Performance of Graphene and Diamond Nanoparticles on EMHD Peristaltic Flow Model with Entropy Generation Analysis

Diamond and graphene are carbide nanoparticles that have valuable biomedical applications in cancer therapy and drug delivery. The aim of this study is to analyze the couple stress nanofluid (blood–graphene/diamond) flow in an asymmetric channel with the effect of viscous dissipation, electromagnetohydrodynamics (EMHD), Joule heating, velocity slip and convective boundary conditions. The mathematical model is solved analytically under the assumptions of long wavelength and low Reynolds number. The impact of various parameters on velocity, temperature, heat transfer coefficient, pressure gradient, trapping, shear stress, and entropy generation are depicted pictorially. The results obtained indicate that diamond-based nanofluid has higher velocity than grapheme-based nanofluid, Bejan number enhances with increasing Brinkman number through entropy generation, and an increase in the couple stress parameter reduces the bolus size.


INTRODUCTION
Peristalsis is the mechanism of fluid transport by expansion and contraction of muscles due to the propagation of waves along the channel. Peristalsis is the process of propelling and mixing of fluid in an anterograde direction of wave propagation. It has wide range of applications in industry, environmental and bioengineering fields. To be more specific, peristaltic concept is useful in physiological phenomena like blood flow in vessels, chyme motion in the gastrointestinal tract, urine transport from kidney to bladder through the ureter, sperm pumping in ducts, swallowing of food through esophagus, and heartlung machine. Latham [1] in 1966 experimentally introduced the peristaltic pumping mechanism. After initiation of this concept many researchers discussed peristaltic propulsion in various geometries. Tripathi and Beg [2] studied peristaltic motion in a symmetric channel. Reddy and Makinde [3] discussed peristaltic propulsion in an asymmetric channel. Nadeem et al. [4] considered peristaltic flow through eccentric cylinders. Ali et al. [5] theoretically observed peristaltic flow in a curved channel. Akbar and Butt [6] investigated peristaltic propulsion through a radially symmetric plumb duct.
In the last few decades researchers have been focusing on the study of nanofluid due to the vast applications of nanofluids in biomedical sciences like vivo therapy, photodynamic therapy, protein engineering, and drug delivery. Nanofluid means merging of nanoscale particles into conventional fluid. This term was initially introduced by Choi [7]. Generally, nanoparticles are composed of metals, carbides, and oxides. Among nanoparticles, graphene nanoparticles have the advantageous properties like high thermal conductivity, good electrical conduction, and reduced clogging. Graphene was discovered by the experimental work of Novoselov et al. [8] in 2004. Graphene is a carbon-based nanomaterial. It is widely used in biomedicine as an anticancer agent, water purifier, a sensor for blood sugar, blood pressure levels, for prosthesis and dental implants. Feng and Liu [9] discussed various biomedical applications of graphene nanoparticles. Sandeep and Malvandi [10] in-vestigated the flow of graphene nanoparticles suspended with non-Newtonian fluids (Jeffrey, Maxwell and Oldroyd-B fluids) and concluded that their outcomes can be helpful in designing heat exchanger devices. Shit and Mukherjee [11] studied the graphenepolydimethylsiloxane (PDMS) nanofluid flow and revealed that their observations have applications in biomedical engineering and powder technology. Khan et al. [12] theoretically studied the Carreau nanofluid flow consisting of graphene nanoparticles and determined that their results are helpful for thermal conductivity and design of coating processes. Aman et al. [13] defined graphene/water nanofluid as another source of solar energy in thermal engineering. Rashid et al. [14] discussed the heat transfer flow of water/graphene nanofluid with distinct nanoparticle shapes. Wang et al. [15] experimentally explored that the thermal conductivity of graphene-based nanofluid is higher than that of the base fluid water. Diamond nanoparticles belong to carbon-family materials. They were initially found in the 1960s by detonation in the USSR [16]. Diamond nanoparticles are utilized in bio-applications such as anti-bacterial and anti-viral treatments, drug delivery vehicles, and therapeutic agents for diagnostic probes [17]. Sani et al. [18] studied the diamond/graphite-ethylene glycol nanofluid flow and their results are applicable in solar energy. Xie et al. [19] discussed the diamond-based nanofluid flow and concluded that the thermal conductivity of nanofluid increases with the volume fraction of diamond nanoparticles. A few more researchers discussed the diamond-based nanofluid flow [20,21].
