Mixed convection stagnation point flow of the blood based hybrid nanofluid around a rotating sphere

In this new world of fluid technologies, hybrid nanofluid has become a productive subject of research among scientists for its potential thermal features and abilities, which provides an excellent result as compared to nanofluids in growing the rate of heat transport. Our purpose here is to introduce the substantial influences of magnetic field on 2D, time-dependent and stagnation point inviscid flow of couple stress hybrid nanofluid around a rotating sphere with base fluid is pure blood, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TiO}}_{2} \,\,{\text{and}}\,\,{\text{Ag}}$$\end{document}TiO2andAg as the nanoparticles. To translate the governing system of partial differential equations and the boundary conditions relevant for computation, some suitable transformations are implemented. To obtain the analytical estimations for the corresponding system of differential expression, the innovative Optimal Homotopy Analysis Method is used. The characteristics of hybrid nanofluid flow patterns, including temperature, velocity and concentration profiles are simulated and analyzed in detail due to the variation in the evolving variables. Detailed research is also performed to investigate the influences of relevant constraints on the rates, momentum and heat transport for both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TiO}}_{2} + {\text{Ag}} + Blood$$\end{document}TiO2+Ag+Blood and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TiO}}_{2} + Blood$$\end{document}TiO2+Blood. One of the many outcomes of this analysis, it is observed that increasing the magnetic factor will decelerate the hybrid nanofluid flow velocity and improve the temperature profile. It may also be demonstrated that by increasing the Brownian motion factor, significant improvement can be made in the concentration field of hybrid nanofluid. The increase in the nanoparticle volume fraction from 0.01 to 0.02 in the case of the hybrid nanofluid enhances the thermal conductivity from 5.8 to 11.947% and for the same value of the nanoparticle volume fraction in the case of nanofluid enhance the thermal conductivity from 2.576 to 5.197%.

www.nature.com/scientificreports/ flow towards the stretching tube was explored by Yousefi et al. 15 . Subhani and Nadeem 16 studied the behavior of Cu -TiO 2 /H 2 O(hybrid nanofluid) over the stretching surface. Dinarvand et al. 17 introduced CuO -Cu/blood (hybrid nanofluid) circulation on a permeable stretching sheet, which is an advanced density concept, that can be a favorable model in medical sciences, particularly in cancer therapy and drug delivery. In an investigation for cancer cell therapy, Liu et al. 18 examined that Pt -TiO 2 and Au -TiO 2 are the best nanocomposites. Through their laboratory analysis they found in the involvement of TiO 2 or Au -TiO 2 nanoparticles, after two hours of treatment with UV irradiation the remaining fraction of cancer cell was diminished in both situations. In summary, this has been reported that most cancer cells will destroy using metal-TiO 2 nanocomposite particles than TiO 2 nanoparticles single which illustrates the need to use nano-materials in medicines. In combination with its special characteristics, Ag (silver) has various biomedical uses, such as permeability, strength, electrochemical and anti-bacterial properties. For anti-bacterial activities against the slew of micro-organisms such as microbes, protozoa, fungi, even the latest viruses, Ag and Ag-related substances are used. Their contributions show the anti-tumor effect of Ag particles which indicate that they could be a cost-effective cancer therapy then another treatment 19 . Some recent studies regarding hybrid nanofluids flows with different geometries are investigated by Zainal et al. [20][21][22] and Dinarvand [23][24][25] .
