Electro-Blood Circulation Fusing Gold and Alumina Nanoparticles in a Diverging Fatty Artery

Emerging bio-microfluidic systems are embracing electroosmosis mechanisms to supervise ionised physiological fluids. Impelled by the exploration of electro-micro-pumping mechanisms in bio-engineering domains, in the present inquisition, a research policy associated with hemodynamical deportment of blood circulation under suspension of gold and alumina nanoparticles in a diverging fatty artery subject to electro-osmotic driving forces is conducted. The exact solutions of the resulting model equations associated with the proposed wall conditions are acquired in terms of Bessel functions under larger wavelength and a smaller ratio of inertial and viscous force. Numerical integration is computed through the NDSolve technique. Plotting and elaboration are performed to explore the physical insight into hemodynamics under diverse parametric aspects. Crucial findings of the study include that a raised electro-osmotic parameter elevates the intensity of blood streaming in the core zone of the artery and it established a declining pattern in the close vicinity of the arterial wall. The parameters linked with electroosmosis contribute to thermal emaciation of blood streaming in the artery. The heat transmission rate across the arterial wall sharply declines for ascending values of the electro-osmotic parameter and Helmholtz-Smoluchowski velocity. The structure of blood boluses is also outlined under the impact of involved parameters in brief. Pure blood offers less size and number of blood boluses near the arterial wall compared to hybrid nano-blood. From a bio-engineering point of view, the modelling study can contribute significantly to biomechanical and medical processes and therapeutic implications.


