Impact of temperature dependent viscosity and thermal conductivity on MHD blood flow through a stretching surface with ohmic effect and chemical reaction


 A study has been carried for a viscous, incompressible electrically conducting MHD blood flow with temperature-dependent thermal conductivity and viscosity through a stretching surface in the presence of thermal radiation, viscous dissipation, and chemical reaction. The flow is subjected to a uniform transverse magnetic field normal to the flow. The governing coupled partial differential equations are converted into a set of non-linear ordinary differential equations (ODE) using similarity analysis. The resultant non-linear coupled ordinary differential equations are solved numerically using the boundary value problem solver (bvp4c) in MATLAB with a convincible accuracy. The effects of the physical parameters such as viscosity parameter 

 
 
 
 
 
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 μ
 (
 
 
 T
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 \left({\mu ({{\tilde T}_b})} \right)
 
 
, permeability parameter (β), magnetic field parameter (M), Local Grashof number (Gr) for thermal diffusion, Local modified Grashof number for mass diffusion (Gm), the Eckert number (Ec), the thermal conductivity parameter 

 
 
 
 
 
 (
 
 K
 (
 
 
 T
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 b
 
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 \left({K({{\tilde T}_b})} \right)
 
 
 on the velocity, temperature, concentration profiles, skin-friction coefficient, Nusselt number, and Sherwood number are presented graphically. The physical visualization of flow parameters that appeared in the problem is discussed with the help of various graphs to convey the real life application in industrial and engineering processes. A comparison has been made with previously published work and present study revels the good agreement with the published work. This study will be helpful in the clinical healing of pathological situations accompanied by accelerated circulation.