Magnetohydrodynamics (MHD) defines the movement of any conducting fluid with an external/induced magnetic field. Magnetic nanofluid has many applications in metallurgy, polymer industry, and medical engineering. MHD nanofluid is of great importance in biomedical area like wound treatment, targeted gene delivery, and magnetic resonance imaging. Akbar et al. [22] discussed the Jeffrey nanofluid flow with the effect of magnetohydrodynamics using the homotopy perturbation method (HPM) and their findings are helpful in nanobiomedicine. Kothandapani and Prakash [23] investigated the Carreau nanofluid flow under a magnetic field using the regular perturbation method and specified that their findings may assist in cancer therapy. Krishna and Chamka [24] investigated the water-Cu/TiO 2 nanofluid flow with the effect of magnetohydrodynamics using perturbation approximation; their outcomes contribute to biomedical applications aimed at de-stroying cancer cells. Mosayebidorcheh and Hatami [25] analyzed an incompressible nanofluid flow with the impact of magnetohydrodynamics by the analytical least square method using Maple mathematical software. Nisar et al. [26] illustrated the magnetic Eyring-Powell nanofluid flow using the NDSolve tool from Mathematica.
Industries and academics alike may benefit the mesomechanical aspects of computational modeling for various established theories known in the field of mechanics. The mesoscopic concept assumes that a continuum is a mixture composed of different components and this concept is widely applicable in many areas like magnetorheological fluids, thermodynamics, and entropy analysis. Research work is underway in this area. For instance, Dong et al. [27] performed numerical simulations on the fluid flow and heat transfer characteristics in a fin channel under the condition of laminar flow. Chen [28] discussed the mathematical modeling of a magnetorheological fluid. Rudyak et al. [29] studied the heat transfer processes in nanofluid flows. Regarding the above literature, this study mathematically investigates the peristaltic propulsion of a couple stress nanofluid consisting of diamond and graphene nanoparticles with the effects of electroosmosis, Joule heating, viscous dissipation, velocity slip and convective boundary conditions. The dimensionless governing equations are solved analytically under the assumptions of low Reynolds number and long wavelength to obtain velocity, temperature, heat transfer coefficient, and streamlines. Numerical discussion is carried out through pictorial representation.

FORMULATION OF THE PROBLEM
Peristaltic propulsion of an unsteady incompressible couple stress nanofluid in an asymmetric channel of width d 1 + d 2 is considered. The flow is assumed to be with different amplitudes and phases but with the same constant speed c. The Cartesian coordinate system is chosen in such a way that the ξ axis is considered along the center line and the η axis is transverse to it. B 0 is the extrinsic magnetic field acting along the η axis; the induced magnetic field is neglected by taking a very small magnetic Reynolds number. The geometry of the flow is described in Fig. 1. The equations for the upper and lower walls are mathematically defined by and c are the wave amplitude of the upper wall, wave amplitude of the lower wall, wave length, wave direction, phase difference, dimensional time and wave speed, respectively.
The governing equations for mass, primary and secondary momentum, and energy are as follows [30][31][32]: Thermophysical properties are considered as [33,34]: The following transformations are introduced to transform from the fixed frame to the wave frame: The nondimensional quantities are given below to nondimensionalize the momentum and energy equations: where δ is the dimensionless wave number, ζ is the nondimensional heat source/sink parameter, θ is the dimensionless temperature, Re is the Reynolds number, Pr is the Prandtl number, Ec is the Eckert number, Br is the Brinkman number, γ is the Joule heating parameter, U HS is the Helmholtz-Smoluchowski velocity, β is the couple stress nanofluid parameter, Rn is the radiation parameter, and M is the Hartmann number.