Introductions
Transferring heat using nanofluids in engineering and bioengineering processes has been a topic worth researching in recent years. Nanofluids (NFs) are cooling fluids that contain solid nanostructures of exceedingly small diameter below 100 nm. Nanoparticles (NPs) have the potential to bring about a revolution in science and technology sectors and open up new avenues of investigation in the field of medicine. NPs have been the subject of several studies showing their usefulness in human tumour targeting, diagnostics, and therapy. A particular NP may give therapeutic elements to tumour cells in large doses without contaminating healthy cells. This is possible because the particles are so small. Gold nanoparticles (Au NPs) possessing optoelectronic properties, low toxicity, high surface-to-volume ratio, and biocompatibility are getting approval in biomedical engineering, diagnostics procedures, pharmaceutical processes, therapeutic agent delivery, gene delivery, photodynamic therapy, chemotherapy, cancer diagnostics, immunological modulation, targeted drug delivery, implants, prosthesis, and waste remediation [1]. Alumina nanoparticles (Al 2 O 3 NPs) have impressive utilities in biomedical procedures, photo imaging, photothermal therapy, antimicrobial, nanosensors detect biomolecules, environmental factors, food packaging, and food safety [2]. In the year of 1995, Choi [3] was the first researcher who introduced the notion of nanofluids to enhance the thermal efficiency of cooling fluids with low thermal conductivity. He examined and proved that the thermal execution of working fluids is notably improved after adding nano-sized particles. In the beginning, Buongiorno [4] investigated the convective transportation of nanofluids and noticed the thermal enhancement because of the volumetric concentration of nano-size solid particles. Narla et al. [5] executed a research study associated with the electro-osmotically induced peristaltic propulsion of a nanofluid in a curved domain to get better efficiency of the curved pump. The biomimetic flow of a nanofluid in a curved passage is explored by Hayat et al. [6]. Thus far, numerous scholars [7][8][9] have exposed thermophysical features of nanomaterials under various thermal and physical constraints.
The investigators are now attempting to disperse nanomaterials compositions in conventional fluids. Mixing two or more nanomaterials into a single conventional fluid produces a hybrid nanofluid (HNF). Hybrid nanofluids improved thermal efficiency, hydrodynamic characteristics, and thermophysical features compared to single nanofluids (SNFs). HNFs have been found to be massive applications in biomedicine and bioengineering, such as diagnostics procedures, therapeutics, anticancer therapy, chemotherapy, and drug transportation systems. Sarkar et al. [10] probed the heat transmission and possible applications of hybrid nanofluids. They claimed HNFs are very promising for enhancing heat transmission. Ijaz et al. [11,12] provided a computational investigation for blood streaming and heat transmission using hybrid nanoparticles in a diseased artery. Their core findings include that the blood flow impedance on the disease arterial wall significantly reduces by suspending more nanoparticles. Sadaf et al. [13] theoretically investigated the impact of infused hybridised nanoparticles on blood pumped by the peristaltic mechanism through an endoscopic arterial canal. According to their observation, the volume concentration of hybridised nanoparticles significantly contributes towards the thermal state of blood streaming. Ali et al. [14] investigated the peristaltically induced hybrid nano-blood (Jeffrey model) flow in a ciliated artery with electromagnetic phenomena. In a study of Abdelsalam et al. [15], the electro-magnetised hybrid Casson nanofluid flow in a sinusoidal channel led by electroosmosis was discussed in the provision of larger wavelength. References [16][17][18][19] entail recent studies with regard to the contribution of hybrid nanoparticles on the arterial blood streaming with various geometrical configurations.
The significance of heat sources in hemodynamics has captured the attention of many modellers because of its diverse medical uses like the transmission process in thermal therapy and cancer therapy. Wall slip conditions are crucial for defining a problem and are concurrent of principal significance in computational hemodynamics. These conditions signify fluid mobility over the wall surface. The slip condition augments the fluid motion adjacent to the wall surface. For that reason, it is essential to consider the slip condition as a more appropriate model collated with a no-slip condition. Solicitations of slip condition conditions can be found in various microfluidics devices, including micropumps. In recent years, inspection of hemodynamical flow and heat transfer problems accompanied by slip conditions has been a subject of interest to many scholars [20][21][22].
Another aspect that works as an effectual tool to pump biofluids through muscular organs and tubes without using an external mechanical force is peristalsis. The phenomenon of peristalsis involves biophysics, physiology, biomechanics, biomedical industries, and bioengineering. The functioning of such a mechanism can be perceived primarily in human organs and organisms. This profound mechanism has an imperative role in blood transportation (heart to entire body), urine transportation (kidney to bladder), transportation of chyme, cilia and sperm, and food bolus movement in the stomach. Additionally, lots of bio-industrial instruments are manufactured, such as heart-lungs machine, dialysis machine, diabetic pumps, biomimetic displacement pumps, electro-pneumatics and roller pumps, reptilian breathing, stethoscope, and cardiopulmonary bypass machine by using the peristaltic code. The experimental concept of peristalsis was first reported by Latham [23] in 1966. The theoretical concept was first uncovered by Shapiro et al. [24]. After the exemplary initiation of Latham, the versatility of the peristaltic phenomenon has been discussed by many researchers. Jaffrin and Shapiro [25] studied the peristaltic flow of biological (seminal) liquid under lubrication and long wavelength assumptions. Their study provided the theoretical basis for numerous examinations, including retrograde, trapping, reflux, and other phenomena. Abdelsalam et al. [26] performed a theoretical analysis of the peristaltically driven pulsating flow of nano-blood in a narrow artery. Large-sized blood bolus is generated near the arterial wall for haemodilution. The peristaltic driving mechanism in a micro-channel packed with couple stress biofluids under electromagnetic forces was demonstrated by Tripathi et al. [27]. They noticed a significant alternation in the pressure distribution in the arterial domain due to electroosmosis. Sunitha [28] emphasised the peristaltic heat and mass transport of non-Newtonian blood with the Jeffrey fluid model fusing with gold nanoparticles in an arterial channel under the occurrence of double-diffusive convection and infrared radiation of heat. It was reported that the occurrence of infrared radiation in the blood flow conveying gold nanoparticles raises the blood temperature. Khan et al. [29] have scrutinised the peristaltic phenomena in a curved micro-channel packed with non-Newtonian biofluid subject to static electric forces. The simulated findings showed that enclosed boluses disappear with intensified biofluid viscoelastic behaviour. The related similar works are listed in Refs. [30][31][32][33][34][35].
The electro-osmotic transport mechanism (or electroosmosis) is a unique electro-kinetics mechanism. A highly efficient electro-transport tool is widely acknowledged for transporting physiological fluids in bio-micro-fluidics devices that are massively used in a wide variety of bioengineering disciplines. In the year 1809, Reuss [36] unravelled the intrinsic science of electroosmosis phenomena. Later, the mathematical theory for this transport mechanism was introduced by Wiedemann [37]. Electroosmotic phenomena occur when an ionic fluid medium comes in contact with a charged solid surface. A negatively charged surface attracts positive charges towards itself to get electric neutrality. As a result, a thin layer of ionic species near the solid surface is founded. This layer is named an electric double layer (EDL). On the imposition of an electric field, ionic species in EDL move towards the direction of the applied electric force that induces the driving force on the bulk fluid. Electroosmosis mechanism covers the broad area of uses in engineering and biomedical sectors, such as soil testing, DNA testing, diagnostic procedures, surgical procedures, examination of abnormal cells and cellular anomalies, nanomaterial devices, and delivery of drugs by employing diagnostic kits. Recently, medical researchers have used electro-osmotic transport techniques to anticipate the behaviour of blood circulation throughout the human body. Both theoretically and practically, many researchers have extensively researched the impact of electric fields on blood circulation. Chaube et al. [38] predicted the rheology of micropolar fluids in microchannels subject to electro-osmotic forces. The electro-osmotically regulated streaming of blood fusing with nanoparticles through a microfluidics channel under the influence of heat radiation was theoretically examined by Prakash et al. [39]. Tripathi et al. [40] explored the flow pattern and heat transport phenomena impinged by electroosmosis in a microchannel under the functioning of gravitational forces. The blood mobility manner due to electroosmosis through microvessels was disclosed by Prakash et al. [41]. Abo-Elkhair et al. [42] conducted an examination on the electromagnetic flow of a dielectric fluid through a sinusoidal wavy channel packed with porous materials. They found that the critical reflux pressure is greater for a nonelectrified fluid. The flow and heat transport phenomena of ionic Casson fluid fusing nanoparticles in a microchannel subject to electromagnetic forces have been disclosed by Das et al. [43]. This examination envisaged that elevated electro-osmotic parameter values significantly develop the momentum of ionic liquid. Akhtar et al. [44] have observed the rheology and hemodynamical characteristics of non-Newtonian blood streaming in an artery with multiple stenoses under an electric double layer. The research outcomes revealed that the seed and temperature of blood streaming in an artery could be regulated by controlling the characteristic thickness of the electric double layer. Zaher et al. [45] have revealed the examination of electro-osmotically progressed Williamson fluid with the suspension of gyrotactic microorganisms through a Darcy-Forchheimer medium. Their findings revealed that intensified electroosmotic's parameter values cause an enhancement in the fluid velocity. The inspection unfolded by Abdelsalam et al. [46] exhibited the swimming of an electrically conducting sperms through cervical canal driven by electromagnetic field. Outcomes admitted that the sperms' velocity inside the cervical canal is declined due to increased electric field parameter. Abdelsalam1 and Zaher [47] have given out an analysis of sperms swimming through a ciliated cervical canal followed by the dual action of peristalsis and electroosmosis. It was observed that higher values of electroosmotic parameter and Helmholtz-Smoluchowski velocity account for the enhancement of mucus velocity upsurges in the upper and lower spaces of swimming sheet. Akram et al. [48] have focused on the electro-osmotic aspects of an aqueous solution doped with single-walled carbon nanotubes (SWCNTs) flowing in a vertical channel subject to magnetic force, thermal radiation, and wall slip condition. Simulated results conveyed that a depletion in EDL thickness tends to boost the velocity and temperature profiles. Das et al. [49] have presented a modelling study associated with the electro-blood streaming carrying hybrid nanoparticles in a diverging endoscopic annulated region.
Their key findings include that the blood bolus's size in the endoscopic conduit amplifies due to EDL formation. Very recently, Karmakar et al. [50] have mathematically described the electro-osmotically regulated streaming attributes of copper-gold-alumina/blood circulation in an eccentric arterial annulus in the placement of a small endoscopic tube. They have claimed that varying concentrations of suspended trihybrid nanoparticles in the bloodstream can be very supportive in regulating hemodynamical profiles and quantities. Other recent explorations on the electroosmosis stimulated blood flow and heat transmission using hybridised nanoparticles via arterial segment are found in Refs. [51][52][53][54][55][56][57].
The above literature assessment identifies that no one has yet to describe the streaming characteristics of blood doped with hybrid nanoparticles in a diverging artery under an electric field's influence. The current research study attempts to close this research gap, and for that reason, it is believed to be original and entails new significant components that other scholars have not reported. Accordingly, our modelling study aims to disclose the hemodynamical attributes of the blood circulation carrying gold and alumina nanoparticles in a diverging fatty artery accounting for the consequence of electric double-layer formation nearby the arterial wall. Blood is used as the base fluid, and gold and alumina are as nanoparticles infused into blood. The energy equation embodies energy contribution due to viscous dissipation, Joule heat generation, and the heat source. Debye-Hückel linearisation approach and lubrication theory (larger wavelength and smaller ratio of inertial and viscous force) are considered to study the hemodynamical aspects of blood streaming in an artery. An analytical strategy is applied to get the solutions of the emanating model equations in the form of the Bessel functions. Graphs and tables are utilised in demonstrating the behaviour of the hemodynamical variable and quantities (axial velocity profile, temperature distribution, pressure gradient, pressure rise per wavelength, and heat transfer coefficient) against cardinal parameters. The phenomenon of trapping is also outlined through graphical illustrations. The reliability of the analytical solution is verified by comparing it with the available literature.
The novelties of the proposed model are concisely stated as follows: • Mathematical modelling and theoretical simulation of electro-osmotically endorsed circulation of blood fused with hybrid nanoparticles in a diverging fatty arterial tract are performed. • Heat energy occurring due to Joule heating and viscous dissipation is hypothesised to amend the blood circulation in the arterial segment. • Analytical methodology via Mathematica coding is deployed to capture the closed-form solutions. • Simulated outcomes of this model are novel and play a crucial role in bioengineering and biomedical disciplines.