Introduction
The analysis of boundary layer ow over a stretching surface has received so much attention from researchers and scientists due to its wide usage in the chemical, food, petroleum industry etc. It also plays a crucial role in the bio-medical eld because there is a signi cant variation in human anatomy, physiology and stenosis. Therefore, predicting accurate blood velocity, shear stress, etc., is more important for clinical healing. Misra et al. [1] introduce the theory of the formation of boundary layer at the entry section of the vessel wall. They also described that this is happening due to the presence of a plasma layer in the vessel wall. A theoretical analysis of wall shear stress (WSS) in the stenosed coronary artery with the help of laminar boundary layer theory has been proposed by Back et al. [2]. This study reveals that results for WSS obtained through boundary layer theory are in good agreement with the results obtained from the Navier stokes equation.
The fundamental study of the magnetohydrodynamic ow is to understand the dynamical aspects of the magnetic eld in the circulation of the conductive uids owing to its induced current. The presence of plasma in the blood instigates the magneto uid properties and the movement of the plasma particles a ects the ow eld in the circulatory system. The utilization of MHD in the diverse areas is to repair the nervous tissues or regeneration of the cells product [3], bone graft [4], and fracture healing [5]. The mixed convective heat transfer through the porous medium was studied by Makinde et al. [6] and transporting the mass in the presence of a magnetic eld with constant wall suction and heat absorption through uid past a vertical porous plate. Hamad et al. [7] pioneered the study investigating the free convection of nano uid through a semi-in nite vertical plate. Bhattacharyya and Chandan Kumawat, Department of Mathematics, Birla Institute of Technology and Science Pilani, Rajasthan, India, E-mail: chandankumawat000@gmail.com Pop [8] employed the shooting method to solve the equations governing the boundary layer ow deals with the MHD due to an exponentially shrinking sheet. To stimulate the future experimental work, the theoretical investigations were undertaken by Sharma et al. [9,10], which examined the transfer of heat and mass in magneto uid ow of blood under the in uence of rst-order chemical reaction, magnetic eld through tapered/non-tapered blood vessels with constriction. Adopting the strategy of boundary layer approximation and similarity transformation, Misra and Shit [11] formulated the equations governing the unsteadiness of MHD ow and heat transfer through a capillary with permeable walls and perceived that the introduction of the external magnetic eld controls the ow velocity, which may be useful in the treatment of the vascular diseases. The transportation of heat and mass on the MHD blood ow through the permeable stretching surface was discussed by Reddy et al. [12] due to the presence of a magnetic eld in the transverse direction of the ow pattern of the uid. They reported that the increment in the value of Hartmann number leads to growth in Lorentz force and hence results decay in the ow velocity. A Taylor-series based method (di erential transformation technique) was introduced by Kumar [13] to simplify the non-linear governing equation of MHD Je ery uid, passes through the vertical stretching surface.
In the current scenario, due to the growth of pollution and harmful materials in our environment, humans are harshly infected from cancer and other respiratory diseases. The heat transfer aspect is used to produce radiation at the forefront of the infected area. Heat transfer plays a vital role in many industrial and chemical processes such as concrete heating, die temperature control, stream generators, batch reactors etc. Apart from these applications, transfer of heat also useful in treatment techniques for dieases such as hyperthermia, laser therapy , physiotherapy, and cryopreservation. Barozzi and Dumas [14] employed the numerical approach to compare convective heat transfer with experimental data in the blood vessel of the circulatory system and reported that the heat transfer rate is high for small blood vessels.They also observed that the pulsatile nature of ow increases the heat transfer rate compared with a steady ow. A theoretical observation of heat transfer for unsteady boundary layer ow with stretching permeable surface and the time-dependent surface temperature has been completed by Ishak et al. [15] and reported that the heat transfer rate enhances with rising in unsteadiness parameter. Tsai et al. [16] examined the in uence of space-dependent heat source for a quiescent uid medium passes through an unsteady stretching sheet. They found that non-uniform heat source plays a signi cant role for heat transfer rate. An analytic study of thermal radiation over a stretching surface has been proposed by Ali et al. [17]. In this study, the behaviour of unsteady third-grade uid observed, and state that the oscillatory frequency ratio of the sheet is responsible for diminishes in the amplitude of uid velocity. Many Researcher [18][19][20][21][22][23][24][25][26][27] described the heat transfer along various type of ow problems with stretching surface.
