The study of non-Newtonian nanofluid with hall and ion slip effects on peristaltically induced motion in a non-uniform channel

In this study, we considered the unsteady peristaltic motion of a non-Newtonian nanofluid under the influence of a magnetic field and Hall currents. The simultaneous effects of ion slip and chemical reaction were also taken into consideration. The flow problem was suggested on the basis of the continuity, thermal energy, linear momentum, and nanoparticle concentration, which were further reduced with the help of Ohm's law. Mathematical modelling was executed using the lubrication approach. The resulting highly nonlinear partial differential equations were solved semi-analytically using the homotopy perturbation technique. The impacts of all the pertinent parameters were investigated mathematically and graphically. Numerical calculations have been used to calculate the expressions for the pressure increase and friction forces along the whole length of the channel. The results depict that for a relatively large value of the Brownian parameter, the chemical reaction has a dual behaviour on the concentration profile. Moreover, there is a critical point of the magnetic parameter at which the behaviours of the pressure increase and friction forces are reversed for progressive values of the power law index. The present investigation provides a theoretical model that estimates the impact of a wide range of parameters on the characteristics of blood-like fluid flows.


Introduction
During the recent years, the study of peristaltic ow has become an increasing interest for various researchers due to its efficient phenomena for the transport of uids in different biological systems. It is a phenomenon in which a sinusoidal wave arises due to the proportional shrinkage and relaxation of smooth muscles in a human body. In particular, the peristaltic ow is involved in the transport of urine through the kidney to the bladder, transport of cilia, locomotion of spermatozoa (in the male reproductive tract), motion of chyme (in the gastrointestinal tract), motion of ova (in the fallopian tubes), and in the vasomotion of tiny blood vessels. In industry, peristaltic mechanism is very benecial in transporting different biological uids such as sanitary and corrosive uids. For this purpose, many devices, such as heat lung machines, roller pumps, cell separators, and nger pumps, have been introduced in biomedical engineering that follow the fundamentals of peristaltic mechanism. Due to the numerous applications of the peristaltic ow, several researchers investigated the mechanism of peristalsis in different media. For instance, Mekheimer 1 studied the motion of couple stress uid due to the peristaltic waves through a non-uniform channel. Later, Mekheimer 2 extended the previous problem inuenced by the magnetic eld considering blood as a couple stress uid and obtained the exact solutions. The nonlinear peristaltic ow under the inuence of a magnetic eld through a uniform planar conduit was discussed by Hayat et al. 3 Ellahi et al. 4 investigated the peristalsis of three-dimensional motion of a non-Newtonian uid in a rectangular canal. The peristaltic ow of the non-Newtonian Williamson uid with compliant walls was investigated by Ellahi et al. 5 He further analysed the impacts of the wall tension and damping and obtained the series solution with the help of the perturbation technique. Nadeem et al. 6 studied the three-dimensional peristaltic motion of a Jeffrey uid in a duct having exible walls and obtained the exact solutions. Mekheimer et al. 7 studied the inuence of the relaxation time of a Maxwell uid together with the MHD peristaltic transport in a microchannel. Some further similar investigations on this topic can be found in the literature. [8][9][10] In uid dynamics, a new branch has been introduced namely nanouid dynamics, which has many applications in biology, medical science, energetics, and engineering processes. Nanotechnology introduces the creation and usage of numerous substances having the nanoscale dimensions from 1 to 100 nm. Basically, a nanouid (NF) is a uid that is amalgamated by scattering the nanoparticle (NP) in the base uid such as body uids, natural/articial lubes, and water. Choi 11 was the rst who initially concluded that the impact of the nanouid phenomena was to enhance the energy performance. Although the basic concept of NFs was introduced in the 19 th century by a well-known scientist James Clark (a Scottish theoretical physician), later the term nano-uid was introduced officially by Choi. Lee et al. 12 investigated the room-temperature thermal conductivity of NFs as a new class of uids that was organized by dispersing NPs in water and ethylene glycol (EG). The non-Newtonian NF was investigated by Ellahi et al. 13 with Reynolds' model and Vogel's model using the homotopy method. Nanouid particles are made up of metals, oxides, nitrides, or carbides having very small diameters (<100 nm). The base uids can be the EG, lubricants, propylene glycol (PG), biouids, coolants, emulsion, water, or silk broin. Fig. 1 shows the combination of different NPs with base uids. Nanouids are applicable and helpful in the understanding of various phenomena such as the enhancement/minimization of the magnitude of heat transfer systems, minimal clogging, miniaturization of the systems, and microchannel clogging.