In different engineering fields, the fluid flowing around hot rotating frames has many implementations, including gravitational chemical processing, advanced technology, spin-stabilized rocket temperature control, electric pumps and generators, thermal plasma manufacturing, and painting spray processes. Takhar and Nath 26 have investigated the self-similar approach for passing flow at the stagnation point through a spinning sphere in the presence of magnetic force influences. In another work, Anilkumar and Roy 27 scrutinized time-dependent variable viscosity movement in the zone of stagnation point a spinning sphere under which the free-stream momentum and angular speed of the revolving sphere very gradually with time. The variational magnetic, thermal convectional flow via a spinning cone in the transversely isotropic porous surface was investigated by Beg et al. 28 . Chamkha and Ahmad 29 calculated the numerical result of time-dependent MHD free and mixed convectional motion and heat transport via a spinning sphere. More recently, Mahdy et al. 30 introduced a mathematical model to determine the Casson fluid flow of due to rotating sphere in the presence of conjugate MHD, entropy generation and convective boundary conditions. Although the base fluid is "pure blood" in our current model, it must be remembered that the blood viscidness is not really consistent from the medical perspective, and it will be changed not only through vessel diameter but also by temperature hematocrit factor and stress. In numerous investigations, this has been also specified that Newtonian blood performance's presumption is suitable for high shear rate motion, for instance, inflow across blood vessels. Usually, as the blood circulation improves, the blood thickness decreases, improving the speed of blood flow and reducing clotting factors 31 . The curved flexible artery through which the blood's viscosity is presumed to be temperature-dependent was studied by Akbar and Nadeem 32 . The statistical blood flow model with circular-based viscosity was examined by Gupta et al. 33 . Ijaz et al. 34 analyzed the theoretical results of hemodynamics with unique features utilizing Cu -CuO/blood, hybrid nanofluid flow in stenosized arteries. Ellahi et al. 35 conducted a numerical investigation of the peristaltic flow of a couple stress-based gold nanofluid among the hole of dual co-axial cylinders with various outlines and structures. Chahregh and Dinarvand 36 examined the circulation of TiO 2 -Ag/blood, hybrid nanofluid across a vessel for blood as well as drug transport applications in the respiration process.
Also, to the best of the authors' knowledge in the relevant literature, no one has ever tried to investigate the movement around a rotating sphere using pure blood as a based liquid and hybrid nano-materials to explain many real-life biomedical applications and medication filed and transportation. Supported by the said research findings, this article analyzes an appropriate framework of couple stress hybrid nanofluid movement of mixed convection across a spinning sphere in the immersion of MHD, thermophoresis and Brownian motion effect. The hybrid nanofluid is illustrated by maintaining two separate Titania (TiO 2 ) and silver Ag nanoparticles in the base fluid blood. The governing equations including constraints are transformed into a boundary value problem of differential equations based on the similarity transformation. Using the OHAM (Optimal Homotopy Analysis Method) in Mathematica, the system of equations is then explained analytically. Moreover, the consequences of several other variables' characteristics of flow and heat transport are seen numerically and graphically.

Mathematical formulation
In the stagnation point zone around a rotating sphere, we assume a time-dependent flow of mixed convection of the couple stress hybrid nanofluid TiO 2 -Ag/blood with the Buongiorno's mathematical model flow along with convective boundary conditions. Figure 1 demonstrates the working model and related coordinate structure of the problem. The model is totally based on the theoretical analysis and the experimental data in Table 3 is used from the existing literature reference 36 . The working problem is presented on some specific assumptions: i. The x-axis is determined on the surface of a sphere and the y-axis is normal to it. ii. The magnetic field B(t) = B 0 t −1 2 is considered perpendicular to the flow field. iii. It is supposed that the temperature and concentration at the surface of the sphere have T w , C w where T ∞ , C ∞ the ambient temperature and concentration. iv. The viscus dissipation terms are negligible. v. Many mechanisms, including thermophoresis, Brownian motion and the MHD impacts are taken into consideration. vi. The nanoparticles are supposed to be in thermal equilibrium.
In view of such flow assumptions, the resulting equations interpreted as 30 (ρcp) f heat capacity ratio with ρc p f is the blood heat capacity, ρc p hnf is the solid nanoparticles heat capacity, (D B , D T ) are the diffusion coefficients of (Brownian, thermophoresis), B 0 is the magnetic field, g is the gravitational acceleration,u, v, w are velocities element in x, y and z directions, η 0 is the couple stress parameter, µ hnf , ρ hnf , σ hnf , k hnf and ρC p hnf are viscosity, density, electrical and thermal conductivities and specific heat, where hybrid nanofluid refer hnf . All such relations 11 and defined in Tables 1 and 2. Initially, as the first nanoparticle we scattered TiO 2 (Titania) into the blood (base fluid) to produce a TiO 2 /blood(mono-nano liquid). In our current TiO 2 /blood(base liquid), Ag (Silver) is then dispersed as a supplementary nanoparticle to construct the appropriate TiO 2 + Ag/blood (hybrid nanofluid). In this case, the subscript S 1 denotes TiO 2 (Titania nanoparticles), although subscript S 2 denotes to Ag (Silver nanoparticles) and subscript f mentions pure blood (base fluid). In Tables 1 and 2, φ 1 and φ 2 are refer the volume fraction of TiO 2 and Ag nanoparticles, where φ 1 = φ 2 = 0 refer normal fluid.