In the lubricating uid system, viscosity of the uid is a ected by heat generated due to internal friction of uidparticle. Therefore, to predict the precise ow behavior, variation in viscosity is allowable. Initially, Rand et al. [28] experimental examined the e ect of temperature on the human blood viscosity with normothermic and hypothermic conditions, and reported that enhancement in temperature leads decaying in viscosity and hence blood velocity. Further, this study was carried forward by Snyder [29] to analyzed the combined e ect of temperature and hematocrit on the blood viscosity. In this study, they select lizards due to having a natural ability to control uctuation in body temperature. They reported that with increment in hematocrit, both oxygen capacity and viscosity of blood rises also nd that reduction in body temperature leads to the decaying erythrocytes in the blood. Makinde et al. [30] proposed a mathematical model for steady, fully developed Couette ow with temperature-dependent viscosity and state that after enhancement in viscosity exponent, the motion of uid getting slow. Impact of varying thermal conductivity over unsteady boundary layer ow on vertical surface was examined by Vajravelu et al. [31]. In the absence of magnetic eld and ohmic heating, a comparative study to identify the combined e ect of viscosity and thermal conductivity in a variable form on the uid ows through a vertical stretch surface is proposed by Cai wenli et al. [32]. Further, this study is carried forward by Mekheimer et al. [33] for the peristaltic ow, ows in the vertical asymmetric channel. In this study, they adopted the perturbation technique to nd the solution of governing equation by considering both viscosity and thermal conductivity parameters as a perturbation parameter. This study also reveals that the variation in the velocity pro le becomes high for non-zero positive values of the viscosity variation parameter.
Ohmic heating is a process of converison of an electric current into thermal energy, which arise due to friction of uid particles. There are so many industrial and medical applications of ohmic heating such as portable fan heater, incandescent bulb, PCR reactors and electroporation (electropermeabilization). Pliquett [34] has done the experimental study to analyze the impact of ohmic heating on solid tissue during eletroporation therapy and found that, ohmic heating evelates the temperature that plays a signi cant role to develop stable permeable structure. El-Amin [35] developed the mathematical model for the ow through a non-isothermal horizontal cylinder to examined the combined e ect of joule heating and viscous dissipation. This work considered that wall temperature is not constant, and it varies with the non-isothermal exponent parameter. This study is extended by the Aydin and kaya [36] for uid ows through the permeable vertical at plate. The Keller box method is used to solve boundary layer equations. This work gives the Richardson number concept and states that both ow velocity and temperature show the reverse e ect with it. Further, Pal and Talukdar [37], extended the study of [36], and analyzed the impact of chemical reaction on mass transport with thermal radiation. They assumed that free stream velocity rises exponentially. A numerical study to observe the physical importance of ohmic heating on the MHD nano uid which ows through a horizontal stretching surface was discussed by Elazem [38]. In this study Chebyshev pseudospectral technique was used to nd non-dimensional governing equations solution and states that this technique is more accurate than the nite di erence method. Sharma et al. [39,40] discussed MHD blood ow considering joule e ect and porous medium.
The occurrence of mass transport is recognizable in daily life and also have so many industrial applications such as a drop of a dye in water, distillation, liquid extracting, and drying. Bio-engineering design such as blood oxygenators, respirators, and arti cial kidneys involves mass transport. Abel et al. [41] observed the Non-Newtonian ow along the stretching surface with mass transfer, also they analyzed the heat transfer for two di erent models. Rashidi et al. [42] observed the mass transfer through the vertical sheet in presence of porous medium along with buoyancy e ect. A chemical reaction is generally considered as either homogenous or heterogenous on whether the reaction appears as single-phase reaction or at surface where a reaction occurs. The modi ed homogeneousheterogeneous reactions were reported by Khan et al. [43] for MHD stagnation ow. Tripathi and Sharma [44][45][46] analyzed the e ects of rst-order chemical reaction with concentration pro le over two-phase model of blood ow.
All the above-published work neglected the combined e ect of ohmic heating, temperature-dependent viscosity and thermal conductivity on the blood ow passes through the stretching surface. Therefore, to ful ll the gap in existing research, an e ort has been made to analyze the physical signi cance of variable thermal conductivity, viscosity, and ohmic heating on the stretching surface in the present work. To demonstrate the phenomena of mass transport, e ect of the chemical reaction has been considered. The governing equations are transformed into system of ODE's by the similarity transformation, and solved numerically using "MATLAB BVP4C Solver". In uences of physical parameters (for e.g. magnetic eld, permeability, viscosity, chemical reaction, thermal conductivity etc) on velocity, temperature and concentration pro le are analyzed graphically. A comparison with the previously published work has been demonstrated graphically and gets the acceptable agreement with published work. Present study has many practical application in the industrial and medical eld.