Various researchers studied the combined mechanism of peristalsis with NF through different geometrical aspects. For instance, Akbar et al. 14 studied the peristaltic NF ow in an irregular tube. Akbar et al. 15 examined numerically the peristalsis of Williamson uid in an asymmetric channel. Nadeem et al. 16 discussed the NF peristalsis in an eccentric conduit with heat and mass transfer. Ellahi et al. 17 studied theoretically the peristaltic mechanism of Prandtl nanouid through a rectangular duct. Nadeem et al. 18 presented a mathematical formulation for the peristaltic motion of a non-Newtonian uid with NPs. The peristaltic ow of NF having carbon NPs through a permeable channel under the inuence of an induced magnetic eld was investigated by Akbar et al. 19 Few more relevant studies can be found in the literature. [20][21][22] The existing fashion in the applications of MHD is towards the strong magnetic elds to take the effect of electromagnetic force into consideration. Consequently, the Hall and ion slip effects are crucial since they have exceptional inuence on the current density. The Hall effect and ion slip effect have numerous applications, especially if incorporated with heat transfer, such as in Hall accelerators, refrigeration coils, heating elements, MHD accelerators, and power generators. Moreover, the study of the inuence of the magnetic eld along with the Hall and ion slip effects on the blood ow in an artery has been found to be very helpful and applicable in magnetic resonance angiography (MRA). This helps to create the images of arteries to explore the existence of stenosis or any other conditions in the arteries of the brain, abdomen, thorax, and kidneys. Magnetic resonance imaging (MRI) is also involved in other applications that involve pumping of blood, hyperthermia, cancer therapy, and magnetic drug targeting. Since the implementation of these applications provides exclusive capabilities to improve the mechanism of peristalsis uses, researchers have devoted much effort towards studying the peristaltic nano-uid with the magnetic eld in different conduits. For instance, El Koumy et al. 23 studied the peristaltic motion of a Maxwell uid under the inuence of a strong magnetic eld and accordingly the Hall effect through a conduit. Asghar et al. 24 investigated the simultaneous effects of Hall and ion slip along with the ohmic and viscous heating on the peristaltic motion through different ducts. Hayat et al. 25 considered the Hall and ion slip effects on the peristaltic phenomenon of a non-Newtonian Carreau-Yasuda uid model. Abbasi et al. 26 considered the peristaltic motion of a silver-water nanouid with the Hall and ion slip effects. They, in addition, considered the ohmic heating and wall characteristics such as the tension of elasticity and damping phenomenon. More comprehensive treatments about magnetic eld models can be found in the outlined ref. 27-33. Considering the abovementioned discussion, the primary motivation of the present study was to extend our interest in studying the peristaltic motion of a hyperbolic tangent uid with the effects of Hall and ion slip through a non-uniform channel taking the chemical reaction into consideration. To the best of our knowledge, this model has not been investigated in any of the referenced state-of-the-art reviews before. The system of equations describing the problem is formed by following the approach of the long wavelength and creeping ow regime. The resulting governing nonlinear partial differential equations have been solved by means of the HPM (homotopy perturbation method). The impacts of all the emerging parameters have been discussed in details with the help of the graphs.

Mathematical formulation
We considered the peristaltic ow of a blood-like incompressible, hyperbolic tangent, and electrically conducting NF under the effect of an externally applied magnetic eld. A ow through the 2D non-uniform channel is induced due to the propagation of sinusoidal waves along its walls. The hydromagnetic ow of the nanouid is considered unsteady and irrotational. We choose the Cartesian coordinate system in a way thatx-axis is taken along the channel length andỹ-axis is normal to it, as shown in Fig. 2. The geometry of the peristaltic walls can be described by The generalized form of Ohm's law taking the Hall and ion slip effects into consideration can be written as Solving eqn (2), we obtain and where b e ¼ u e s e . The equations of motion governing the ow along with the thermal energy, continuity, and nanoparticle fraction for the blood NF can be written as 34 The stress tensor for hyperbolic tangent uid is dened as In the abovementioned equation, we have considered h N ¼ 0 and G _ g\1: Accordingly, the stress tensor can be rewritten as Dening the dimensionless quantities as We followed the creeping ow proposition such that the half-width of the conduit was taken small as compared to the peristaltic wavelength. We further speculated that the Reynolds number is low. These assumptions are extensively used in many peristalsis analyses. 30,32,35-38 These approximations are considered in many biological tracts such as in the transport of enzymes to the duodenum. Using eqn (12) in eqn (3)-(10), we obtained the reduced system of equations in the following form: with corresponding boundary conditions as where

Method of solution
In this section, we attempted to solve the aforementioned non-linear couple of partial differential equations by means of the HPM. The homotopy for eqn (13)-(15) can be written as The linear operators L 1 , L 2 , and L 3 are suggested in the next forms Moreover, we dened the initial guess for the abovementioned linear operators as Dening the following expansions By substituting eqn (27)À(29) into eqn (18)À(20) and matching the like powers ofq, a linear system of differential equations, along with their corresponding boundary conditions, was obtained. With reference to the scheme of HPM, we deduced the solution asq / 1, and we obtained The solutions of the temperature prole, nanoparticle concentration, and velocity prole can simply be written as The instantaneous volume ow rate can be determined through the expression Thus, we can obtain the expression of the pressure gradient, dp/dx, aer solving the latter equation. Hence, the dimensionless forms of pressure increase, Dp L , and friction force, Df L , by the wall are given by where L is the non-uniform channel length.