Geometrical Description
We account for a 2D streaming of electrolyte nano-blood suspension (Au-Al 2 O 3 /blood) influenced by electroosmosis in a diverging sinusoidal wavy arterial tube travelling with a wave speed c. The hybrid nano-blood is composed of blood as a base fluid and gold and alumina NPs having a low volume fraction in size. A co-ordinate R, Z is designated the flow layout with R is oriented to the radial direction, and Z is taken along the arterial tube axis. The blood flow model is depicted in Fig. 1a. An external electric field of unvarying strength E z in terms of electroosmotic force is employed on the blood streaming along axial direction (i.e. the positive z-direction). The arterial wall is under hydrodynamic slip phenomena and assumed with the constant temperature T 0 . The subsequent suppositions are taken into account when developing the blood flow model: • Hybr id nano-blood (Au-Al 2 O 3 /blood) is an incompressible one-phasic electrolyte suspension. • The adopting nanoparticles' shape is spherical. • The arterial wall is fatty.
• The base fluid (blood) and hybridised nanoparticles (Au, Al 2 O 3 ) are in the state of thermal equilibrium.
The geometrical form of the arterial wall is proposed as [20,49]: where a Z = a 0 + KZ , a 0 is the radius of inlet, K the constant depends on the arterial tube length, b sinusoidal wave amplitude, λ the wavelength, and t the time.

