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1 Introduction

In 1966, Eringen [12] introduced the theory of the micropolar fluid model and later extended it into a thermo-micropolar fluid. The non-Newtonian micropolar fluid is characterized by microstructures, rigid and randomly focused particles on microrotation of fluid components. Such fluid is taken into account through macroscopic views such as polymeric fluids, liquid crystals, colloidal suspensions, animal and human blood. Blood is a nonhomogeneous fluid that streams inside the human artery system and delivers the cells with nutrients due to the blood’s microstructure, and its flow inside a human carotid model can be taken as micropolar fluid. Eringen have established the micropolar fluid model for viscoplastic and viscoelastic medium containing through a classical continuum and thermodynamics. He formulated a much complex structure for simulating effective microrheological properties and areas of applications in various industrial and engineering fields like chemical engineering, arterial blood flows, synovial fluid, biofluids, semicircular canal fluids, gastric liquids, slurry technologies. Nowadays, a micropolar theory has been successfully performed in various branches in the medical field [5, 7, 13, 20, 25]. Sheikholeslami et al. [27] discussed the effects of substantial parameters like microrotation and angular velocity, Reynolds number, Peclet number then solved laminar flow of steady micropolar fluid on porous walls of two-dimensional channel. Rashad et al. [24] probed the heat transfer and convective flow of micropolar nanofluids on a cylinder in a soaked penetrable medium. Kabeir et al. [10] investigated the hydromagnetic flow of micropolar ferrofluid with thermal radiation and partial slip effects. Mahdy et al. [17] scrutinized the convective flow of a micropolar hybrid nanofluid with the presence of internal heat generation, magnetic field, and thermal radiation. Rashad et al. [23] studied the coupled heat and mass transfer and the influence of chemical reaction through a combined convectional micropolar fluid. Bhattacharyya et al. [3] analytically investigated the heat transfer on the flow of carbon nanotubes between two expandable coaxial rotating disks subject to the Cattaneo–Christov heat flux model by employing optimal homotopy analysis method. Motivated by the aforesaid studies, it is noticed that there is a gap related to the survey of the pulsating flow of micropolar nanofluid into a channel with the presence of applied magnetic field, Joule heating, viscous dissipation. So that, in this analysis, we made an attempt to study the simultaneous impacts of viscous dissipation and Joule heating on pulsating magnetohydrodynamic (MHD) flow of a micropolar nanofluid with the influence of heat source/sink and thermal radiation.

The investigation of pulsating flow in a channel turns out to be the aim of scientific exploration due to its significance on biological demands, which is corresponding to hemodynamics, and the several researchers have attracted due to their technological systems like circulatory systems, gaseous diffusion, transpiration cooling, and respiratory system [2, 6, 18, 19, 21, 33]. The pulsatile flow, which is the rate of fluid flow that stimulates by a periodic oscillation or pressure gradient and the superimposed periodical time-varying component, is composed into a steady flow component. The mathematical analysis is divided into two sections, one dealing with higher-order outcomes for the microinertia, microstretch, and micropressure fields, as well as the other with solutions for the velocity and microrotation fields. The pulsatile flow of microstretch fluids in circular tubes is studied as a result of a sinusoidally changing pressure gradient. Venkatesan and Reddy [31] analyzed the effects of pulsatile flow of blood-alumina nanofluid in a channel with the existence of viscous dissipation and Joule heating. Bitla and Iyengar [4] examined the pulsating flow of noncompressible micropolar fluid in the middle of porous walls with an angled magnetic field by adopting the perturbation method. Srinivas et al. [28] investigated the chemical and Soret impacts on the hydromagnetic pulsatile flow of Casson fluid into the permeable channel.

In the present days, the study of nanofluid is greatly essential due to its significant implementations in electronic, optical, and biomedical fields. Nanofluids are a comparatively current category of heat transfer that has fascinated much observation of researchers from previous generations around the globe [22,30]. The section of nanofluids was first established by Choi [9] in 1995. Nanofluid is a fluid that contains nanosized (less than 100 nm) solid particles in the base fluid. Many scientists have been observed extraordinary thermal characteristics in nanofluids and developed a new model. The good compilation of issued papers on nanofluids can be discovered in the book by Choi [8]. Selimefendigil et al. [26] performed the numerical study of forced convective nanofluid flow in a bifurcating channel with the existence of a uniform magnetic field by adopting the Galerkin weighted residual finite element method. Hashmi et al. [14] analytically discussed the problem of MHD squeezing flow of a nanofluid between two disks employing homotopy analysis approach.Vijayalakshmi et al. [32] investigated the hydromagnetic pulsating flow of nanofluids between parallel walls, and the authors treated water as base fluid, and silver, alumina, titanium dioxide, and copper oxide are indicated as nanoparticles. Hatami et al. [15] pointed out the features of gold-blood nanofluid flow through a hollow vessel in the existence of a magnetic field. Elgazery [11] presented the flow of a blood-based nonNewtonian nanofluid with alumina and gold nanoparticles via the non-Darcian penetrable medium. Reddy et al. [29] highlighted the thermal properties of Au/Cu-blood nanofluid flow through a permeable channel with stretching or shrinking walls for its heat transfer properties. Abo-Dahab et al. [1] interrogated the hydromagnetic Casson nanofluid flow in a permeable medium with the transversal magnetized field and suction/injection effects. Kumar et al. [16] exhibited the pulsatile flow characteristics of Casson nanofluid into an allowable channel with help of the perturbation method.