Numerical results and discussion
In this part, we have discussed the theoretical signicance of the developed physical expressions that are involved in the problem based on the current study. A Mathematica toolbox has been used to explore the outcomes arising due to the existence of the Brownian parameter N b , thermophoresis parameter N t , chemical reaction parameter g, ion slip parameter b i , Hall parameter b e , basic density Grashof number G rF , thermal Grashof number G rT , power law index n, magnetic parameter M, Weissenberg number We, and the average time ow Q into the ow eld. More specically, we investigated their inuence on the distributions of temperature q, concentration F, and velocity u, as well as the pressure increase Dp L and friction force Df L . In the subsequent gures, the red, blue, and green coloured curves represent the variations of the given variable with the indicated parameter in an ascending order. Same is applicable to the variations of the solid and dashed lines where the solid line indicates a smaller value of the parameter under consideration. We consider that Q(x,t) is the instantaneous volume ow rate, which is cyclic having (x À t) cycle and hence can be written as where Q expresses the average of the time ow over one wave cycle. Fig. 3À6 provide insight into the changes in the behaviour of the temperature and concentration distributions on the nano-uid that occur due to changes in the values of the chemical reaction parameter g, the thermophoresis parameter N t , and the Brownian parameter N b . Fig. 3 examines the dependence of q that is plotted with y for different values of N b and N t . The gure shows that the temperature prole q is semi-parabolic, and there is a considerable increase in q upon increasing both N b and N t . The possible reason is that the Brownian motion causes the nanoparticles to rearrange, forming a blend, which increases the thermal conductivity. Further, eqn (14) and (15) show that the temperature prole is commensurate with the thermophoresis parameter. The latter result may be important in the event that a treatment requires an increase in the temperature of the tissues such as in the case of magnetic hyperthermia treatment where the major aim of hyperthermia is to increase the temperature of malignant tissues above 42 C.
On the other hand, it is shown in Fig. 4 that q is reduced with an increase in g for distinct values of N b . Further, it is shown that as N b increases, the impact of g becomes more signicant. In this case, it must be taken into account that an increase in the Brownian parameter may result in an increase in the chemical reaction of the nanouid temperature prole. Fig. 5 depicts the inuence of N b and N t on the concentration of nanouid, F. It is noticed that N b has a decreasing effect on F for different values for N t , whereas N t shows a quite opposite effect on F for various values of N b . It is further shown that the Brownian parameter increases the impact of the thermophoresis in the nanouid concentration prole. Fig. 6 illustrates the behaviour of F with various values of the chemical reaction g at different values of the Brownian parameter. In this gure, it is shown that for a small value of N b (¼1), the chemical reaction seems to weakly affect F in the narrow part of the channel where y˛[0, 0.48], whereas g tends to reduce the concentration prole aerwards. Conversely, for a larger value of N b (¼1.5), there is an obvious dual behaviour of g on the concentration prole. That is, the concentration is enhanced in the region of y > 1.3, where g enhances the uid density, whereas it is reduced in the region of y < 1.3 due to the reduction in viscosity.