Electrohydrodynamics
An ionised blood (Au-Al 2 O 3 /blood) is taken up, and its electric potential dispersion is mathematically described by the Poisson-Boltzmann equation as [39,54,49,55]: where Φ denotes the electric potential, ε 0 the dielectric permittivity of the medium, ρ e the density of ionic charge, which is expressed as [14,49]: where e represents the electron charge, z v valence of ions, n − number density of negative ions, and n + number density of positive ions. Invoking no-EDL overlapping, the Boltzmann ionic distribution gives [39,49,55]: where the bulk density of ionic species in ionised blood is symbolised by n 0 , the average temperature by T a , and the Boltzmann constant by k B .

Streaming Governing Equations
Based on the mentioned assumptions, the model equations in the fixed frame are as follows [20,49,54]: Fig. 1 a Physical structure of blood flow. b Comparison of axial velocity profile with Abbas et al. [20] for where ( U , W ) stands for the components of velocity along R and Z directions, E z the axial electric field, Q 0 the heat source coefficient, T the temperature, P the pressure, ρ hnf the density, μ hnf dynamic viscosity, (ρc p ) hnf the heat capacity, σ hnf the electrical conductivity, and k hnf the thermal conductivity of hybrid nano-blood (Au-Al 2 O 3 /blood). In the fixed frame, the wall conditions for the proposed blood streaming are as follows [39,20,49]: where L denotes the hydrodynamic slip coefficient.

Thermo-Physical Properties of Blood and Nanoparticles
The thermo-physical models for SNF and HNF are presented in Table 1. The values correspond to the thermo-physical properties of nanoparticles along with blood are furnished in Table 2.

Solution Methodology
The exact solution of Eqs. (24), (26), and (27) subject to the wall conditions (28) is captured as follows [49,54]: where J 0 designates the zeroth order Bessel function of first kind, I 0 the zeroth order modified Bessel function of first kind, and F the generalised hypergeometric function. The constants w 0 , w 1 and w 2 , θ 0 , θ 1 , ⋯, θ 6 are given in the Appendix. Equation (25) discloses that the blood pressure p is a function of z only.

Physical Quantities of Interest
The volumetric flow rate across the arterial tube is formulated as below [20,21,49,54]: On computation of (32), the pressure gradient is derived after simplification as: where 0F1 denotes the regularised confluent hypergeometric function, and the constants p 1 and p 2 are given in the Appendix.
The mean volumetric flow rate over one period of the peristaltic propulsion is calculated as [20,21,49]: The dimensionless pressure rise ΔP over one period of the peristaltic propulsion is formulated as [20,54]: Inserting the pressure gradient expression in (33), ΔP is numerically evaluated via numerical integration scheme in Mathematica computational software.
The dimensionless heat transfer coefficient (HTC) at the arterial wall is formulated as [14,20,21,49,54]: where where the constants h 0 , h 1 , and, h 2 are given in the Appendix.
On integrating r = rw subject to the condition ψ = 0 at r = h, we find the stream function ψ as [21,49,54]: in which the constants Ψ 0 , Ψ 1 , and Ψ 2 are provided in the Appendix.

Reliability of Analytical Solution
This part intends to verify the reliability of our analytical solutions. The modelling study of [20] is ditto the same after deputising κ = U hs = ϕ 1 = ϕ 2 = 0 in the present analytical solutions. In addition, the axial velocity curve obtained from the current model is graphically compared with the corresponding results of Abbas et al. [20], as depicted in Fig. 1b. The set of curves demonstrates coherent agreement. Thus, the obtained analytical solutions are confirmed to be reliable.