Propelled by the above literature survey, it is noticed that no attempt related to the survey of the pulsating flow of micropolar nanofluid into a channel with the applied magnetic field, Joule heating, viscous dissipation effects has made so far. The current work aims to look at the simultaneous impacts of viscous dissipation and Joule heating on pulsating MHD flow of a micropolar nanofluid with the influence of heat source/sink and thermal radiation. The present study may be useful in the study of physical aspects of biomedical, drug delivery, or radio therapy using nanoparticles. In the present work, blood is taken as micropolar fluid (base fluid), and gold (Au) is chosen as a nanoparticle. Gold nanoparticles are efficient in drug carry and drug delivery vehicles because they can encapsulate large quantities of therapeutic molecules. Here the perturbation technique is utilized to convert a set of partial differential equations (PDEs), which are governing current flow into a set of ordinary differential equations (ODEs) and then numerically solved by employing the shooting technique and the Runge–Kutta fourth-order procedure. The velocity, temperature, and rate of heat transfer distributions of micropolar nanofluid are discussed via pictorial representations in detail.

2 Formulation of the problem

We consider an incompressible and laminar pulsatile flow of electrically conducting micropolar nanofluid through a channel. The schematic diagram of the current flow model is exhibited in Fig. 1. Assume that the bottom wall is parallel to the -axis and that the -axis is perpendicular to it. Micropolar fluid is taken as blood (base fluid), and gold (Au) as a nanoparticle. The Joule heating, viscous dissipation, thermal radiation, and heat

source/sink are taken into account. An applied magnetic field of magnitude is enforced uniformly orthogonal to both the walls. Since no external electric field is applied and the effect of polarization of the ionized fluid is negligible, we can assume that the electric field is zero. When the external electric field is zero and the induced electric field is negligible, the magnetic Reynolds number is very small; so that effects of both the Hall effect and the induced magnetic field are negligible compared with the applied magnetic field. The bottom and top walls are retained at consistent temperatures and , respectively. Under these hypothesis, the governing equations are given by [4,5,27,32]:

Here is velocity along with in -direction, is the fluid pressure. and represent density, dynamic viscosity, electrical conductivity, effective specific heat, and thermal conductivity of nanofluid, accordingly, and the subscripts and . denote the nanofluid, base fluid, and solid nanoparticles, respectively. is the temperature of nanofluid, is coupling parameter, is microrotation vector, is microinertia parameter, is the radiative heat flux, and is the coefficient of the heat source/sink.

The corresponding boundary conditions (BCs) are

Here is the distance between the walls.

The characteristics of nanofluid are defined as [15,29]

Here is volume fraction of nanoparticles. The thermal features of base fluid and nanoparticles are shown in Table 1.

Now, with the help of Rosseland approximation for the radiative heat flux , Eq. (3) becomes

By utilizing the ensuing nondimensionl parameters,

Equations (1), (2), and (6) become

Here

is the prandtl number, is the Eckert number, is the frequency parameter, .is Hartmann number, is coupling parameter, is the radiatiom parameter, is gyration parameter, is microinerta parameter, is thr heat source/sink parameter.

The appropriate BCs are

3 Solution of the problem

The pressure gradient of the form is supposed to induce the pulsating flow

Since the pressure gradient conferred in Eq. (12) causes the flow of velocity, microrotation and temperature are declared as

By substituting Eqs. (12)–(13) in Eqs. (8)–(10) and equating the coefficients of like power of one can acquire

The corresponding BCs are

Further, the nondimensionl heat transfer rate (Nusselt number) at the walls is assigned as

The system of Eqs. (14)–(19) is solved with boundary conditions (20) by adopting the shooting technique with Runge–Kutta fourth-order approach. The step size is fixed as accuracy is fixed for the convergence criteria.

The comparison with the results of Venkatesan and Reddy [31] for and for Al₂O₃ nanoparticle is presented in Table 2. The comparison shows that there is a good agreement between the present results and the results of Venkatesan and Reddy [31]. Further, to check the correctness of the present results, we made a comparison between the present results and the results obtained by NDsolve using MATHEMATICA software, which are given in Table 3. It is observed that there is a good agreement between the present results and the results obtained by NDsolve.

Advantages and limitations of shooting method

The shooting method works by considering the boundary conditions as a multivariate function of initial conditions at some point, reducing the boundary value problem to

finding the initial conditions that give a root. The advantage of the shooting method is that it takes advantage of the speed and adaptivity of methods for initial value problems. The limitations of the method is that it is not as robust as finite difference or collocation methods: some initial value problems with growing modes are inherently unstable even though the BVP itself may be quite well posed and stable.

