MHD hybrid nanofluid flow comprising the medication through a blood artery

The current study focuses on the laminar flow of copper and copper oxide (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Cu/blood}}$$\end{document}Cu/blood and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Cu}} + {\text{CuO/blood}}$$\end{document}Cu+CuO/blood) hybrid nanoliquid, considering blood as a carrier fluid in a rectangular domain between two permeable channels. This study may manipulate for the purpose such as the drug delivery process, flow dynamic mechanism of the micro-circulatory system. In the proposed model, MHD and heat source/sink on the flow pattern have been studied. Furthermore, the sides of each channel are permeable, allowing the nanoliquid to escape, filter, squeezing and dilating with a fixed velocity. Appropriate transformations are incorporated to convert the governing partial differential equations and the boundary conditions suitable for computation. The elegant homotopy analysis method (HAM) is used to obtain analytic approximations for the resulting system of nonlinear differential equations. The features of flow characteristics such as velocity, and temperature profiles in response to the variations of the emerging parameters are simulated and examined with a physical explanation. The magnetic field plays a vital role in the blood flow and therefore the existing literature has been extending with the addition of magnetic field. Among the many outputs of the study, it is found that the pressure distribution decline with the accumulated values of the magnetic parameter at the center of the flow regime. The augmentation in the temperature distribution estimates the pH values and electric conductivity. Therefore, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Cu}}\,\,{\text{and}}\,\,{\text{CuO}}$$\end{document}CuandCuO hybrid nanofluids are used in this study for medication purposes. The magnetic field has an important role in the blood flow and therefore the extending study has been extending using the magnetic field. The heat emission/absorption term is added to the energy equation to maintain the homogeneous temperature for the blood flow. We expect that this work will provide efficient outputs for medical purposes such as drug delivery.

www.nature.com/scientificreports/ oncology, immunology and cardiology 2 . Many research articles exist associated with the nanoliquid in which the enhancement of the heat transmission rate is elaborated with the standard carrier liquids [3][4][5][6][7][8] . The silver Ag and aluminum Al have high reactivity as well as excellent thermal properties stability. Recently, a researcher has presented a new sort of nanoliquid to overcome the above limitation and enhances thermal characteristics by making a compound of different elements in a base fluid known as hybrid nanoliquids. This new emerging variety of nanoliquid has been utilized in several fields' aspects as heat transmission, energy, medical, nano-electronics, and transportation in the naval system etc.
There are several of scientific articles related to hybrid nanoliquid are reported, which govern the enhancement in thermophysical property [9][10][11][12][13] . Ali and Sajid 14 discussed the thermal conductivity of a hybrid nanoliquid through experimental and computational research. It the problem has been concluded that the rising trend of mass and energy fields increases the thermal conductivity. Furthermore, an appropriate choice of nanoparticles performs a mandatory role in hybrid nanoliquid stability. Rostami and Dinarvand 15 studied the medical model such as liquid flow in the capillary drug delivery, blood oxygenation and particularly in the micro-circulatory system. They examined an electrically conducting hybrid Cu + CuO/blood nanoliquid laminar flow passing through a horizontal surface. Later on, they rebound in another problem, the flow of hybrid TiO 2 + Ag/blood nanoliquid through a porous tube 16 . Their study provided a basic model for blood circulation in the respiratory system and drug delivery.
Ali et al. 17,18 investigated the hybrid nanofluid peristaltic flow considering the TiO2-Cu and Cu-Al2O3 hybrid nanomaterials together with MHD and Jeffery fluid. The size and shape dependence effect of the heat transfer for cooling/heating purposes were studied by Saleem et al. 19 . The hybrid nanofluid tends to enhance the mass and heat transfer which have many applications in the pharmaceutical industries, preservation and the circulation of blood, and so on. Many researchers used the nanofluids/hybrid nanofluids in the verities of geometries [20][21][22][23] .