Simulated Results and Discussion
This segment of the modelling study provides an elaboration of the simulated results achieved from analytical solutions of the streaming equations via Mathematica symbolic scheme. The graphics are designed for diverse governed parameters to impart their physical impact on hemodynamical variables, and quantities of concern blood flow. The nominated values of embedded parameters are as [20,54,21,49]: In addition, (ϕ 1 , ϕ 2 ) = (0, 0) reduces to pure blood model, (ϕ 1 , ϕ 2 ) = (0.01,0) relates to nano-blood (Au-blood) model, and (ϕ 1 , ϕ 2 ) = (0.01,0.02) represents hybrid nano-blood (Au-Al 2 O 3 /blood) model. All figures have been plotted for Au-Al 2 O 3 /blood (solid lines) and Au-blood (dashed lines).

Electric Potential Profile
This subsection proposes to discuss the electric potential distribution Φ against radial coordinate r for several values of electro-osmotic parameter κ, and amplitude ratio φ.  Figure 3 a-h present the profile of the dimensionless velocity w deportment in response to disparate parameters viz., electro-osmotic parameter κ, Helmholtz-Smoluchowski velocity U hs , slip parameter L, mean flow rate Q, amplitude ratio φ, non-uniformity parameter ζ, and volume fractions (ϕ 1 , ϕ 2 ) for both hybrid nano blood (Au-Al 2 O 3 /blood), and nano blood (Au-blood). The result of electro-osmotic parameter κ on the graph of w is explained in Fig. 3 a, b when U hs < 0 (applied electric field is oriented in the blood flow direction, i.e. positive z-direction) and U hs > 0 (applied electric field is aligned along in the reverse direction of blood streaming, i.e. negative z-direction). For the case of U hs < 0, with amplified values of κ, the axial velocity gets intensified in the centre region of the arterial tract and diminished in the area close to the arterial wall, while a contrary behaviour is recorded for the case of U hs > 0. Physically, negative values of U hs govern the assisting electro-osmotic body force (−κU hs Φ) that effectively indulges the blood motion. In the case of positive values of U hs , the reverse trend is manifested due to the activation of resisting electro-osmotic body force (−κU hs Φ). The occurrence of electro-osmotic forces in the blood-streaming domain gently aids the movement of blood in the arterial tract, thereby leading to an upliftment in blood velocity. Figure 3c characterises the pattern of blood motion in response to Helmholtz-Smoluchowski velocity U hs . It is clear in this figure that higher axial velocity attached to U hs < 0, whereas the lower axial velocity is associated with the case of U hs > 0. Helmholtz-Smoluchowski velocity is the velocity of ionised blood induced by the drag of ions' density on blood molecules. Physically, negative Helmholtz-Smoluchowski velocity prolongs the peristaltic propulsion, which augments the blood mobility in the centre zone of the arterial tract. The blood mobility is inhibited for positive Helmholtz-Smoluchowski velocity resisting the peristaltic propulsion. U hs = 0 signifies the negation of the electric field in the flow realm. The blood flow induces due to purely peristaltic pumping for the case of U hs = 0. Thus, the axial blood velocity for U hs = 0 is less dominated than that of U hs < 0. The profile of w for a range of slip parameter L is sketched in Fig. 3d. The w profile gets declined in the core region of the arterial tube and a contrary trend near the arterial wall by the augmentation of L. From the physical perspective, frictional forces in the close vicinity of the arterial wall are emaciated by larger estimations of L, which gives additional support to the momentum development, and the outcome is a boost in the blood flow near the wall. exhibits an analysis of the influential role of the mean flow rate Q on the graph of w. The axial velocity is realised to improve for escalating variation of Q. In a physical sense, larger Q amplifies the intensity of flow current, which results in an augmentation of blood mobility. To investigate the consequence of increasing variation in amplitude ratio φ on the graph of w, Fig. 3f is prepared. The graph of w is significantly augmented, leading to an increment in φ's values. Larger wave amplitude possesses inertial forces that decently incite the movement of suspended NPs. This outcome results in a boost in blood motion. Figure 3g demonstrates the alteration of the blood-streaming pattern under the consequence of non-uniformity parameter ζ. With an increasing variety of ζ, the streaming profile shows enhancing behaviour in the arterial tube. In the divergent arterial tract (ζ > 0), the axial velocity of blood manifests dominant behaviour as compared to the convergent arterial tract (ζ < 0) and the uniform arterial tract (ζ = 0). This happens due to the greater inertial flow and a wide flow space, providing a pressure that boosts blood mobility. The impression of volume fractions (ϕ 1 , ϕ 2 ) on the axial velocity profile is expounded in Fig. 3e. Expansion of volume fractions (ϕ 1 , ϕ 2 ) is ruled to an emaciation in the axial velocity graph in the arterial tract's centre zone. In contrast, the insignificantly opposite nature is noted in the wall position of the artery. The case of ϕ 2 = 0 signifies the Au-blood model (i.e. Al 2 O 3 NPs are absent in blood) with higher velocity than Au-Al 2 O 3 /blood model. The blood velocity attains its maximum at (ϕ 1 , ϕ 2 )=(0, 0) (nanoparticles free blood). The graphical panel of the axial velocity exposes that the maximum of the velocity curve is observed to attain at the centre line (r = 0), and the curves take a parabolic and symmetric form with regard to significant parameters. On the other hand, the velocity curves for Au-blood model are marginally larger than Au-Al 2 O 3 /blood model. This is because Au-Al 2 O 3 /blood is thicker than Au-blood due to the dispersion of NPs. A similar nature was reported by Das et al. [49].

