Darcy–Forchheimer couple stress hybrid nanofluids flow with variable fluid properties

The current study provides a detailed analysis of steady two-dimensional incompressible and electrically conducting magnetohydrodynamic flow of a couple stress hybrid nanofluid under the influence of Darcy–Forchheimer, viscous dissipation, joule heating, heat generation, chemical reaction, and variable viscosity. The system of partial differential equations of the current model (equation of motion, energy, and concentration) is converted into a system of ordinary differential equations by adopting the suitable similarity practice. Analytically, homotopy analysis method (HAM) is employed to solve the obtained set of equations. The impact of permeability, couple-stress and magnetic parameters on axial velocity, mean critical reflux condition and mean velocity on the channel walls are discussed in details. Computational effects show that the axial mean velocity at the boundary has an inverse relation with couple stress parameter while the permeability parameter has a direct relation with the magnetic parameter and vice versa. The enhancement in the temperature distribution evaluates 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}$$SWCNTs\,\,{\text{and}}\,\,MWCNTs$$\end{document}SWCNTsandMWCNTs hybrid nanofluids are used in this study for medication purpose.

The study of electrically conducted incompressible fluid flows is termed as magnetohydrodynamic (MHD). The forces such as fluid's Lorentz force have significant observations like the global magnetic field effects upon the Earth. This type of arena is certainly created with the help of strong Lorentz forces that are mainly present in Earth liquid core. Since the Lorentz forces are less common in our routine life observations that make the concept of magnetohydrodynamics difficult to understand. In present study, the role of Lorentz forces on fluids has established by taking electrically conducted fluid and propulsion of a magnetohydrodynamic ship. In review of 19 , this propulsion technique 20 is attractive in several characteristics, as magnetohydrodynamic (MHD) propulsion does not need any movable parts. There are numerous applications of MHD propulsion as far as the high speed ships are concerned for naval submarines 21 . Baumgartl et al. 22 have inspected the evaluation of MHD effects upon time dependent and independent fluid flow. In this work an exterior weak magnetic field has applied to the flow system. A detail analysis 23,24 has carried out with main focus upon the impact of MHD fluid flow using different flow conditions and geometrical view of the flow system. The magnetohydrodynamic propulsion could be created in several techniques. As electro hydrodynamics is the study of electrically conducting ionized particles motions or atoms and their transportations with the neighboring liquids and electric fields. These particles, molecules or atom and liquid transportation process consists of electro-osmosis, electro-kinesis, electro-phoresis and electro rotation fusing metals 25 in nuclear reactor and electric heater. Andersson 26 has originated closed form solution for the incompressible fluid flow over the surface which was stretching. Over an exponential surface, the numerical and analytical solution for the incompressible fluid flow with the exponential jump of temperature has explained by Magyari and keller 27 . Partha et al. 28 have studied the mutual influence of dissipation and convective flow past a stretched surface. The energy transportation and numerical simulation of viscous fluid flow past a stretching sheet has designed by Elbashbeshy 29 . Ellahi et al. 30 have applied HAM method for the investigation of 3D flow of Carreau liquid using magnetic effects. Rashidi et al. 31 have investigated the Burger's model for nanofluid flow under the impact of magnetic effects.
In 1856 the Henry Darcy has investigated the flow of homogeneous fluids through madium, consisting of void spaces or pores (termed as porous medium). The work has carried out for small velosities and low permeable media. Later, Forchheimer 32 has overcome the drowbackes of Darcy work by inserting the square of flow term in flow eqiuation. Muskat 33 has identified the addtional term as 'Forchheimer' for the first time. Afterwards, a number of investigation have been conducted by differnt people using differnt geometries for fluid flow and heat transfer through porus media. Pal and Mondal 34 have discussed the the mixed convection flow past a permeable medium with differnt flow conditions. Ganesh et al. 35 have inspected the nanofluid flow past a shrinkng and stretching porous surface with application of second order slip condition. Seddeek 36 has discussed the combined effects of themophoresis and viscous dissipation for investigation of mixed convective Darcy Forchheimer flow through a permeable surface. It has observed in this study that, the flow has reduced with augmentation in inertia cofficient and porosity parameter. Hayat et al. 37  In daily life, most of the physical phenomen are non-linear rather some are highly nonlinear phenomenon. Solution of such complex and complicated physical problems is very difficult, even in some cases it becomes impposible to obtaine the analytical solution. In order to solve such problems most of the investigators are emloying different numerical or analytical techniques. Out of such techniques, HAM [39][40][41][42][43][44] is also useful for solution of such problems.
Principal aim of this research is to inspect the heat transmission and the influence of electro-magnetic effects upon MHD flow of a couple stress hybrid nanofluids over a Darcy-Forchheimer model in a symmetric flow with variable viscosity. The equations investigating the electro-magneto hydrodynamic of MHD flow for a hybrid fluid have converted to non-dimensional notation with suitable variables. The semi-analytical technique HAM is employed to solve the obtained set of equations. The impact of permeability, couple-stress and magnetic parameters on axial velocity, mean critical reflux condition and mean velocity on the channel walls are examined in details. The augmentation in the temperature distribution assesses the pH values and electric conductivity. Consequently, the SWCNTs and MWCNTs hybrid nanofluids are utilized in this study for medication purpose.