Solution methodology
The newly introduced BVPh 2.0 package of OHAM 37,38 has been used to find the solution of the nonlinear problem. This package has the tendency to obtain the outputs of the modeled problem in short time. The recent solution has been achieved using the above-mentioned package. The trail solution for the modeled problem is obtained as: The square residual error for each one Eqs. (9-11) is obtained through: The sum of the total residual errors ε t m has been attained from the velocity, temperature and concentration profiles.

Result and discussion
This portion of the current study is concentrated throughout the theoretical analysis of blood base hybrid nanofluid flow around a rotating sphere. From these analyses certain important insights are produced from the graphical configurations of the velocity, temperature, and concentration profile, taking into account various groundbreaking parameters. In the presence of distinct nanoparticles, the whole theoretical analysis is based on comparing    Table 3. The comparison of the published with the present work is displayed in Table 4. Similarly, Table 5 displays the variation of both C f for TiO 2 /blood (nanofluid) and TiO 2 + Ag/blood (hybrid nanofluid). Since C f are directly related to k * , M, ( φ 1 , φ 2 ) . As a consequence, it is dug out that the remarkable nature is investigated for TiO 2 /blood (nanofluid) and TiO 2 + Ag/blood. In addition, it was clear from Table 5 that C f in the case of TiO 2 + Ag/blood (hybrid nanofluid) demonstrated superiority when equated to the TiO 2 /blood (nanofluid). The results of the current analysis for the Nu (heat transfer rate) are exposed in Table 6. It is worth mentioning that Nu is directly related to the M, N t , (φ 1 , φ 2 ) . So Nu enhances due to the intensification of M, N t , (φ 1 , φ 2 ) parameters. The percentage (%) enhancement in the heat transfer rate has been observed with the increment of the nanoparticle volume fractions φ 1 , φ 2 . Increase in the φ 1 , φ 2 from 0.01 to 0.02 in case of the TiO 2 + Ag enhance the thermal conductivity 5.8% and 11.947% respectively. Furthermore, the same value of the nanoparticle volume fraction in case of TiO 2 enhancing the thermal conductivity 2.576% and 5.197% respectively as shown in Table 6. Also, the convergence of OHAM-BVPh 2.0 package up to 25 orders of approximation is displayed in Tables 7 and 8 for TiO 2 /blood (nanofluid) and TiO 2 + Ag/blood (hybrid nanofluid) respectively. Detailed results of the model problem have been achieved and its corresponding specifications are graphically presented for component of F ′ (η) and G(η) (primary and secondary velocities), θ(η) temperature field and �(η) nanoparticle concentration filed in Figs Table 5. Evaluation of the Skin friction coefficients F ′′ (0), G ′ (0) for selected values of A = 1.6, φ 1 = 0.04.  Table 6. Evaluation of the heat transfer rate for selected values of Pr = 21, A = 0.9, using the % formula % Increase = With Nanoparticle Without Nanoparticle × 100 = Result, Result − 100 = % enhancment. www.nature.com/scientificreports/ behavior of TiO 2 /blood and TiO 2 + Ag/blood on primary velocities F ′ (η) for variations in several variables present in the equation of motion. Such as k * , φ 1 , φ 2 , M and * on flow is investigated graphically. Figure 2 depicts a particular image of the F ′ (η) (primary velocity) profile for various values of k * (couple stress variable). It is predicted from the figure that F ′ (η) (primary velocity) for both cases of fluid ( TiO 2 /blood and TiO 2 + Ag/blood ) declines as enhancing the value of k * . The explanation is so clear. By strengthening k * variable slow down the motion of both fluid ( TiO 2 /blood and TiO 2 + Ag/blood ) due to an increasing the drag force which is equal to an obvious reduction in the fluid viscosity. From such parabolic images, one should observe that the velocity of TiO 2 + Ag/blood case gives frequent decline relative to the case of TiO 2 /blood nanofluid.              