#### Abstract

A mathematical model is developed to examine the behaviors of a peristalsis flow with nanoparticles in a symmetric channel under the magnetic environment. Here, the nanofluid is electrically conducted through an external magnetic field. Thermal radiation and Joule heating effects are also retained in the present analysis. Under the lubrication approach, the reduced nonlinear systems are obtained. Then, they are solved very efficiently by means of a homotopy analysis method-based package BVPh 2.0. The influences of important physical parameters on the flow behaviors are presented. Analysis of the entropy generation is illustrated. It is found that the Brownian diffusion and the thermophoresis are the two most important nanoparticle slip mechanisms in the Jeffery fluids as well. Besides, the Hartman number, the type of the Jeffery fluid, the Brinkman number, and the thermal radiation parameter play important roles on flow behaviors. Results show that the temperature profile enhanced but the nanoparticles’ volume fraction profiles lowered with increase in the Hartman number. However, using the Jeffery nanofluid induces effect on the velocity distribution that decreases with the increase in the Jeffery fluid parameter. It is also found that the generated total entropy increases with an increase in the Brownian motion parameter but with a decrease in the thermophoresis parameter.

#### 1. Introduction

Peristaltic transport in a tube/channel is a decisive type of flow which plays an important role in many engineering and industrial processes. Practical and potential applications of peristalsis could be found in transport of urine from kidney to bladder, motion of chyme in various sizes of the intestine, passage of cilia, blood circulation in arterioles, movement of egg in uterus, roller and finger pump’s design used for pumping fluids without being contaminated from exposed pumping components, powder technology, and so on. Firstly, Latham [1] and then Shapiro et al. [2] put forward the theoretical and experimental studies on peristalsis of viscous fluids. From then on, some researchers examined various aspects of peristaltic transport with different types of fluid flows including magnetohydrodynamics, heat and mass transfer, and wall properties. Recent studies on peristaltic transport have been done by Brasseur et al. [3], Hayat et al. [4], Kothandapani and Srinivas [5], Abd-Alla et al. [6] Hayat et al. [7], and Ramesh and Devakar [8].

Recently, nanofluids have received much attention owing to their great capability in heat transfer enhancement. Several mathematical models for the description of nanofluid behaviors have been suggested to estimate the behaviors of nanofluids such as the homogeneous flow model [9], the dispersion model [10], and Buongiorno’s model [11]. Buongiorno’s model has gained plausible attention since it explains the slip mechanism between the base fluid and nanoparticles. Nanofluids have been found to have potential applications in various industrial and engineering procedures including hybrid power engines, pharmaceutical processes, nuclear power plants, coolants, lubricants, and petroleum industries such as enhanced oil recovery, drilling, and anticorrosive coating technology. Some nanoparticles such as , , and are helpful for friction reduction in conventional lubrication oils [12]. Moreover, many nanoparticles have been testified to be used to enhance thermal conductivity. Some investigations on nanofluids with thermal behaviors have been performed by Xing et al. [13], Raees et al. [14], Kosmala et al. [15], Xu and Cui [16], and Khan et al. [17].

Mangnetohydrodynamics (MHD) has an important role in cancer therapy, cardiac surgeries to control blood flow, drugs transportation, blood pumping machines, etc. Peristaltic MHD flow can be used to describe the motion of a conducting physiological fluid such as the blood, which is precisely useful in modern drug delivery systems. In the presence of an applied magnetic field, the drug delivery system guides drugs suspended with nanoparticles to the tumor site. The undesired tissues could be destructed by the mechanism of hyperthermia and cryosurgery. Therefore, it is expected that nanofluid MHD flow could be an important replacement for traditional cancer treatment methods in modern drug delivery systems. On the contrary, the Hall effect is of prominent significance in the design of electric transformers, Hall accelerators, refrigeration coils, etc. Various theoretical studies have been carried out towards understanding the transportal characteristics of those MHD flows. For example, Xu and Liao [18] studied the unsteady mangnetohydrodynamic viscous flows of non-Newtonian fluids over an impulsively stretching plate. Muthuraj and Srinivas [19] examined an MHD flow of a micropolar fluid near a vertical porous space. Misra et al. [20] examined the peristaltic transport of a physiological fluid under the effect of a magnetic field in a porous asymmetric channel. Ellahi et al. [21] made an analysis on the peristaltic MHD flow of the Jeffrey fluid in a rectangular duct through a porous medium. Sheikholeslami et al. [22] studied a nanofluid MHD flow and heat transfer with consideration of two-phase model.