The two-dimensional and three-dimensional behaviour of the velocity prole are displayed in Fig. 7À10 for distinct values of the emerged parameters. Fig. 7 helps to elaborate the effects of the slip parameter, b i , and the Hall parameter, b e , on the velocity distribution. This gure reveals that the ow is accelerated for progressive values of b i till a certain turning point at y ¼ 0.8 of negligible slip effect from which the ow decreases substantially aerwards. Similarly, the impact of the Hall parameter is seen to accelerate the ow till the same point before it begins to lag. Fig. 8 demonstrates the inuence of the basic density Grashof number G rF and thermal Grashof number G rT on the velocity prole. Physical interpretation of the behaviour of G rF suggests that beyond a certain critical point (y ¼ 0.78), the ow of the nanouid decelerates with an increase in G rF . This has been predictable since a reduction in the Grashof number in the narrow part of the channel implies an increase in the viscosity causing deceleration in the velocity prole and vice versa. Conversely, the inuence of G rT is observed to reduce the velocity distribution till y ¼ 0.78 from which the velocity increases markedly aerwards. Fig. 9 illustrates the impact of the power law index n and the magnetic parameter M on the velocity prole. The examination shows that the ow of nanouid decelerates when the value of M increases in the narrow part of the channel where y˛[0, 0.82]. The effect is quite opposite aerwards where the uid ow is seen to be substantially increasing with an increase of M. It is also noticed that in the narrow part of the channel, the magnitude of velocity is higher in the absence of the magnetic parameter. Thus, the nanouid velocity can be reduced in this part by the application of a strong magnetic eld on the ow. Contrariwise, the inuence of n on the velocity distribution is seen to be increasing in the narrow part and decreasing aerwards. Fig. 10 helps to demonstrate the inuence of We and Q on the velocity prole. The inspection of this graph reveals that the ow decelerates for progressive values of We in the narrow This journal is © The Royal Society of Chemistry 2018 part of the channel until y ¼ 0.82, where its effect on u is negligible. Later on, the behaviour of the ow is reversed with an increase in We. However, the velocity prole is seen to be remarkably accelerated with an increment in Q .
To study the inuence of the pertinent parameters on the pressure increase Dp L , Fig. 11À14 have been plotted. Fig. 11 demonstrates the effect of the slip parameter b i and the Hall parameter b e on the pressure increase. It is deduced from this   graph that Dp L is reinforced with escalating both b i and b e . It is further noticed that Dp L is almost unperturbed by the variations in b i and b e in the interval t˛[0.22, 0.3], whereas the pressure increase attains its maximum value subsequently at t ¼ 0.34. The fact that pressure increase is small in some intervals can be interpreted as the ow can facilely pass without imposition of large pressure, whereas to retain the same ux, large pressure is required. Fig. 12 and 13 are shown to investigate the effects of G rF , G rT , M, and n on the pressure increase Dp L . A close look to the graphs reveals that Dp L increases with the progressive values of G rT , whereas it decreases with an increase in G rF and M with a maximum value occurring at t ¼ 0.32. It is also shown that M ¼ 0 causes n to have a decreasing effect on the pressure increase till it reaches a critical value of M (¼ 0.5) where n weakly affects Dp L . Aerwards, the behaviour of Dp L is totally reversed to be increasing with n at a higher value of M. It is recognized that the pressure increase is higher in the absence of M. This phenomenon symbolizes the fact that the pressure can be controlled by suitably applying the magnetic eld on the ow. This is an important factor in the use of magnetic eld in physiology as any abrupt change in the intensity of the applied magnetic eld can cause severe changes in the systolic/diastolic readings of the patient exposed to the magnetic eld. Fig. 14 shows the inuence of We and Q on Dp L . Evidently, We has an increasing effect on Dp L , whereas Q has a decreasing effect on it. Further, the pressure increase attains its maximum value at t ¼ 0.35.      Fig. 16. It is observed that G rF has an increasing effect on the friction force, whereas G rT has a decreasing effect on it. It is also noticed that Df L attains its minimum at t ¼ 0.33. Fig. 17 helps to demonstrate the inuence of the Hartmann number and power law index on Df L . It is concluded that Df L increases with an increase in M. Moreover, it is shown that M ¼ 0 causes n to have a decreasing effect on the friction force till it reaches a critical value of M (¼ 0.5) where n weakly affects Df L . Aerwards, the behaviour of Df L is totally reversed to be increasing with n at a higher value of M. In addition, it is seen that the friction force is lower in the absence of M, and the minimum value of Df L takes place at t ¼ 0.33. Fig. 18 depicts the variations in Df L due to the changes in We and Q. Obviously, the friction force is seen to be enhanced noticeably for the higher values of Q, whereas it decays with an increase in We. It is also shown that Df L attains its minimum at t ¼ 0.33.