Mathematical formulation
Considered heat and mass transfer convectiven unsteady, incompressible two-dimensional blood ow over permeable vertical stretching artery in the presence of porous medium. Generally, blood exhibits both type of Newtonian and non-Newtonian nature which depends upon the lenght of artery and shear stress. In the present work, lenght of the artery is considered su cient large, so that blood is considered as an incompressible Newtonian uid. The Cartesian coordinate system has been consider as shown in are gure 1. The x-axis is taken along the ow direction and y-direction is normal to the ow. Let us consider a region of ow which barred for y > .The magnetic Reynolds number is considered very small in comparison with acting magnetic eld, so that the induced magnetic eld can be neglected. Magnetic eld applied in the positive y-direction is taken in the form of The ohmic heating and magnetic eld along with chemical reaction also studied in uid ow. All the variables are independent of the y-direction. Stretched artery where d and γ are constants and dimensions of these constants are (sec) − with the condition d > , γ ≥ and γt < . Temperature and Concentration of the arterial wall arẽ Where T * ∞ and C * ∞ represents temperature and concentration of the outside the boundary layer, respectively and T * and C * are constants . ν = µ * ∞ ρ and µ * ∞ are the kinematic and dynamic viscosity of ambient uid respectively.Ṽ * w indicates velocity of mass transfer of arterial wall at surface and represented as for v < ,Ṽ * w is suction velocity and for v > ,Ṽ * w is the injection velocity. It is assume that the temperature-dependent viscosity µ(T b ), changes linearly with temperature and is written as where ∆T * =T * w − T * ∞ andT b represents blood temperature, ϵ is small viscosity variation parameter. The thermal conductivity of the uid is assumed as temperature dependent and, vary linearly with temperature as where ϵ is small thermal conductivity variation parameter, K * ∞ is thermal conductivity of ambient uid. Under Boussinesq's approximation and using above assumptions the governing equations for blood ow are .
and Boundary conditions for the ow arẽ Whereũ b &ṽ b represent velocity components in x and y direction, respectively . β * T and β * C represent coe cient of thermal and concentration expansion. k is porous medium permeability and represent as k = k ( − γt) . k is initial permeability, g is the gravitational acceleration, Cp is speci c heat at constant pressure, D * m is molecular di usivity and R * represents parameter of chemical reaction. q * r expresses radiative heat ux which characterized by Rosseland approximation [32]: where σ * represents Stefan-Boltzman constant and k * indicates Rosseland mean absorption coe cient. Temperature di erence inside ow is assumed adequately meager, so by Taylor series expansion ofT b about T * ∞ and avoid higher order terms, then we getT Now introducing similarity transformations are - Here ψ * (x, y, t) represents stream function and expressed in form where prime indicates the di erentiation with respect to η. Here f =ũ b /Ũ * w , θ & ϕ are the dimensionless velocity, temperature and concentration, respectively. After substituting the similarity transformation into equations (4)- (6) ,we get The boundary conditions corresponding equations (12)- (14) are In above equation if fw < , fw > and fw = symbolize the injection, suction and impermeable nature of sheet, respectively. Introduce following non-dimensional parameters- x (local Grashof number for mass di usion), There are three type of physical quantities of our interest Skin-friction coe cient C f , rate of heat transfer and mass transfer which are given by are the skin friction, heat ux and mass ux. After substituting these coe cient, then equation (16)can be written as,