Mathematical modeling
Take a two-dimensional electro-hydrodynamic flow of viscous liquid couple stress nanofluid past a stretching surface. The fluid is stabilized by the collective effects of electric and magnetic fields. The effects of Joule heating and viscous dissipation have also been considered for the current flow system in order to control its thermal characteristics. The mathematical expression for Lorentz force is described as � J × � B in which the magnetic field is represented by B while current density by J . Mathematically the expression for J by Ohm's law is described as Where ' E ' stands for electric field such that � E = 0 , while the field electrical conductivity is given by ' σ ' . The temperature of nanofluid at surface of wall is T w , whereas at the free stream it is T ∞ . The complete geometry of the current model is shown in Fig. 1.
By applying the above suppositions, resultant equations are: The subjected conditions at boundary are: is Non-uniform inertia coefficient of porous medium. The kinematic viscosity is ν hnf , heat diffusivity is α hnf , heat capacity is ρc p hnf , dynamic viscosity is given by µ hnf , and Q 0 is heat source. C denotes nanoparticle concentration, k r chemical reaction. Thermophysical properties. The mathematical expression for thermophysical properties of base and hybrid nanofluid are described as 4-8 : The following appropriate variables are suggested: Using Eq. (10) in Eqs. (1)(2)(3)(4)(5) we have With inter-related boundary conditions: In above equations the dimensionless couple stress parameter is k , the Prandtl number is Pr , M is magnetic parameter,Q is heat flux parameter,E is electric field parameter and Ec is Eckert number. The mathematical descriptions for these parameters are given as follows: Engineering quantities. The Skin friction, heat and mass fluxes have numerous uses in engineering field.
The mathematical expression for these quantities is given as: .