Nowadays, heat transfer effects with entropy generation have been explored intensively since the pioneer work by Bejan [23]. Entropy generation associated with various aspects such as friction forces, chemical reactions, and viscosity could result in loss of energy within the thermodynamical system. It has been known that entropy generation plays a vital role in different manufacturing processes including chemical vapor deposition instruments, turbo machinery, combustion, heat exchangers, solar collectors, and electronic cooling devices. Khan and Ali [24] investigated the recent developments in modeling and simulation of entropy generation for dissipative cross material with quartic autocatalysis. Many researchers [25–28] have devoted themselves to investigate entropy generation in various nonlinear flow and heat transfer problems.

The objective of this work is to consider jointly the effects of thermal radiation, nanoparticles, peristalsis, and Ohmic heating on a fully developed convection flow in a symmetric horizontal channel. In the literature, few studies have been performed on analyzing behaviors of peristaltic nanofluid flow, especially, the combined influences of external magnetic field and thermal radiation in the presence of an electrically conducting nanofluid. In our analysis, Buongiorno’s [11] nanofluid model is to be employed to describe nanolfuid’s behavior and the Rosseland approximation [22] is introduced for simplification of governing equations based on the assumptions of long wavelength and low Reynolds number. The resulting governing equations are to be solved analytically by means of a homotopy-based computational package BVPh 2.0. The entropy generation analysis will be performed based on the entropy generation minimization method.

#### 2. Problem Statement

Here, we incorporated Buongiorno’s nanofluid model [11] in the Jeffery fluid for consideration of a two-dimensional thermal peristaltic Jeffery nanofluid flow between two horizontally placed symmetric plates in the presence of a transverse magnetic field . The physical sketch is shown in Figure 1. Here, the Cartesian coordinate system (*x*, *y*) is chosen with the *x*-axis being measured along the channel walls and the *y*-axis being measured in the normal direction to the channel walls. The incoming sinusoidal waves are electrically conducting and propagate parallel to the channel walls with a constant speed *c*. The channel walls are convectively heated with thermal radiation aspect being also accounted for in the present analysis.

With the assumptions that the fluid is incompressible and the flow is laminar and steady, the Navier–Stokes equations describing the conversations of the total mass, momentum, thermal energy, and nanoparticles volume fraction are written bywhere is the flow velocity vector, is the extra stress tensor, is the ion density of current, is the magnetic field, *P* is the pressure, *T* is the temperature, *C* is the nanoparticles volume fraction, *t* is the time, *ρ* is the density of the fluid, *ξ* is the specific heat of the fluid, *k* is the thermal conductivity, is the density of nanoparticles, is the specific heat of nanoparticles, is the coefficient of Brownian motion, is the coefficient of thermophoretic diffusion, is a reference temperature, is the wall heat flux, and *σ* is the electrical conductivity.

The extra stress tensor for a Jeffery fluid is defined, based on the study of Kothandapani and Srinivas [5], aswhere *μ* is the fluid dynamic viscosity, is the ratio of relaxation to retardation time, is the retardation time, and is the shear rate defined by

Note that when in equation (5) is chosen, the current analysis is reduced to that in a viscous fluid.

In our analysis, the symmetric channel walls are assumed to be electrically insulated. In the case that the electrically conducting fluid flows across the channel in presence of a magnetohydrodynamic field, the generalized Ohm’s law with the Hall aspect can be written, based on the study of Hayat et al. [4], aswhere *e* is the electric charge of ions and *n* is the number density of electrons.