In this study, we consider The volumetric flow rate is defined between the fixed and wave frame of references as follows: Here,

Entropy Generation Analysis
Entropy generation defines the irreversible nature due to heat transfer and viscosity effects within the fluid and at the solid boundaries. A larger rate of entropy generation in any thermal system reduces the efficiency of the system. The volumetric rate of entropy generation for the nanofluid is given by [36,37] The nondimensional form of entropy generation can be written as (1 Rn Pr) where the temperature ratio and the characteristic entropy generation number are 2 0 f 0 g 2 2 1 , .
The total entropy generation N s is the ratio of actual entropy generation rate to reference volumetric entropy generation. The total entropy generation in Eq. (30) can be written as Here, N hi is the heat transfer irreversibility, N fi is the fluid friction irreversibility, and N mi+Ji+gi is the irreversibility of the combination of magnetic field, Joule heating effect, and heat generation parameter. The Bejan number defines the fraction of heat transfer irreversibility over total irreversibility: In the above expression, the Bejan number varies in the range 0 ≤ Be ≤ 1. Be = 1 is the case of dominating heat transfer irreversibility, Be = 0.5 corresponds to entropy generations due to heat transfer and fluid friction with the magnetic field, Joule heating and the heat source/sink are equal, and Be = 0 is the case of irreversibility dominated by the combined impacts of fluid friction, Joule heating, magnetic field, and heat generation irreversibilities.

RESULTS AND DISCUSSION
In this section, pictorial analysis of the analytical solutions is provided for nanofluid velocity, temperature, heat transfer coefficient, shear stress, streamlines, and entropy generation by variations of various parameters. The thermophysical properties of the base fluid and nanoparticles are specified in Table 1 and the shape factors of nanoparticles are presented in Table 2  the axial velocity decreases with increasing Hartmann number. In the presence of the magnetic field, a force called the Lorentz force arises in the flow domain and causes the retardation of the fluid flow. Figure 2b elucidates that an increase in the velocity slip parameter leads to a decrease in the axial velocity in the core part of the channel, and an opposite trend is observed near the peristaltic walls. A similar behavior is observed in [41]. Figure 2c illustrates that the Helmholtz-Smoluchowski velocity shows different behavior near the lower and upper peristaltic walls: the axial velocity increases near the lower channel and decreases at the upper channel. Figure 2d provides a comparison of the axial velocity between blood/diamond and blood/graphene nanofluid. One can see that the axial velocity of blood/diamond nanofluid exceeds that of blood/graphene nanofluid. Figures 3a-3d are drawn to understand how the temperature of the couple stress nanofluid consisting  of diamond and graphene nanoparticles varies depending on the parameters like thermal radiation Rn, couple stress parameter β, heat generation parameter ζ, and electroosmosis parameter κ. Figure 3a indicates that an increase in the value of thermal radiation reduces the blood-graphene nanofluid temperature due to heat loss. The highest temperature is achieved in the absence of thermal radiation. A similar result was reported in [42]. Figure 3b is plotted to observe the impact of the heat generation parameter on the couple stress nanofluid consisting of diamond nanoparticles. The temperature is found to rise due to friction caused by the enhancement of heat generation parameter. Figure 3c portrays that with increasing electroosmosis parameter the temperature rises of blood-graphene nanofluid due to reduction in electric double layer (EDL). From Figure 3d it is observed that temperature of the couple stress nanofluid consisting of diamond nanoparticles decreases with increasing couple stress parameter. According to Figs. 3a-3d, the couple stress nanofluid with spherical nanoparticles exhibits the highest temperature in the middle part of the channel compared to the nanofluid with cylindrical, platelet-or blade-like nanoparticles. A reverse trend is observed at the peripheral part of the walls.