Temperature Profile
For both hybrid nano-blood (Au-Al 2 O 3 /blood) and nanoblood (Au-blood), the dimensionless temperature evolution θ under the assorted values of controlling parameters like electro-osmotic parameter κ, Helmholtz-Smoluchowski velocity U hs , Joule heating parameter S, Brinkmann number Br, heat source parameter χ, mean flow rate Q, non-uniformity parameter ζ, and volume fractions (ϕ 1 , ϕ 2 ) is publicised in Fig. 4 a-h. Figure 4a stipulates the repercussion of varying electro-osmotic parameter κ on the blood temperature θ. The blood temperature drops with upgraded values of κ. This is because escalation in κ reduces the electric double layer thickness, due to which the blood's viscosity declines, which leads to lower blood temperature. The impression of Helmholtz-Smoluchowski velocity U hs on the profile of θ is shown in Fig. 4b. It is demonstrated that with changing values of U hs , the graph of θ diminishes. Physically, negative Helmholtz-Smoluchowski velocity (U hs < 0) boosts the blood motion, due to which the suspended nanoparticles and blood molecules move rapidly. Consequently, kinetic energy converts to thermal energy, enhancing the blood temperature. For positive Helmholtz-Smoluchowski velocity (U hs > 0), the blood mobility enfeebles and, therefore, a descending profile of blood temperature envisages. As expected, lower temperature attains for positive U hs . Figure 4c scrutinises the significant contribution of Joule heating parameter S on the θ-profile. It is examined that the graph of θ upsurges with incremental values of S. This can be comprehended from the actuality that as the Joule heating parameter grows, mechanical energy due to Joule heating brings about resisting force on ionic species to move through the electrolyte blood, which generates more heat, thus leading to an upsurge in the blood temperature. These findings are matched by Das et al. [49] in the hybrid nano-blood model. The reflection of Brinkmann number Br on the thermal behaviour of blood is signified in Fig. 4d. It can be seen from the resulting graph that an upsurge in the values of Br leads to augmentation in the thermal distribution of blood. The Brinkman number measures the heat dissipated by viscous dispersion to the heat supplied by blood molecules/suspended hybrid nanoparticles. A raised Brinkmann number tends to enhance the viscous effects that increase thermal energy; therefore, a significant amount of heat is produced via viscous dissipation. Consequently, for larger Br, the temperature distribution inflates. Figure 4e elucidates the consequence of heat source parameter χ on the θ-profile. This figure shows that θ upgrades with an escalation in χ. This happens because greater heat source strength within the system during the blood streaming leads to a boost in blood movement. Thus the temperature rises. A nearly similar estimation has been achieved by Das et al. [49]. The temperature profile behaviour regarding the mean flow rate Q is demonstrated in Fig. 4f. With the elevation in Q, the temperature profile intensifies. Physically, for larger values of Q (larger flow rate across the arterial segment), the blood flow substantially expedites, leading to an upsurge in the temperature graph of blood. Variation in the profile of θ with non-uniformity parameter ζ is displayed in Fig. 4g. As expected in this figure, higher temperature belongs to the case of the diverging arterial tract (ζ > 0), whereas lower temperature corresponds to the issue of converging arterial tract (ζ < 0). The case ζ = 0 represents the uniform arterial tract. From the physical perspective, energy in a diverging arterial tract is rapidly distributed in the form of heat from NPs. This, in turn, amplifies the blood temperature following the outcome of frictional heating.  Figure 4h presents the curve regard to the blood temperature θ when varying volume fractions (ϕ 1 , ϕ 2 ). With a higher estimation of (ϕ 1 , ϕ 2 ), the blood temperature in the arterial path dwindles. Adding more gold and alumina nanoparticles enhances heat absorption within the arterial domain, consequently dropping the hybrid nano-blood temperature. A key factor affecting the thermal profile of blood streaming is the dispersion of hybridised nanoparticles throughout the blood streaming. On the other hand, lower temperature relates to Au-Al 2 O 3 /blood while higher temperature belongs to Au-blood. This is because of the significant improvement in the thermal conductivity of Au-Al 2 O 3 /blood than Au-blood. These results verify that the hybrid nano-blood model can achieve desirable thermal control with more nano-blood. The lower temperature associated with HNB implies that the hybrid composition of nanomaterials can assist in improving the thermal attributes of the fluid.