www.nature.com/scientificreports/ The influences of the first TiO 2 and the second Ag nanoparticle volume fraction (φ 1 , φ 2 ) on the primary velocity field F ′ (η) is graphically presented in Fig. 3. The primary velocity field F ′ (η) is clearly seen to be remarkably declined against the enhancing value of the nanoparticle volume fraction TiO 2 and Ag (φ 1 , φ 2 ) of fluids. Physically, the higher values of the nanoparticle volume fraction of TiO 2 /blood and TiO 2 + Ag/blood causes the thinning behavior of the momentum boundary layer. This should also be observed that the sharp decline in nanofluid TiO 2 velocity is lower than that for hybrids nanofluid TiO 2 + Ag/blood . The effect of the M(magnetic parameter) on the TiO 2 + Ag/blood and TiO 2 + Ag/blood F ′ (η) primary velocities is investigated and the fundamental physics is visually illustrated in Fig. 4. Figure 4 indicates that when the M(magnetic field) is raised, then the velocity field of TiO 2 /blood and TiO 2 + Ag/blood fluid sluggish. This declining influence on the F ′ (η) velocity of the TiO 2 + Ag/blood is more prominent than TiO 2 /blood . Such decline state of velocities happens owing to the production of resistant type force identified as Lorentz force. The strength of such force enhances with the rising strength of M which counteracts the motion of fluid in boundary film and drop the viscosity of boundary film. The effect of * (mixed convection parameters) on TiO 2 /blood and TiO 2 + Ag/blood fluid of F ′ (η) primary velocity is demonstrated in Fig. 5. In order to increase the * (mixed convection parameters) the F ′ (η) primary velocity is significantly accelerated. A F ′ (η) velocity enhancing is reported, through a greater magnitude of the * which could be due to the high buoyancy force. The supporting buoyancy force works as a desirable pressure gradient and significantly speeds up F ′ (η) . This accelerating influence on F ′ (η) of the TiO 2 + Ag/blood is more prominent than that of the nanofluid. Figures 6, 7, and 8, we have plotted to describe the physical behavior of TiO 2 /blood and TiO 2 + Ag/blood on the secondary velocity G(η) field for variations in several of model equations. The physical nature of these model factors φ 1 , φ 2 , M and on flow is graphically studied. The influence of the M(magnetic field) on the both ( TiO 2 /blood and TiO 2 + Ag/blood) nanofluid G(η) secondary velocities is also examined and the essential physics is visually demonstrating in Fig. 6. Figure 6 specifies that when the magnitude of M is more elevated than the G(η) velocity field of TiO 2 /blood and TiO 2 + Ag/blood fluid slow down. Such diminishing effect on the G(η) of the TiO 2 + Ag/blood is much more prominent than the TiO 2 /blood . Such decay condition of G(η) velocities happen due to the creation of protected type force identified as Lorentz force. The power of such force improves with the growing strength of M which offsets the speed of both fluids within boundary layer and drop the thickness of the boundary layer. The G(η) vs first TiO 2 and second Ag nanoparticle volume fraction (φ 1 , φ 2 ) behaviors are seen in Fig. 7, that indicates that G(η) of both ( TiO 2 /blood and TiO 2 + Ag/blood) decreases as the (φ 1 , φ 2 ) increases. Substantially, the higher values of both (φ 1 , φ 2 ) of nanofluid and hybrid nanofluid cause the thinning behavior of the momentum boundary layer. This should also be observed that the sharp decline in nanofluid TiO 2 velocity is lower than that for hybrids nanofluid TiO 2 + Ag/blood . The effect of the (rotation parameter) of TiO 2 /blood and TiO 2 + Ag/blood fluid on G(η) secondary velocity components is seen in Fig. 8. In order to upsurge the magnitude of , the G(η) secondary velocity components are significantly accelerated. It is clear from the diagram that the rotation becomes much more intense by increasing