The magnetic field is only applied in the *y* direction, i.e., . On the contrary, the flow velocity is two-dimensional, which is given as . Hence, we getwhere is the Hall current parameter.

The additive aspect of radiation in the energy equation is taken along the *y* direction as one-dimensional heat flux, which is defined, based on the Rosseland assumption of Sheikholeslami et al. [22], aswhere is the Stefan–Boltzmann constant whose numerical value is 1.380648 and is the coefficient of Rosseland mean absorption. By assuming that the temperature field inside the liquid is small, which indicates the term is taken as a linear function of temperature, we expand it by Taylor series about , obtaining

Ignoring all higher-order terms of , we obtain

Hence, is reduced to

The geometry of the symmetric channel walls is assumed to be of the following form:where *a* is the mean half width of the channel, *b* is the amplitude of wave, and *λ* is the wavelength.

#### 3. Nondimensional Analysis

With the above assumptions, the governing equations (1)–(4) are written, in Cartesian coordinates, assubject to the boundary conditions where is the heat transfer coefficient, is the mass transfer coefficient, and , and , are the temperatures and concentrations at the center and the upper channel walls, respectively.

Furthermore, to simplify the governing equations (14)–(18), the Galilean transformations are applied to transform the fixed frames to the periodic ones, which are given as

Then, we define the following dimensionless scaling transformations:and we introduce the stream function *ψ*where is the wave number.

Substituting the above transformations (20), (21), and (22) into the coupling nonlinear system (14–18), the continuity equation (14) is automatically satisfied and other governing equations becomewhere is the Reynolds number, *M* is the Hartman number, is the Prandtl number, is the Brinkman number, is the Brownian motion parameter, is the thermophoresis parameter, and is the radiation parameter, which are defined bywhere *ν* is the kinematic viscosity, *α* is the thermal diffusivity, is the Eckert number, and *τ* is the ratio of heat capacity of nanofluid to base fluid, given by

On the contrary, the extra stress tensor components () through equation (5) are expanded as

#### 4. Similarity Reduction

In the limiting case the inertialess flow corresponds to the Poiseuille-like longitudinal velocity profile. In laboratory frame, the pressure gradient is only a function of *x* and *t* since there is no streamline curvature to produce pressure gradient in the transverse direction when . As a result, it is able to employ the assumptions of the long wavelength and low Reynolds number, which indicates and . Note that this theory was applied by Srivastava and Srivastava [29] for the description of transport of chyme in the small intestine of a male. In their pattern, , cm/min, and are prescribed, which shows that the ratio (wavelength) is greater than half width of the intestine. In a similar vein, Lew et al. [30] noticed that the Reynolds number is small as fluid moves in a small intestine. On the contrary, the intrauterine fluid flow due to myometrial contractions in a cavity can also be deemed as a type of peristaltic transport. Eytan and Elad [31] simulated intrauterine fluid transport in a narrow channel surrounded by two fairly parallel walls. The width of their channel is 1–3 mm, which is very small as compared to its length 50 mm. The physical parameters used in the analysis such as wavelength, frequency, velocity, and amplitude of the fluid wall induced by typical wave contractions were found by selected values 10–30 mm, 0.01–0.057 Hz, 0.05–0.2 mm, and 0.5–1.9 mm/s, respectively.

Therefore, we are able to reduce equations (23) and (24) towhere . Differentiating equation (30) with respect to *y* and equation (31) with respect to *x* and then equating them, we obtain

Similarly, equations (25) and (26) are transformed to

The associated boundary conditions of dimensionless forms arewhereand is the amplitude ratio, is the Biot number of convective heat transfer, is the Biot number of convective mass transfer, and is the flow rate in wave frame with *Q* being the absolute flow rate in the laboratory frame. Based on the study of Ramesh and Devakar [8], *F* can be computed by

#### 5. Entropy Generation

The volumetric entropy generation of nanofluid is modeled in fixed frame as modeled in the study of Hayat et al. [26]:

The above equation comprises three parts of physical quantities. One part represents the entropy generation due to heat transfer and thermal radiation effect, and the other parts comprise the combined effect of nanoparticles and the entropy generation due to magnetic field effect. The characteristic entropy generation is defined according to the study of Rashidi et al. [25] as

Thus, the entropy generation in the nondimensional form iswhere is the temperature difference parameter, is the concentration difference parameter, is the ratio of temperature to concentration parameters, and *γ* is defined as follows:

#### 6. Solution Method and Error Analysis

The homotopy analysis method- (HAM-) based package BVPh 2.0 is employed to give accurate analytical approximations to equation (32)–(34). With this approach, the linear operators, the initial guesses, and the appropriate convergence-control parameters are needed to be selected. Since the boundary conditions fall into a finite region [0, h], it is very convenient to use power series functions as the solution expressions. Therefore, the initial guesses can be chosen as

Note that the initial guesses must satisfy all known boundary conditions (35). The linear operators are then chosen, based on the boundary conditions (35), as

The crucial convergence-control parameters in the HAM technique are determined by the minimum variance integral method. In doing so, we prescribe values to all physical parameters and HAM auxiliary functions except the values of the convergence-control parameters , , and . Then, the following error evaluating functions, based on equations (32) and (33), are defined:

Substituting various order results into the above error evaluation functions, they are only the functions of the convergence-control parameters , , and so that we can determine their values from the least errors of , , and , respectively (i.e., to use the minimize command in mathematica to determine them). For example, for , , , , , , , and , those values can properly be chosen, as shown in Table 1.

Then, we are able to check the validity and precision of the solutions. By doing so, we define the absolute residual error function:where *N* is the order of HAM computation. Substituting the various order computational results into the error evaluation function (45), the corresponding errors can be obtained. For example, with those determined convergence-control parameters, in the case of , , , , , , , and , the maximum computational errors are obtained, as shown in Table 2. It is found from the table that all computational errors decrease very quickly and evenly as the computational order increases. This guarantees the precision of the solutions. We also notice that 20th order computations are good enough to keep the accuracy and perform an analysis.

#### 7. Results

In this section, we present a series of analysis about the influences of various emerging parameters on the thermal nanofluid flow behaviors like velocity, temperature, nanoparticle concentration, and entropy generation. We firstly examine the influence of the Hartman number *M* on the velocity profile. Note that the Hartman number is frequently encountered in fluid flows through a magnetic field and represents the ratio of electromagnetic force to the viscous force. It is observed from Figure 2 that the velocity profile decreases as the Hartman number enlarges because here the magnetic force is normal to the flow direction which dominates the viscous effect and as a result velocity of the fluid decreases. The effect of the Hartman number on the temperature profile is shown in Figure 3. It is seen that the increase in the Hartman number is helpful to enhance the temperature profile. Physically, this is due to the fact that the magnetic field retards the body force (they are in opposite direction) so that it abbreviates the motion of the fluid and consequently increases the temperature of the fluid. Figure 4 reveals the variational trend of the nanoparticle volume fraction with the Hartman number *M*. It is seen that the nanoparticle volume fraction profile decreases as *M* increases. In a magnetic field with high strength a small change in its strength could cause a big change in the thermal gradient, which alters the nanoparticles motion in an important manner.

The behaviors of the Jeffery fluid could have a significant effect on the distributions of various profiles. As illustrated in Figure 5, the increase in the Jeffery fluid parameter causes the reduction in the velocity of the fluid. This is due to the enhancement in shear stress as grows, which is imparted into the boundary layer that causes the velocity loss. It is found in Figure 6 that the fluid temperature enhances with the increase in near the upper wall but remains almost unchange near the lower wall. Physically, in the vicinity of the upper channel wall, the loss of flow velocity results in the slowness of the transportal rate of heat transfer, more thermal quantity is retarded in the boundary layer. As a result, the flow temperature enhances accordingly. Note that the convective heat transfer boundary condition on the upper wall causes the temperature variation near this wall to be obviously larger than that near the lower wall owing to more thermal energy being imparted into the nearby area of the upper channel wall. The trend of the variation of the nanoparticle volume fraction with is, on the contrary, the nanoparticle volume fraction decreases as evolves. The latter imparted nanoparticles on the upper wall make the changing trend of nanoparticle volume fraction near the upper wall greater than that near the lower wall, as shown in Figure 7.