Pressure Gradient Profile
For both hybrid nano-blood (Au-Al 2 O 3 /blood) and nanoblood (Au-blood), Fig. 5 a-h demonstrates the variation of the dimensionless pressure gradient p z inside the arterial tube against diverse values of κ, U hs , L, Q, φ, ζ, ϕ 1 , and ϕ 2 . In the case of U hs < 0 (electric field's imposition towards the blood flow direction), Fig. 5a shows that the pressure gradient is a descending function of κ. On the contrary, for the case of U hs > 0 (electric field's imposition towards the opposite direction of the blood flow), the pressure gradient augments with incremental values of κ, as revealed in Fig. 5b. The higher resisting pressure difference associated with the case of U hs > 0 is due to the intensification of opposing electro-osmotic body force to the motion of blood. Figure 5c paints the pressure gradient deportment with regard to varied Helmholtz-Smoluchowski velocity U hs . It is evident in this figure that with the changing values of U hs , p z significantly alters. As expected, the imposition of an electric field along the flow direction (U hs < 0) induces a higher assisting pressure difference in the flow tract when compared to the other cases. The remarkable consequence of slip parameter L on the graph of p z is illustrated in Fig. 5d. With increasing estimations of L, the resisting pressure gradient develops. Figure 5e expounds that the resisting pressure difference can be emaciated by depreciating the mean flow rate Q. This is to be expected since enhancement in the discharged blood volume due to higher Q results in the emaciation of the resisting pressure difference. Figure 5f renders the fluctuating trend of p z in response to amplifying amplitude ratio φ. As expected, amplifying values of φ result in larger wave progression along the arterial wall that induces such a trend in the pressure gradient. Figure 5g brings out the variation of the pressure gradient against the different geometrical configurations in terms of non-uniformity parameter ζ. In the case of a converging arterial tract (ζ < 0), the figure discloses that the assisting pressure difference is higher when compared to the diverging arterial tract (ζ > 0) and uniform arterial tract (ζ = 0). In Fig. 5h, it is to be perceived that the pressure gradient drops with enlarging values of (ϕ 1 , ϕ 2 ). This implies that the assisting pressure difference can be debilitated by including more NPs in the blood flow system. The negative value of p z signifies the pressure difference, which assists the forward motion of blood (i.e. blood flows from the reign of higher pressure to a reign of lower pressure). The bloodflow noticeably accelerates for larger negative values of the pressure gradient.

Pressure Rise Profile
The pressure rise ΔP (i.e. the pressure difference in one wavelength) is an emergent physical attribute in the pumping phenomena. The pressure rise is uttered in integral form, which is numerically evaluated via the Mathematical NDSolve routine. ΔP is plotted against the mean flow rate Q in response to varied κ, U hs , L, φ, ζ, (ϕ 1 , ϕ 2 ) for both hybrid nanofluid (Au-Al 2 O 3 /blood), and nanofluid (Au-blood). With upgraded electro-osmotic parameter κ, ΔP dwindles in the case of U hs < 0, while an opposite manner is documented for the case of U hs > 0, as shown in Fig. 6a, b. The variation of applied electric field orientation endorses this reflection. One can observe from Fig. 6c that the pressure rise is lower related to the case of U hs < 0 when compared to the other instances of U hs = 0 and U hs > 0. This phenomenon results from stronger/ weaker electro-osmotic body forces. Figure 6d clarifies that ΔP is proportional to slip parameter L. This reveals that the slip parameter tends to elevate blood pressure rise. Figure 6e discloses that with higher φ, a rising trend in the pressure rise profile is evident in the peristaltic pumping zone (−1 ≤ Q ≤ − 0.4), and an opposing trend is exposed in the augmented pumping zone (−0.4 < Q ≤ 1). For the converging arterial tube (ζ < 0), ΔP is lower as compared to the diverging arterial tube (ζ > 0) and uniform arterial tube (ζ = 0), as demonstrated in Fig. 6f. Figure 6g displays the variation of the pressure rise for pure blood, Au-blood, Au-Al 2 O 3 /blood. The hybrid nano-blood (Au-Al 2 O 3 /blood) is more sensitive to improving the pressure rise than Au-blood and Au-Al 2 O 3 / blood. It is also witnessed that ΔP for Au-blood is higher and minimal is connected to Au-Al 2 O 3 /blood.