Total entropy generation with respect to various emerging parameters is visualized through Figs. 4a-4d, and the effect of various parameters on the Bejan number is depicted through Figs. 5a-5d. Figure 4a is drawn to observe the influence of the velocity slip parameter against the total entropy. It is revealed that an increase in the velocity slip parameter α reduces entropy generation throughout the channel, as was also observed in [43]. Figure 4b shows the impact of the Joule heating parameter γ on the total entropy. It indicates that the temperature rises with increasing Joule heating parameter, leading to an increase in the total entropy, as was also reported in [36]. According to Figs. 4c and 4d, the total entropy decreases with increasing Prandtl number Pr and thermal radiation Rn, respectively; a reduction in the total entropy is due to a decrease in the nanofluid temperature. Figures 4a-4d give results for a blood-based nanofluid with blade-shaped nanoparticles, for which the least entropy generation is observed compared to the nanofluid with platelet-like, cylindrical or spherical nanoparticles. Figure 5a is plotted to explore the Bejan number against the Brinkman number Br. It is found that the Bejan number increases with increasing Brinkman number. A similar trend is observed in Figs. 5b-5d, which means that the Bejan number in-      movement. An increase in the Brinkman number leads to an increase in the heat transfer coefficient in blood/graphene nanofluid (Fig. 6a). According to Fig. 6b, the heat transfer coefficient is higher in the absence that in the presence of the Prandtl number. The heat transfer coefficient increases in magnitude with increasing Joule heating parameter (Fig. 6c), but decreases with increasing Helmholtz-Smoluchowski velocity (Fig. 6d). Figures 6a-6d also demonstrate that the couple stress fluid with spherical nanoparticles has the maximum heat transfer coefficient as compared to the fluid with cylindrical, platelet-and blade-like nanoparticles. Figures 7a, 7b and 8a, 8b illustrate the trapping phenomenon of the couple stress nanofluid flow with respect to emerging parameters M and β. The trapping phenomenon defines the composition of the circulation bolus within the fluid by streamlines. The increasing magnetic field M reduces the bolus size in the upper and lower parts of blood/ graphene nanofluid (Figs. 7a and 7b). With increasing couple stress parameter β, the bolus size of blood/ diamond nanofluid decreases (Figs. 8a and 8b).
Figures 9a-9d are plotted to investigate the shear stress at the upper peristaltic wall depending on various parameters. Each plot demonstrates oscillatory behavior with peaks and troughs, caused by peristaltic propulsion. Figure 9a shows that the shear stress increases with increasing velocity slip parameter α, but tends to its lowest value with vanishing velocity slip parameter. Figures 9b and 9c illustrate the shear stress behavior with respect to increasing couple stress parameter β and electroosmosis parameter κ, respectively; in both cases, the shear stress increased. An increase in the Hartmann number leads to a decrease in the shear stress (Fig. 9d). The growing Hartmann number generates the Lorentz force, due to which the shear stress decreases. Figures 10a-10d portray the pressure gradient variation with respect to ξ depending on various parameters. Each plot demonstrates oscillatory behavior due to peristaltic propulsion. In general, in the wider portion of the channel there is no need to apply a larger pressure gradient because the fluid can flow easily, while in the narrow portion of the channel a larger pressure gradient is necessary to preserve constant flow. Figures 10a and 10b illustrate that the pressure gradient increases with increasing wave amplitude b and couple stress parameter β, respectively. An increase in the Hartmann number M reduces the pressure gradient (Figs. 10c), and an opposite behavior is shown in

CONCLUSIONS
A theoretical model was developed to analyze the couple stress nanofluid flow in an asymmetric channel aggravated by peristaltic propulsion under the effect of electroosmosis, magnetic field, viscous dissipation, and slip boundary conditions. The nanofluids considered contain diamond and graphene nanoparticles with four types of geometries. Expressions for the streamlines, axial velocity field, pressure gradient, temperature distribution, and entropy generation are obtained analytically. The important results of this study can be concisely summarized as: The velocity of blood/diamond nanofluid is higher than that of blood/graphene nanofluid.
The temperature of blood/graphene nanofluid decreases with increasing thermal radiation.
Entropy generation increases near the peristaltic walls with increasing heat generation parameter.
Irreversibility due to heat transfer is high near the channel wall with increasing Hartmann and Brinkman numbers.
The heat transfer coefficient of the couple stress nanofluid is high in the case of spherical nanoparticles.
The couple stress parameter reduces the size of the trapped bolus.
The shear stress increases with increasing values of the electroosmosis parameter.