The liquid type substance, which circulates throughout the body and animals, is recognized as blood. Since the fluid in the present study is considered the purest blood, it is observed that the blood viscosity is not static according to the medical concept, and can control body pressure, hemoglobin ratio, and temperature and vessels/ artery size as well. The large blood vessel is considered in this study. So, considering blood nature as suitable as a newtonian fluid. Politis et al. 24 has implemented the numerical method by considering the Newtonian blood flow conditions and stable state, to investigate the several blood stream configurations. They concluded that the hemodynamic factors could be affected via different features. Koriko et al. 25 has introduced the gold-blood concept over a horizontal upper side of a paraboloid. Later, Ijaz et al. 26 scrutinized theoretically the CuO/blood transportation through the stenosis artery with the special characteristics of hemodynamics. The numerical calculation of blood in veins and arteries using non-Newtonian and Newtonian approaches with heat flux for biomedical sciences is simulated by Foong et al. 27 . The comparative investigation of copper/blood and copperoxide/blood nanoliquids flow through the asymmetric channel is made by Bharathi and Prakash 28 .
The erythrocyte cell is an important magnetic factor, which is mainly possible that it will affect blood flow through arteries. Chen et al. 29 has conducted a theoretical result by assuming magnetic effects on blood flow. The conducting fluids are mainly used for several applications in the biomedical field, including cellular separation, magnetic resonance imaging (MRI) or even as a contrast agent, as an active material quantification, treatment of tumor infected cells, MHD micropump and biomedical sensors 30 . Similarly, a magnetic role is also being examined for electromagnetic devices, cancer therapy, hydro-magnetic motors are also representing its importance. The existing and coming new development in MHD technologies has vast and real practical life uses 31 . Shamlooei and Sheikholeslami 32 have studied the upshot of radiation on MHD nanoliquid by manipulating the finite element volume method. The MHD based hybrid nanofluid flows have been studied in [33][34][35][36][37] .
The study of the flow via contracting/enlarging the permeable object like channel or pipe has received the concentration of many researchers. They take part in many technical, biophysical and engineering issues. The role of such flow comprises heat exchanger, combustion chamber, and porous walled flow system and exhaust nozzles. Furthermore, such fluid flow in and animal bodies can be simulated as a flow through a contracting substance with pores allowing essential particles. Akinshilo 38 presented the concept of nanoliquid flow passes through a permeable channel with contracting and expanding walls, which play an important role in shear stresses and heat transmission control. Ahmad et al. 39 examined the carbon nanotubes based nanoliquid flows in squeezing and dilating rectangular porous wall channel. Several mathematicians and scientists have recently investigated such type flow models, which are elaborated through [40][41][42] .
The extension in the existing study 16 has been pointed out as: • The magnetic field has an important role in the blood flow and therefore the extending study 16 has been extending using the magnetic field. • Health acquired infections (IACs) are a main public health issue worldwide. Whereas Cuo nanofluid plays its important role as antimicrobial. ( CuO ) properties have a strong antimicrobial perspective and CuO nanofluids are used in the Escherichia coli culture to assess their antibacterial potential. The enhancement in the temperature distribution evaluates the pH values and electric conductivity. Therefore, the Cu and CuO hybrid nanofluids are used in this study for medication purposes. • The variation in the temperature distribution is a regular phenomenon in the blood flow and in this study the energy equation is extended with the heat omission/absorption parameter. • To tackle the solution, we applied the homotopy analysis method (HAM) [43][44][45][46][47][48][49][50] . The present study has a significant role in the physiological side. It is mostly applicable to the circulation of blood and drug delivery.
Mathematical formulation. The copper oxide ( Cu + CuO/blood ) hybrid nanoliquid flow with couple stress through a permeable channel in a rectangular domain is studied. In Fig. 1, the flow mechanism is shown.
Here, η , ρ hnf and µ hnf is density and dynamic viscosity, σ hnf and k hnf is the electrical and thermal conductivity of the hybrid nanofluid. ρC p hnf is the heat capacitance, Q 0 is a heat source/sink substrate, P is the pressure and h nf manifest hybrid nanoliquid.
The boundary conditions are 16,38 : The blood is considered as the base fluid. In which, we have disseminated Cu nanoparticles in the carrier fluid to synthesis mono-nanoliquid ( Cu/blood ). Then CuO nanoparticles are scattered in copper solution Cu/blood , to get the desired hybrid nanoliquid Cu + CuO/blood.