the λ, and the G(η) secondary momentum helps more through the swirl effect that retards the secondary movement G(η) . From this parabolic plot, one should perceive that the velocity of TiO 2 + Ag/blood case gives frequent increase relative to the case of TiO 2 /blood nanofluid. The Figs. 9, 10, 11, 12, and 13, afterward demonstrating the effect of Pr, M, A, (φ 1 , φ 2 ) and N t on θ(η) for both ( TiO 2 /blood and TiO 2 + Ag/blood). To scrutinize the trend of Pr on θ (η) , Fig. 9 is plotted for both cases of ( TiO 2 /blood and TiO 2 + Ag/blood). It is evident from the plot that the θ(η) field shows the diminishing role for Pr . Consequently, it is defensible owing to the basic reason that thermal conductivity of the fluid relative lesser with an intensified magnitude of Pr and thus reducing the temperature of both ( TiO 2 /blood and TiO 2 + Ag/ blood). It's also reported herein that in the scenario of TiO 2 + Ag/blood the θ (η) temperature field reached its maximum value as compared to the TiO 2 /blood nanofluid case. The variance in M (magnetic parameter) on temperature θ(η) profile is depicted in Fig. 10 for both ( TiO 2 /blood and TiO 2 + Ag/blood). Form this drawing it is detected that θ(η) and thermal layer thickness enhances via growing magnitude of magnetic parameter M . The magnetic parameter and density of ( TiO 2 /blood and TiO 2 + Ag/blood) are inversely related with each other. Hence, the strengthening value of M shrinking the density and as a consequence the thermal nature of fluid upsurges. From this outline, one can observe that the temperature for TiO 2 + Ag/blood case tends to decrease more quickly relative to the caseof TiO 2 /blood . The effect of the A (unsteadiness parameter) on both ( TiO 2 /blood and TiO 2 + Ag/blood), θ(η) temperature profiles is revealed in Fig. 11. This can be seen from a plot that are www.nature.com/scientificreports/ rising the magnitude of A leads to a reduction in the temperature field. Physically, with increasing values of A , the θ(η) field is reduced, indicating that the thermal boundary surfaces are diminished with the growing magnitude of A . Therefore, the boundary layer is cooler, so additional temperature and nanoparticles are moved to the surface of the sphere (wall) with a growing magnitude of A . It's also described herein that in the scenario of TiO 2 + Ag/blood the θ(η) temperature field reached its maximum value as compared to the TiO 2 /blood nanofluid case. The effect of (φ 1 , φ 2 ) on θ(η) is also studied and the basic reason is obviously portrayed in Fig. 12.
From the drawing it is observed that the θ(η) is enhanced for the increase in the volume concentration of both nanoparticles TiO 2 /blood and hybrid TiO 2 + Ag/blood , which can be attributed to the collisions between the suspended TiO 2 , Ag nanoparticles. (φ 1 , φ 2 ) is basically link to both TiO 2 , Ag nanoparticles, therefore, the effect of both nanoparticles on θ(η) is observed. Mostly, the stimulation of thermal boundary film viscosity to (φ 1 , φ 2 ) is linked to upgraded thermal conduction of both ( TiO 2 /blood and TiO 2 + Ag/blood ). In fact, superior amount of thermal conduction is reinforced by improved thermal diffusivity. Actually, θ (η) field is directly related to both (φ 1 , φ 2 ) in case of contraction and in this scenario, it displays extra inconsistency closed to the wall of the sphere. Due to evolving N t (thermophoresis parameter), this paragraph is dedicated to capturing the variation in the θ(η) temperature profile for both nanoparticles TiO 2 /blood and hybrid TiO 2 + Ag/blood . Figure 11 is depicted in order to observe the way that θ (η) temperature is affected. Figure 13 illustrates that the magnitude of both ( TiO 2 /blood and TiO 2 + Ag/blood ) temperature grows by enhancing N t . Basically, it is how the thermophoresis force allows the particles to shift from the hot zone to the cold area, improves as N t raises and the thermophoresis