Conclusions
In this study, the peristaltic ow of a blood-like non-Newtonian (hyperbolic tangent) NF is investigated in a non-uniform channel to study the mathematical results under an external magnetic eld, nanoparticle concentration, chemical reaction,     Hall current, and ion slip conditions. The governing equations along with the boundary conditions are modelled under the long wavelength assumption. The solutions are obtained analytically using the homotopy perturbation technique, and the physical interpretation of the pertinent parameters is discussed. Diagrammatic sketches are given for the physical expressions with the relevant parameters considered in the ow eld. The primary ndings can be outlined as follows: (i) An increment in the Brownian parameter N b causes an increase in the thermal conductivity q, but causes a decrease in the concentration prole F.
(ii) For a relatively large value of N b , there is a dual behaviour of the chemical reaction g on the concentration prole.
(iii) The effect of the thermophoresis N t is to enhance q and F.
(iv) The chemical reaction g has a decreasing effect on both q and F.
(v) Unlike the effect of the basic density Grashof number G rT , Hartmann number M, and Weissenberg number We, the slip parameter b i , Hall parameter b e , thermal Grashof number G rF , and power law index n serve to boost the velocity distribution markedly before a certain critical point.
(vi) The pressure increase Dp L is enhanced with an increase in b i , b e , We, and G rT , whereas it decreases with an increase in G rF , M, and Q. (vii) In the absence of the magnetic eld, the pressure increase attains the highest value, and the friction force attains the lowest value.
(viii) There is a critical point of M at which the behaviours of the pressure increase and friction force are reversed for the progressive values of n.
(ix) Contrary to the inuence of b i , b e , We, and G rT on the friction force Df L , G rF , M, and Q are seen to enhance Df L prominently.
(x) Setting n ¼ 0, We ¼ 0, M ¼ 0, and G rT ¼ G rF ¼ 0 in our analysis, the intrinsic equations governing the ow of Gupta 39 are recovered.
(xi) Our study agrees with that reported for Newtonian uid by Srivastava and Srivastava 36 in the absence of the magnetic eld, basic and thermal Grashof numbers, and Weissenberg number.
(xii) Upon solving our model in plane or axisymmetric geometries for n ¼ 0, We ¼ 0, M ¼ 0, and G rT ¼ G rF ¼ 0 implies to a consistent physical situation as discussed by Shapiro and Jaffrin. 35 (xiii) Upon adopting the analysis of Mekheimer 2 for the Newtonian uids, our fundamental equations will be in a perfect match with his aer setting n ¼ 0, We ¼ 0, M ¼ 0, and G rT ¼ G rF ¼ 0 in our analysis.
(xiv) Choosing n ¼ 0, M ¼ 0, and g ¼ 0 in the present investigation, our system of equations coincides with that of Abbas et al. 40 for the Newtonian uids.
(xv) If n, b i , and b e vanish in the leading equations that govern the current ow eld, the system is reduced to that of Rashidi et al. 41 if performed in a non-porous medium in the absence of the radiation parameter, heat source/sink, and Weissenberg number. Magnetic eld (in Tesla) We

Conflicts of interest
Weissenberg number Q Volume ow rate (m 3 s À1 ) T, F Temperature (K) and concentration T 0 , T 1 Temperature at the center and at the wall F 0 , F 1 Nanoparticle fraction at the center and at the wall q Perturbation parameter M Hartmann number g Acceleration due to gravity (m s À2 ) D B Brownian diffusion coefficient (m 2 s À1 ) D T Thermophoretic diffusion coefficient (m 2 s À1 ) E Electric eld (V m À1 ) V Fluid velocity (m s À1 ) j Current density (A m 2 ).
Greek characters g Chemical reaction parameter k Nanouid thermal conductivity (W m K À1 ) b Heat source/sink parameter m Viscosity of the uid (N s m À2 ) q Dimensionless temperature prole F Nanoparticle concentration s Electrical conductivity (S m À1 ) _ g Second invariant tensor. d Wavenumber (m À1 ) c p Effective heat capacity of nanoparticle (J/K) n Nanouid kinematic viscosity(m 2 s À1 ) (r) p Nanoparticle mass density (kg m À3 ) r f Fluid density (kg m À3 ) r f 0 Fluid density at the reference temperature (T 0 ) (kg m À3 ) z Volumetric expansion coefficient of the uid (rc) f Heat capacity of uid (J/K) l Wavelength (m)