After putting
The cross differentiation has been used to neglect Pỹx from Eqs. (2, 3).  Transformation parameters. For unsteady blood flow the non-dimensional distance used as in 16 and 38 : Now, the similarity transformation are: The physical conditions are depleted as Here, Re explains the Reynolds number that is deliberating positive and adverse for injection and suction respectively.
Consequently, to dimensionless the velocity, we introduce: The above-mentioned substrates in Eq. (10) are used to get the momentum equation as:

And
Here S = A l A u defines the permeability parameter. On behalf of the above transformations the energy equation is transformed as:
The normal pressure is defined as: The wall deformation rate, heat source variable and Magnetic variable define as: The simplified form of the drag force and rate of heat transfer for the hybrid nanofluid fluid at the both the lower and upper walls are defined as 16 : www.nature.com/scientificreports/ Solution procedure. The HAM opts to model equations for analytic results, which was presented by Liao 42,43 . To show the convergence, the sum of square residual error is achieved by using the BVP 2.0 package [44][45][46][47][48][49][50] . The initial approximation for velocity f 0 and temperature θ 0 are given as The linear operators are presented as: The expand form of The idea of the BVPh 2.0 package as defined by Liao 42 applied to the modeled Eqs. (13,15) with the initial and boundary conditions in Eqs. (14,16) as: The total sum of the square residual is defined as Liao 41      www.nature.com/scientificreports/ strength of the magnetic field, which causes the nanoliquid flow to reduce inside the channel. It also inversely affects the viscosity of blood. Figure 3 revealed the nature of the velocity profile f ′ (η) versus the suction/injection parameter. We perceived that, by rising credit of parameter α , the velocity profile f ′ (η) gradually reduces, but an opposite seen has been observed as we move towards the hemodynamic boundary layer center. Consequently, it can be noticed that in both situations (injection/suction), when the channel sides expand α > 0 , it improves the velocity f ′ (η) . Physically, when channel walls expand, space is generated near the channel wall. So, to cover that palace the nanoliquid flow move towards the center of the channel walls. On the other hand, when channel walls contract by taking α < 0 . The absolute credit of α leads to an enhancement in the flow field near the side or walls. As a consequence of this, the squeezing, as well as reduction in the flow field, was observed to keep momentum conservation maintain.
The upshots of volume fraction φ 2 ( CuO ) on temperature θ(η) are depicted in Fig. 4. It can be perceived that the fluid θ(η) is rising for the positive increment of copper-oxide volume fraction φ 2 ( CuO/blood ). Physically, the increasing credit of φ 2 reduces the specific heat capacity of nanoliquid, because the specific heat capacity of CuO is less than blood. Furthermore, it also enhances the thermal expansion of nanoliquid, that's why the fluid temperature enhances with the increases of φ 2 . Figure 5 expresses the consequences of the Prandtl number Pr on temperature θ(η) profile. Physically, higher Prandtl fluid reduces the thermal boundary layer as well as has less thermal diffusivity, that's why temperature decline with the effect of the Prandtl number Pr. The effect of S in the case of contraction and injection as well as dilation and injection is drawn in Fig. 6. So, the temperature profile enhances with the rising credit of |S|. However, upper wall temperature is slightly affected by the variation |S|. Because, T l > T u , that's why the upper wall is less affected due to injection, the fluid near the wall carries more heat as compared to the upper wall. Figures 7 and 8 illustrate the trend of θ(η) versus the variation of heat absorption (Q < 0) and heat generation (Q > 0) . Variation in (Q > 0) bringing an enhancement in the θ(η) field, because the generated energy raises the thermal gradient. That's why the growing value of (Q > 0) term augmenting the temperature distribution. Further, the behavior of θ(η) profile against the negative value of Q is illustrated in Fig. 8. The negative values of Q decline the temperature profile. Because the heat absorption factor absorbs the heat from the channel surface which reduces the fluid temperature.
The influence of the magnetic parameter M has been displayed in Fig. 9 respectively. The pressure distribution decline near the center of the two walls with the increasing values of these parameters and boost up away from the center. In fact, during fluid flow in a channel, the pressure distribution is strongest in the center; therefore, the parameter effect is mainly counted at the center of the channel. Also, the rising values of these parameters   Table 1. displays the thermophysical properties of copper Cu and copper oxide, CuO hybrid nanoliquid fluid. Tables 2 and 3 scrutinized the flow model for Cu/blood and CuO/blood respectively. Table 4 illustrates the Skin friction at lower and upper cases versus different values of physical constraints such as Re, M. The increasing value of these constraints boosts up the skin friction at both walls. The skin friction using hybrid nanofluid is more effective. In fact, these parameters enhance the resistive force against the fluid flow, and hybrid nanofluids are comparatively denser than common fluids. The influence of the physical constraints over the heat transfer rate has been detected in Table 5. The rise in the value of the volume fractions φ 1 , φ 2 improves the heat transfer rate. It has been detected that the hybrid nanofluids are more efficient to enhance the heat transfer rate as compared to conventional fluids. The Prandtl number Pr for the blood is initially 21 and increasing the value of Pr decline the heat transfer rate. This effect is also very efficient using the hybrid nanofluid as displayed in Table 5. The parameter Q > 0 raises the heat transfer rate and the effect is more influential using the hybrid nanofluid. The comparison of the present work with the existing literature 16 using the HAM technique has been analyzed in Table 6. The closed agreement of the present study with the existing study authenticates the obtained results.

Conclusion
The flow of the hybrid nanofluid flow through a deformable asymmetric channel of permeable walls has been analyzed. The Cu and CuO nanoparticles are suspended in the blood. The heat transfer rate and skin friction for the proposed model are calculated numerically and discussed. The HAM technique has been used for the solution of the problem. The obtained outcomes are pointed out as:      15,27 .  15,27 .     Table 6. Comparison of the present study and existing literature 16  www.nature.com/scientificreports/ the flow regime. The variation in these parameters can play an important role in controlling the pressure distribution during the drug delivery to sustain the stability of blood flow in the vessels. • It has been noticed that in both situations (injection/suction), when the channel sides expand α > 0 , it improves the blood velocity. On the other hand, when channel walls contract by taking α < 0 . As a consequence of this, the squeezing, as well as reduction in the flow field, was observed to keep momentum conservation maintain. • The CuO nanofluid plays its important role as antimicrobial. ( CuO ) properties have a strong antimicrobial perspective and CuO nanofluids are used in the Escherichia coli culture to assess their antibacterial potential. The focus has been given to the mentioned applications using the variation in the nanoparticle volume fraction. • The enhancement in the temperature distribution evaluates the pH values and electric conductivity and therefore, the Cu and CuO hybrid nanofluids are used in this study for medication purposes. • The variation in the temperature distribution is a regular phenomenon in the blood flow and in this study, the heat omission/absorption parameter plays the role to maintain a reasonable temperature for the blood supply. • The outcomes show that the hybrid nanofluid ( Cu + CuO/blood ) is more effective for the thermal analysis and drug deliveries in the contracting expanding channels.

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