HTC Behaviour
This subsection deals with the behaviour of heat transmission across the charged diverging arterial wall during the blood streaming, which is very crucial in view of hemodynamical aspects. The heat transfer coefficient (HTC) Z * profile over the diverse variation of κ, U hs , S, Br, χ, Q, φ, (ϕ 1 , ϕ 2 ) is imprinted in Fig. 7 a-j. Figure 7 a-i expounds that the THC increases for the improved values of S, Br, χ, Q, φ, ϕ 1 , and ϕ 2 , but depreciates with higher values of κ and U hs . An increase in κ and U hs causes a decremented impact on THC, as shown in Fig. 7a, b. The HTC for Au-Al 2 O 3 / blood is higher than Au-blood and pure blood, as portrayed in Fig. 7j. It is evident from the graphical analysis that the electroosmosis phenomena significantly impact HTC. The electro-osmotic parameter κ associated with electroosmosis phenomena is inversely proportional to the thickness of EDL formed near the arterial wall. A thinner EDL corresponds to a larger electro-osmotic parameter. The electric potential difference in the electroosmosis process figures on the EDL thickness. Less electric potential difference builds up in a thinner EDL, which results in an impairment in HTC. Helmholtz-Smoluchowski velocity U hs is directly linked to the electric field strength. More negative Helmholtz-Smoluchowski velocity upraises the pumping intensity of blood, and more heat is transmitted (in the form of HTC) across the arterial tube via convective mode as compared to conduction mode, as exhibited in Fig. 7b. The positive Helmholtz-Smoluchowski velocity U hs enfeebles the pumping intensity of blood, and consequently, HTC declines. Physically, an elevation in Joule heating parameter S ensures the quick transferring of mechanical energy to heat energy, which picks up the blood temperature and convective heat transfer. Consequently, a notable rise in THC is reflected. Higher heat source parameter χ relates to the intense heat generation, which in turn an upraising of blood temperature and subsequent enhancement in HTC. In Fig. 7h, i, it is ensured that higher loading of hybrid nanoparticles into the bloodstream causes the blood's thermal conductivity to enhance, which significantly impacts HTC at the arterial wall. The visual inspection indicates that suspended hybrid nanoparticles in the bloodstream significantly control heat transfer. The higher HTC associated with hybrid nano-blood (Au-Al 2 O 3 /blood) is because inserting more NPs results in superior energy conversion via circulation with the irregular movement of NPs, as manifested in Fig. 7j.

Trapping Event
In a pumping mechanism, trapping is an exciting manifestation. Trapping relates to the creation of bolus and the parting of streamlines in a wave frame. To demonstrate the trapping event, the streamlines pattern in the blood streaming is painted and analysed for the varied values of κ, U hs , Q, φ, ζ, ϕ 1 , and ϕ 2 via Fig. 8 a-n. Figure 8 a and b explore the streamlines pattern for the raising values of electroosmotic parameter κ. A substantial value of κ results in an increase in the entangled bolus's size and number near the arterial wall. According to observations, a large number of streamlines are looped due to the formation of a thinner EDL (larger κ) coherent to the arterial wall. Figure 8 c and d show the repercussion of Helmholtz-Smoluchowski velocity U hs on the blood bolus formation. Higher U hs reduces the number of blood boluses in the arterial tract. In the case of U hs < 0 (electric field aligned with the blood flow direction), more streamlines are enclosed in the vicinity of the arterial wall, while less number of streamlines are looped for U hs > 0 (electric field aligned the opposite of the blood flow direction). For positive U hs , the size of blood boluses marginally increases. These behaviours result from intensity variation of electro-osmotic body force in the flow realm. Das et al. [49] have documented a similar contribution of electroosmosis phenomena on the streamlines pattern in an arterial tract. The streamlined pattern for multiple values of mean flow rate Q is sketched in Fig. 8e, f. It is evident in these graphs that less number of streamlines are encircled by augmenting Q. The upshot of amplitude ratio φ on streamlines structure is examined via Fig. 8g, h. It can be observed from the plotted contour that more streamlines are circulated near the arterial wall with regard to amplified φ. The trapping mechanism for the divergent, uniform, and convergent arterial tubes is demonstrated in Fig. 8 i-k. The entangled bolus's size shrinks for the divergent arterial tract (ζ > 0). Figure 8 l-n reveal that the enclosed boluses in size and number are climbed for Au-Al 2 O 3 /blood than those for Au-blood and pure blood.

Conclusions
In this current study, a computational research study associated with the electro-circulation of blood-fusing hybrid nanoparticles in a diverging fatty artery with Joule heating participation, heat generation is established. The hybrid nano-blood is prepared by suspending solid NPs of copper and alumina into the blood in a particular order. The model equations are simplified by utilising the practical constraints of Debye-Hückel linearisation, larger wavelength, and small Reynolds number. The succeeding flow equations with the prescribed wall conditions are solved analytically. The physical consequences of emerging parameters on the axial velocity, temperature, and physical quantities of interest are pictured and illustrated graphically. The trapping bolus phenomenon is also debated under the  boluses in the arterial tract. For negative Helmholtz-Smoluchowski velocity, a reduced number of streamlines is encircled near the arterial wall. • As comparative scrutiny, it is perceived that hybrid nanoblood (Au-Al 2 O 3 /blood) plays an efficient role in the intensification of thermal conductance when collated with nano-blood and pure blood.