force improves which upgrades the magnitude of θ(η) temperature field magnitude of nanofluids TiO 2 /blood and hybrid nanofluids TiO 2 + Ag/blood . Figures 14, 15, and 16, we have designed to define the physical characteristics of TiO 2 /blood and TiO 2 + Ag/blood on concentration profile �(η) for variations in several variable present in the model equations. Such as Sc, (φ 1 , φ 2 ) and N b on concentration profile �(η) is investigated graphically. The consequence of the Sc (Schmidt Number) on �(η)(concentration) profiles is portrayed in Fig. 14 for the case of ( TiO 2 /blood and TiO 2 + Ag/blood ). The �(η) magnitude is reduced slightly substantially with rising values of Sc . Basically, the relation of momentum diffusivity to mass (nanoparticle) diffusivity is called Sc . For Sc < 1 , diffusivity in mass dominates diffusivity in momentum, and vice versa for Sc > 1 . Hence, with higher Sc , concentration boundary-layer density for both nanoparticles TiO 2 , Ag is substantially enhanced, while the thermal boundary-layer density is marginally reduced. In order to see the influence of volumetric fractions of nanomaterials (φ 1 , φ 2 ) on the concentration profile �(η) , Fig. 15 has been plotted for both ( TiO 2 /blood and TiO 2 + Ag/blood). It is noticeable that the decline in solute concentration was perceived by growing the volumetric fraction of nano additives (φ 1 , φ 2 ) . In addition, lower values for TiO 2 + Ag/blood, hybrid nanofluid were observed in the �(η) profile when equated to TiO 2 /blood nanofluid. The effect of the �(η) concentration profile of the N b (Brownian motion parameter) is displayed in Fig. 16 for the case of ( TiO 2 /blood and TiO 2 + Ag/blood). An improvement to N b lead to accelerates �(η) as seen in Figure. Physically, it is attributable to disperse through the ( TiO 2 , Ag ) particles, fluid improvements with the maximization of N b . It is therefore understood that in the case of TiO 2 + Ag/blood, hybrid nanofluid, as can be seen in the figure the temperature gets its highest values as compared with TiO 2 /blood nanofluid.

Concluding remarks
The circulation of the TiO 2 + Ag/blood, hybrid nanofluid was examined around a rotating sphere under the action of a uniform applied magnetic field. The velocity, temperature along with concentration distribution, was also scientifically and systematically analyzed by taking the effects of N b ,N t and (φ 1 , φ 2 ) into consideration. The main results of this analysis were the following: • The F ′ (η) component of primary velocity, declined with the boosting value of (φ 1 , φ 2 ) and M , while an accelerating behavior was perceived via enhancing value of * . • Also, the G(η) secondary velocity component, boosted with the enhancing value of , where declinatory conduct was found through increasing values of (φ 1 , φ 2 ) and M. • In contrast to TiO 2 nanofluid, the TiO 2 + Ag/blood (hybrid nanofluid) demonstrated a weak velocity profile of F ′ (η) and G(η). • For the growing values of (φ 1 , φ 2 ), N t and M , temperature elevation was chronicled. In those scenarios, the TiO 2 + Ag/blood (hybrid nanofluid) exhibited the greatest temperature as equated to TiO 2 nanofluid. • With the increasing value of N b , an increase in the �(η) was observed. However, an opposite pattern was reported in increasing (φ 1 , φ 2 ) and Sc.
• Nu and C f demonstrated considerable superiority in the case of TiO 2 + Ag/blood (hybrid nanofluid) as equated to TiO 2 nanofluid. • There was considerable realistic relevance to the current study, particularly in the field of biomedical and chemical industries. • The output shows that the increase in the φ 1 , φ 2 from 0.01 to 0.02 in case of the TiO 2 + Ag enhance the thermal conductivity 5.8% and 11.947% and for the same value of the nanoparticle volume fraction in case of TiO 2 enhancing the thermal conductivity 2.576% and 5.197%.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.