Numerical solution
In order to analyze the in uence of variable viscosity and thermal conductivity on blood ow through the permeable vertical stretching surface, the non-dimensionalize boundary layer equations (12)- (14) subjected to the boundary conditions (15) are solved numerically by using "BVP4C Solver" under Matlab. Equations (12)- (14) are non-linear coupled ODE's, so to apply "BVP4C solver", rst these will be converted into the system of initial value problem (IVP). Let us assume that- By introducing new variables the equations (12)-(14) will be transformed into IVP as and the boundary conditions (15) for the above IVP are converted in the following form:

Results and discussion
In the present study, the e ects of temperature-dependent viscosity, thermal conductivity and ohmic heating for blood ow through the vertical stretching surface have been discussed. E ects of the physical parameters (M, δ, Ec, ξ & Sc ) have been neglected for validating the current study with Wenli et al. [32]. Wenli et al. [32] used the shooting method to simplify the dimensionless governing equation. Therefore in this comparison, the shooting method has used for [32] work and Matlab BVP4C solver for the present study. This BVP4C solver is based on the Rungekutta method of order 4. The iterative process will be terminated when the error involved is < − . Figures 2-3 have been drawn for validating the velocity and temperature pro le of present work with the previous study [32], and these gures replicates the good agreement between present work and previous published work [32] for the uid velocity and temperature. The numerical results are getting by solving equations (12) Figure 4 perceived that a non-linear decay in velocity with the parameter η for all the values of M. A rising in magnetic eld leads to decay in velocity pro le owing to its resistivity occurs due to Lorentz force on the ow of uid. Figure 5 depicts the impact of magnetic eld on the temperature pro le, due to applying ohmic heating in the energy equation. Physically, with the applying ohmic heating in the blood ow, an electromagnetic force appears in the ow, which produces a rise in the blood temperature. Therefore a decay in the surface temperature is observed. Further, it is also concluded that the presence of a magnetic eld generates viscous heating in the ow, which leads to an increase in the ow temperature, which is in good agreement with the published work [38]. Figures 6-7 replicate the variation of velocity with η for the di erent values of local Grashoof number (Gr) and local Grasho number for mass transfer (Gm), respectively. From the both gures it can be analyzed that velocity pro le approaches gradually to zero in non-linear way for all values of Gr and Gm. Also, it can be observed that the velocity pro le enhances due increment in both Gr and Gm. As Grashoof number stated the relation between buoyancy force and viscous force, so it is concluded that    rising in Grashoof number leads to less dominance of viscous force, i.e., the resistivity of the ow decreases, and the velocity of uid ow increases. Figure 8 demonstrates the impact of the variation of temperature-dependent viscosity on the velocity pro le for the di erent values of (ϵ ). An increment in viscosity parameter (ϵ ) leads to enhancement in the velocity of uid due to increament in thickness of boundary layer. From the Figure 8, it can be seen that as the parameter η increases, the velocity decaying monotonically and approaches to 0. As increasing ϵ , initially near the surface of wall velocity decreases, then after velocity increases away from the wall surface due to increment in the thickness of the boundary layer, which is quite similar with the previous study [32] . Figures 9-10 exhibit the e ect of an unsteady parameter (A) with η on the uid velocity, and temperature. Variation of uid velocity for di erent values of A can be analyzed from the gure 9 and noticed that thickness of the boundary layer decreases with an increment in unsteady parameter. As can be seen from this gure, the uid velocity is decaying after enhancing the value of A. Figure 10 describes the variation of temperature pro le for di erent values of A and it can be noted as (η) increases from 0, a non-linear decay occurs in uid temperature, and it approaches to 0. A rise in unsteady parameter leads to diminishes in the thickness of the thermal boundary layer, resultant a downfall in the the temperature pro le can be observed. Figure 11 expresses the variation in ow velocity with parameter η for the di erent values of permeability parameter (β) and it is noted that for all values of (β) velocity decreasing in monotonic way and approaches to 0. It can be perceived that as the permeability parameter enhances, uid velocity decreases, which is relatively similar to the earlier study [47]. Figure 12 described variation in velocity for di erent values fw. In this gure, the velocity pro le is displayed for injection, impermeable, and suction sheet, respectively, which depends on the nature of fw. It is observed that thickness of the boundary layer is high for the injection in comparision with suction and hence, the velocity pro le decreases with enhancing fw. It is also noted that for injection, uid velocity is high in comparison with suction . Figures 13-14 explained the variation in uid temperature with parameter (η) under the di erent values of Prandtl number (Pr) and thermal Conductivity variation parameter (ϵ ), and these gures depict that for Pr and ϵ temperature shows a non-linear decaying pro le and approaches to 0 with increasing in η. Prandtl number illustrates the relation between momentum di usivity and thermal di usivity. High Prandtl number replicates the dominance of momentum di usivity over the thermal di usivity, and hence thickness of the thermal boundary layer reduced. The graphical analysis through the Figure 13 reveals that uid temperature decay slightly with enhancement in Pr. Figure 14 replicates the variation in temperature pro le for di erent values of thermal conductivity variation parameter ϵ . Rise in ϵ leads to slightly growth in uid temperature as depicted in Figure 14, owing to enhancement in the thickness of thermal boundary layer. Figure 15-16 depict the e ects of Eckert number and radiation parameter on the uid temperature with the parameter η, respectively. A non-linear decreasing pro le in uid temperature for all values of Ec and Nr observed. Form Figure 15 it is observed that for high values of Ec, initially temperature increases and after that temperature pro le decreases and tends to 0. A rise in Ec leads to enhancement in uid temperature owing to heat energy is stored in the uid due to friction heating. Figure 16 reveals the e ect of radiation on the uid temperature and noted that with rise in radiation parameter, uid temperature enhances. This is happening due to the release of heat energy in the radiation cycle process, which absorbed by the uid. Therefore, a rise in the uid temperature and thermal boundary layer thickness occurs. Figures 17-18 depict the in uences on concentration pro le for di erent values of Schmidt number and chemical reaction parameter, respectively. It is observed that a decay in concentration pro le with parameter η for all values numeric of Sc and ξ . Schmidt number gives the relationship between the viscous di usion rate and molecular di usion rate. A rising in Schmidt number leads to decline in the thickness of mass-transfer boundary layer, resultant the concentration pro le diminishes. The e ect of chemical reaction on the concentration pro le can be observed through Figure 18. It interpreted that in the presence of chemical reaction (ξ ≠ ), concentration pro le decreases rapidly in comparison with the absence of chemical reaction (ξ = ). It is also analyzed that an enhancement in the chemical reaction parameter diminishes the concentration of species in the boundary layer, that why the thickness of the mass transfer boundary layer decreases. This is happening because ow, consumes the chemical in the chemical reaction process, resulting a decay in the concentration pro le occurs. Figures 19-22 delineate the deviation of skin-friction coe cient with viscosity variation parameter (ϵ ) for the di erent values of parameters M, β, Gr and Gm, respectively. An enhancement in the values of viscosity variation parameter ϵ exhibits the rising in uid viscosity. Therefore, the skin-friction coe cient of uid enhances linearly near to the wall. A rise in M and β, reveal the enhancement in C f and reverse e ect is observed with increment in both Gr and Gm, respectively. From Figure 19 it can be also analyzed that growth rate of C f becomes high when M enhances from 0 to 1,incomparision with 1 to 2. Figures 23-26 delineate the variation in heat transfer coe cient (nusselt number) with thermal conductivity variation parameter ϵ for di erent values of Eckert number, Prandtl number, magnetic number, and radiation parameter, respectively. An enhancement in ϵ , contributes to rising in the thermal conductivity of the uid, resulting nusselt number decreases monotonically. From these gures, it is observed that the rate of heat transfer approaches to 0 for a considerable value of ϵ . A rising in Pr demonstrate the enhancement in heat transfer rate and reverse e ect is noted with rising in Ec, M and Nr, respectively. Deviation in rate of mass transfer coe cient (Sherwood number) with unsteady parameter (A) for di erent values of Sc and ξ are exhibited in Figures 27-28, respectively. A rising in unsteady parameter (A), contributes towards non-linear enhancement in Sherwood number near the wall. A rise in Sc and ξ , reveals the improvement in Shx and also observed from Figure 27 that growth rate of Shx becomes higher when Sc enhances form 0.5 to 1, in comparision with 1 to 1.5.