4 Results and discussion

In the present section, numerical simulations are used to test the dominant features of the flow behaviour of nanofluid. The static values of and are considered for graphical results unless otherwise stated. The current scenario deals with the impacts of several parameters on velocity, microrotation, temperature, and Nusselt number (heat transfer rate) of Au-blood micropolar nanofluid with the help of graphical outcomes, which are presented in Figs. 2–7.

4.1 Velocity distribution

The velocity distributions for various values of coupling parameter gyration parameter Hartmann number nanoparticles volume fraction and frequency parameter are shown in Figs. 2(a)–2(e). Figure 2(a) exhibits the variations of the velocity of nanofluid with different values of coupling parameter From Fig. 2(b) one can notice that the velocity falls with the enhancement of Hartmann number The reason for the fall in the velocity is because of the retarding forces upon the utilization of magnetic field, which act opposite to the flow direction. Hence there is a fall in velocity. Figure 2(c) presents the impact of gyration parameter on the velocity of nanofluid. This figure shows that the velocity is increasing with a rise of the gyration parameter. Figure 2(d) describes that the velocity slows down with the enhancement of nanoparticle volume fraction The reason behind this is the combined influence of thermal

coefficient and density extension of the base fluid with nanoparticle mass density is signified. Figure 2(e) elucidates that there is an enhancement in the velocity of nanofluid with an increment on frequency parameter .

4.2 Microrotation distribution

The influence of coupling parameter Hartmann number gyration parameter nanoparticles volume fraction on microrotation is presented in Figs. 3(a)–3(d). Figures 3(a) and 3(b) describe that the micro velocity is increasing near the bottom wall at the same time it is falling at the top wall by varying and Figure 3(c) demonstrates that by varying the gyration parameter the microrotation velocity is decreasing near a bottom wall when it rising at the top wall. The opposite trend can be noticed from Fig. 3(d) by varying volume fractions

4.3 Temperature distribution

Figures 4(a)–4(d) and 5 are presented to see the impacts of Eckert number radiation parameter heat source/sink parameter coupling parameter Hartmann number on the profiles of temperature distribution of the nanofluid. Figure 4(a) displays that the temperature accelerates with an enhancement in Eckert number. Because is correlated between the heat enthalpy and differences in kinetic energy, when we increase , it leads to boost the kinetic energy of particles by reducing enthalpy factor, which rises the temperature. Figure 4(b) describes that there is a decrease in temperature distributions for the rising values of Such decrease may be due to the physical fact that rising the radiation parameter reduces the thickness of the thermal boundary layer. Figure 4(c) reveals the temperature differences for various heat source/sink parameter values. It is clear that the temperature boosts up with the rising of heat source due to the heat generation in the working nanofluid, while it reduces with enhancing of a heat

sink due to the heat amalgamation. Figure 4(d) shows that the temperature is decreasing, while the coupling parameter increases. Because the coupling parameter is against the flow of temperature profile, this results in the decrease in θ. Figure 5 presents that increasing Hartmann number decreases the temperature, this is maybe due to the Lorentz force effect created by applied an magnetic field, which reduces the thermal boundary layer thickness.

4.4 Nusselt distribution

Figures 6(a), 6(b), and 7(a)–7(c) depict the behaviour of the rate of heat transfer of Au-blood nanofluid at the bottom wall against for various parameters and From these figures it is noticed that the rate of heat transfer is increasing function of Figure 6(a) elucidates that the heat transfer rate is rising

with a rise in . It is due to enhancing viscous dissipation accelerates the rate of heat transfer on the wall. Figure 6(b) presents that the Nusselt number is decreasing with an increment of nanoparticle volume fraction A similar trend is seen from Figs. 7(a)–7(c) with an improvement of Hartmann number, radiation parameter, and coupling parameter, respectively.

5 Conclusion

The current study investigates the pulsating MHD flow of Au-blood micropolar nanofluid in a channel with the presence of Joule heating and thermal radiation. Micropolar fluid is chosen as blood (base fluid) and gold (Au) as nanoparticles. The considered model is important in the study of biological fluid modeling, polymer engineering, sediments in rivers, and nanodrug delivery. The numerical results for dimensionless fluid flow variables are obtainded by hiring a shooting technique with the aid of Runge–Kutta fourthorder scheme. The impacts of relevant parameters on the velocity, microrotation, and temperature distributions of Au-blood nanofluid are investigated in detail by plotting graphs. The main outcomes are outlined as follows:

• Strengthening magnetic field dwindling the current flow due to the Lorentz forces produced by an applied magnetic field, which act opposite to the flow direction.

• The velocity of nanofluid is reducing with an enhancement of coupling parameter and nanoparticles volume fraction.

• The velocity is increasing with an enhancement of the gyration parameter and frequency parameter.

• The temperature profile is rising with the rise of the heat source parameter.

• Magnifying viscous dissipation producing the additional energy, which encourages the temperature.

• The temperature profile is reducing with an increase of radiation parameter, coupling parameter, magnetic field.

• The heat transfer rate of nanofluid against frequency parameter is increasing with an improvement of the Eckert number.

• The heat transfer is decreasing with an increase of

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