,(a),(c),and (e) correspond to the symmetric channel,and (b),(d),and (f ) correspond to the asymmetric channel where the asymmetry of the channel is considered only through the phase difference. From Fig. 7,we can see that when β_e increases,the size of the trapped bolus increases. Moreover, the bolus is symmetric about the center line for the symmetric channel. The trapped bolus tends to shift in the opposite direction of the wave propagation when the channel is asymmetric. This can be attributed to the different phase angles. Similar effects of β_i on the size of the trapped bolus are seen through Fig. 8. However,opposite effects on trapping are observed in Fig. 9 where an increase in Ha yields a decrease in the size of the trapped bolus.

5 Concluding remarks

A heat transfer analysis is performed for the peristaltic transport of an electrically conducting fluid under the influence of the applied uniform magnetic field. The generalized Ohm’s law is used to consider the Hall and ion slip effects. Viscous and Ohmic heating effects are also taken into account. The exact solutions of the governing problems are constructed with the wave reference frame. The main findings obtained from the present article may be summarized as follows:

(i) The maximum pressure,which is required to produce zero flow rate,decreases when β_e and β_i increase while increases when Ha increases.

(ii) When β_e and β_i increase and Ha decreases,the frictional forces F_λs (s = 1,2) increase first,and then decrease after a certain value of the flow rate Q.

(iii) All Q0,Q h1,and Qh2 increase when β_e and β_i are increased or Ha is decreased.

(iv) The magnitude of the pressure gradient dp/dx decreases when β_e and β_i increase while increases when Ha increases.

(v) For small Ha,the amplitude of the velocity u slightly increases when β_e increases. However,for large Ha,the amplitude of the velocity u greatly increases when β_e increases. Similar effects on u are noticed when the ion slip parameter β_i increases.

(vi) The temperature θ decreases when β_e and β_i increase,and increases when Ha and Br increase. These observations are also true for the absolute value of the heat transfer coefficient Z .

(vii) For large Ha,the effects of β_e and β_i are more pronounced on the pressure rise ∆Pλ , the frictional forces Fλs (s = 1,2),the pressure gradient dp/dx,the velocity u,the temperature θ,and the heat transfer coefficient Z at the upper wall. However,the critical value varies at the sharp rate for small Ha.

(viii) The size of the trapped bolus increases when β_e and β_i increase,and decreases when Ha increases.