Discussion of results
In this work we have thoroughly inspected the flow of fluid and transmission of heat for a couple stress nanoliquid inserted among the viscous fluid packed through a horizontal conduit. We shall discuss the impact of different physical factors upon the fluid flow system in following paragraphs with the help of graphical view.
Velocity profile. The influence of different physical parameters such as E, M, , F r , k upon flow field F ′ (η) has presented in Figs. 2, 3, 4, 5 and 6. Figure 2 exposed the effect of electric field on velocity field F ′ (η) . In the presence of electrical effects, the field of velocity decreases and at a certain distance away from the wall, it rises closed to the nonlinear stretching sheet. In Fig. 3 we found that, the magnetic effect has reduced the flow field. Actually, for increasing values of M there is a generation of Lorentz force in opposite direction of flow field that declines the velocity filed. This physical phenomenon augments the thermal and concentration characteristics. The effect of porosity parameter λ shows a decrement in the flow profile as depicted in Fig. 4. Physically, for In this process the flow of fluid declines. Figure 5 depicts the impact upon flow profile for augmentation in inertia coefficient F r . From this figure, it has perceived that an augmentation in F r results in generation of resistive force to fluid motion, that result a reduction in flow profile. Figure 6 calculates the influence of k upon flow profile. It has revealed from this figure that, for augmenting values of k the viscous forces will jump up that acts as a reducing agent to fluid motion. Hence the flow profile declines for growth in k.    Fig. 7. Clearly the electric filed is directly proportional to the rise in temperature. Hence increase in E accelerates the Lorentz force due to which temperature of nanoparticles increases. Figure 8 depicts that hike in magnetic parameter corresponds to an augmentation in temperature profile. Physically, higher values of M pushes the Lorentz force that creates a resistance to the fluid motion. In this phenomenon the thermal profile jumps up. The increase in Eckert number Ec causes an augmentation in thermal flow as portrays in Fig. 9. Actually, due to higher values of Ec the fluid friction amongst nano-      www.nature.com/scientificreports/ profile is displayed through Fig. 11. This figure illustrates that an upsurge in Schmidt number with respect to a weaker solute diffusion allows a deep penetration of solutal effect. As a result, the concentration decreases with increase in Sc. Figure 12 depicts that, rising in the chemical reaction parameter rapidly reduces the concentration profile. The major reason is that, the number of molecules of solute involved a chemical reaction increase with rise in chemical reaction parameter, which results in decrease of concentration field. Moreover, it is verified that the concentration of profile becomes sharp for augmentation in chemical reaction. The Fig. 13 clearly shows the effects of volume friction upon concentration. Physically, the thermal conductivity increases by boosting the volume concentration of nanoparticles and as a result the nanoparticles act as bridge to pass the heat flow. In this physical phenomenon the concentration profile jumps up as depicted in Fig. 13.
Tables discussion. The physical parameters influenced on the drag force for the nanofluid and hybrid nanofluid have been demonstrated in Table 1. The higher magnitude of parameters φ 1 + φ 2 , �, M, k, Fr, increasing the skin friction coefficient and these consequences are relatively larger in the hybrid nanofluids. The electric field parameter declines the drag force for its increasing value and very small difference between the traditional and hybrid nanofluids has obtained. The rate of heat transfer for nanofluid and hybrid nanofluid versus different parameters has been displayed in Table 2. The augmentation in φ 1 + φ 2 , M, Q, Ec is enhancing the rate of thermal transmission while the increasing value for viscosity parameter declines this physical quantity. The characteristics of the Schmitt number Sc , chemical reaction parameter γ and nanoparticle volume fraction φ 1 , φ 2 are displayed in Table 3. It has been revealed that greater values of these parameters have augmented the concentration rate.

Conclusions
Health acquired infections (IACs) is a main public health issue worldwide. Whereas CNTs nanofluid plays its important role as antimicrobial. Carbon properties have a strong antimicrobial perspective and CNTs nanofluids are used in the Escherichia coli culture to assess their antibacterial potential. The improvement in the temperature distribution appraises the pH values and electric conductivity. Thus, the SWCNTs and MWCNTs hybrid nanofluids are used in this study for medication purpose.   The core purpose of this research is to inspect the heat transfer and the effect of electro-magnetic field upon flow of a couple stress hybrid nanofluid over a Darcy-Forchheimer model in a symmetric flow with variable viscosity. Analytically, HAM is employed to solve the obtained set of equations in non-dimensional arrangement. The impact of encountered parameters has also discussed in detail. During this deep discussion, the underlined points have been revealed: • In the presence of electric effects, the field of velocity decreases and at a certain distance away from the wall, it rises closed to the nonlinear stretching sheet.   www.nature.com/scientificreports/ • For increasing values of magnetic effect, there is a generation of Lorentz force in opposite direction of flow field that declines the velocity filed. • For increasing values of porosity parameter the void spaces in the medium augments that offers more resistance to the flow of fluid and declines the flow profile. • Augmentation in inertia coefficient results in generation of resistive force to fluid motion that causes a reduction in flow profile. • For augmentation in Prandtl number the thermal diffusion reduces due to which less heat transfer takes place that ultimately declines the thermal profile. • An increase in electric field accelerates the Lorentz force due to which temperature of nanoparticles increases.
• Augmentation in magnetic parameter corresponds to an augmentation in temperature profile.
• For higher values of Eckert number, the fluid friction amongst nanoparticles generates with more intensity.
In this physical phenomenon the kinetic energy transformed to thermal energy that finally supports the augmentation in thermal profile. • Augmenting values of volumetric friction causes increase in the dense behaviour of the fluid particles that depreciates the fluid flow and appreciates the thermal behaviour of fluid particles. Moreover, the concentration of fluid also rises in this phenomenon. • An upsurge in Schmidt number results a reduction in concentration profile.
• Rising values in the chemical reaction parameter rapidly reduces the concentration profile.