Conclusion
Two-dimensional unsteady MHD convection blood ow with temperature-dependent thermal conductivity and viscosity through a vertical stretching surface is numerically investigated in this paper. The governing equations are reduced in the system of ODE's, and then analyzed numerically with the help of 'MATLAB BVP4C solver'. The physical insight analysis for ow parameters is graphically performed. The present mathematical study of boundary layer ow for blood will be helpful for researchers and clinical engineers to estimate the realistic behaviour of blood ow velocity, temperature etc., in the process of clinical healing. The main results of this study are as follows: • Ohmic heating in the ow, enhance the temperature and thickness of thermal boundary layer.
• The thickness of boundary layer enhances by increasing in value of ϵ while viscosity of blood shows reverse e ect with it. • A rise in thermal conductivity variation parameter (ϵ ) leads to slightly growth in blood temperature and thickness of thermal boundary layer. • By increasing unsteady parameter A, both the velocity and temperature of ow diminish. • The numerical value of heat transfer coe cient (Nusselt number) diminishes with increment in thermal conductivity variation parameterϵ and magnetic eld (M) while the reverse e ect is observed when Ec and Nr enhances.
• The numerical value of coe cient of skin-friction increases with increment in both ϵ and magnetic eld M. • A rise in both unsteady parameter (A) and chemical reaction parameter (ξ ), symbolizes the enhances the rate of mass transfer coe cient.

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Con ict of interest:
The authors state no con ict of interest.