It has been known that the Brownian diffusion and the thermophoresis diffusion are dominant factors for slip mechanism of nanofluids. However, it is seen from the reduced momentum equation (33) that the stream function is not associated with the two parameters. That means that their effect on the flow field could not be measured by this model. We only consider their influence on temperature and nanoparticle volume fraction distributions. In Figure 8, the increase in causes the enhancement in the temperature profile is shown. A similar trend is observed with change in the temperature profile against the Brownian diffusion coefficient , as shown in Figure 9. This means that the thermal motion caused by the Brownian diffusion and thermophoresis diffusion in the flow increase, which results in an increase in the temperature profiles of the flow. On the contrary, it is observed that the variational trends of nanoparticle volume fraction trends are different with the variation of the Brownian diffusion parameter or the thermophoresis diffusion parameter. As shown in Figure 10, the nanoparticle volume fraction profile increases with the increase in the Brownian diffusion parameter. This happens because more nanoparticles are involved as the Brownian motion parameter enlarges, while the nanoparticle volume fraction profile decreases with the increase in the thermophoresis parameter, as illustrated in Figure 11. This is due to the higher temperature gradient due to the fact that the increase in thermophoresis parameter forces the nanoparticles to move elsewhere.

The entropy generation analysis for various values of and is demonstrated in Figures 12 and 13. It is known from these figures that the entropy generation exhibits totally opposite variational trends with or evolving. These trends are similar periodic owing to the influence of periodic peristalsis. Distribution of versus the Brinkman number is shown in Figure 14. It is observed that with the increase in , the entropy generation significantly increases accordingly. The effect of the Jeffery fluid parameter on entropy generation is plotted in Figure 15. It is noticed that decreases as increases. This happens because is a ratio of relaxation to retardation time and retardation phenomena and generally dominates the entropy generation; thus, entropy decreases for growing . Figure 16 elucidates the increasing behavior of entropy generation versus variation of . Increase in thermal radiation parameter results in the decrease of the mean absorption parameter, which refers to the less energy absorption by the fluid, resulting in an increase of entropy. Figure 17 presents a contour of the behavior of the entropy generation with variation in the Hartman number. It is observed that grows as *M* evolves, with the increasing values of . From this figure, we notice that entropy generation enhances from the center toward the walls of the channel and reaches its peak value. This is explained in Figure 3 that an increase in the Hartman number has a corresponding increase in fluid temperature because the applied magnetic field clustered fluid particles together thereby enhancing viscous dissipation. Thus, the temperature rises leading to a rise in entropy generation. Figure 18 shows the behavior of entropy generation versus several values of the Brinkman number with the variation of . Physically, the larger Brinkman number results in the increase of the viscous dissipation effect because of production of energy which enhances the entropy.

#### 8. Conclusion

The peristaltic motion of a non-Newtonian nanofluid flow in a symmetric channel in the presence of the Joule heating and the thermal radiation effect under a magnetic environment have been examined in detail. The governing flow analysis is established based on the lubrication approach. The highly accurate solution is obtained by means of a novel HAM-based package BVPh 2.0 and is based on the set of reduced nonlinear systems. Physical features of pertinent parameters on distributions of various physical fields have been graphically presented and discussed. Analysis of the entropy generation has been performed with different physical parameters. It is found that the Brownian diffusion and the thermophoresis are the two most important nanoparticle slip mechanisms in the Jeffery fluids as well. Besides, the Hartman number, the type of the Jeffery fluid, the Brinkman number, and the thermal radiation parameter also have great effects on flow behaviors.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant no. 11872241).