Magnetohydrodynamic flow of Sisko fluid over a stretching cylinder with variable thermal conductivity: A numerical study

The analysis of non-Newtonian fluids has gained an abundance of attention amongst researchers in last few decades due to its wide use in industry. Since huge numbers of varieties of non-Newtonian fluids are subsist in nature, so many fluid models are suggested to study the physical properties of these fluids. Among these fluid models power law fluid model is most felicitous model to presage attitude of non-Newtonian fluids. But it can predicts fluid properties in the power law region only, while it fails to analyze flow characteristics when shear rate become very small or large. To surmount this difficulty Sisko¹ proposed a model which denominated Sisko fluid model, this fluid model predicts fluid properties in power law region and also at high shear rate. Additionally experimentally it is verified that many realistic fluids such as waterborne coatings, drilling muds, biological fluids, cement slurries, paints etc. follow Sisko fluid model. The most congruous Sisko fluids are lubricating greases which are analyzed by Sisko.¹ Up till now number of researchers have investigated Sisko fluid model under different physical situations. Akyildiz et al.² examined the steady flow of two dimensional Sisko fluid over thin film. The approximate analytic solution was found with homotopy analysis method. The variations in fluid velocity were shown by varying material parameter and power law index for shear thinning fluids. The results perceived that fluid velocity inclines for colossal values of both parameters. Mekhimer et al.³ discussed the blood flow under the influences of chemical reactions through an isotropically tapered elastic artery with time-variant overlapping stenosis by utilizing Sisko fluid model. They concluded that magnitude of velocity is more substantially large for Newtonian fluid than Sisko fluid and also Sisko parameter enhances fluid temperature. Additionally the pressure of Newtonian fluid is extremely immense than Sisko fluid while shear thickening fluids have low pressure than shear thinning fluids. Khan et al.⁴ inspected the steady flow of Sisko fluid model in the annular pipe with heat transfer. The problem was solved analytically by homotopy analysis method. The computed analytical solution was certified by comparing it with numerical solution. Khan et al.⁵ studied the incompressible time dependent flow of Sisko fluid under the influence of transverse magnetic field in annular pipe. The velocity and temperature profiles are computed numerically. The results of this analysis reflects that power law index, material parameter and magnetic field decelerates the motion of fluid. Additionally Newtonian fluid has larger velocity than Sisko fluid. The peristaltic flow of incompressible Sisko fluid in a uniform inclined tube was investigated by Nadeem et al.⁶ The quantities of physical interest i.e. pressure rise, friction force and pressure gradient were calculated numerically. They concluded that pressure rise increases for larger values of angle inclination parameter, Weissenberg number and amplitude ratio. Withal they suggested that pressure rise for square wave gives best pumping features while trapezoidal has worst pumping characteristics. A homogeneous study on Sisko fluid was proposed by Akbar.⁷ She formulated the flow problem of Sisko fluid in an asymmetric channel with nanoparticles. The numerical scheme Runge-Kutta-Fehlberg was applied to solve governing differential equations. She found that Sisko fluid parameter inclines pressure rise in peristaltic pumping while its deportment is contrast in augmented pumping.

The study of non-Newtonian fluids under the impact of transverse magnetic field is intending the attention of peoples due to its various applications in science and technology such as MHD pumps, MHD power generators, nuclear reactors, purification of crude oil, electromagnetic propulsion etc. Withal turbulence comportment of fluid is also handled with magnetic field. Alfven⁸ originated the conception of electromagnetic-hydrodynamic waves in their valuable work. Hsiao⁹ discussed the MHD mixed convection flow of visco-elastic fluid over a stretching sheet. He solved the problem by using finite difference scheme. Hsiao¹⁰ studied the MHD stagnation point flow of visco-elastic fluid on thermal forming stretching sheet with viscous dissipation effect. He found that heat transfer rate decreases by increasing magnetic field. Again, Hsiao¹¹ analyzed the MHD mixed convection flow of visco-elastic fluid over a porous wedge. Nadeem et al.¹² surveyed the fluid properties of MHD Casson fluid past from a linearly stretched sheet with porousity. The non-dimensional velocities were computed numerically with Runge-Kutta-Fehlberg scheme. They perceived that presence of transverse magnetic field causes declination in both velocity components while it inclines friction of wall in both directions. Akram et al.¹³ analyzed the incompressible two dimensional flow of Jeffrey nano fluid in an asymmetric channel with magnetic field. The closed form solutions of governing equations were found with Adomian decomposition method. Recently, boundary layer stagnation point flow of MHD Williamson fluid over stretching cylinder was investigated by Malik et al.¹⁴ The numerical solution of governing flow equations was computed with Runge-Kutta-Fehlberg method. Malik et al.¹⁵ also analyzed the MHD flow of tangent hyperbolic fluid over stretching cylinder. Numerical solution was computed with Kellor-Box method. Malik et al.¹⁶ surveyed the flow properties of MHD Sisko fluid over stretching cylinder and found numerical solution by applying shooting method. Moreover some investigations of MHD flows are presented by Yih,¹⁷ Liao,¹⁸ Cortell,¹⁹ Ali et al.²⁰ and Machireddy et al.²¹ 

The products of industry like paper production, fiber-glass production and polymer sheets extrusion are applications of boundary layer flow over stretching surfaces. Additionally the excellence of product is controlled with heat transfer from stretched surface. Withal heat transfer is paramount for many other practical situations like cooling or heating of heat exchanger, sultry rolling etc. Sakiadis²² is considered as pioneer of boundary layer flow problems. He²² discussed the two dimensional axisymmetric flow over the continuous solid surface. Additionally the work of Crane²³ was a major contribution in boundary layer flows. He examined the two dimensional flow of incompressible Newtonian fluid over a stretched plate and found exact solution of the problem. The problem of heat transfer over stretched surface was firstly encountered by Gupta and Gupta.²⁴ In this investigation heat and mass transfer of viscous fluid from a porous stretching sheet with suction or blowing was studied. Additionally, Dutta et al.²⁵ analyzed the heat transfer of Newtonian fluid flow over a stretching surface by surmising uniform heat flux. Also in some recent years boundary layer flows of Newtonian and non-Newtonian fluids are studied consistently. Zaimi et al.²⁶ discussed the incompressible flow of Newtonian fluid over stretching/shrinking sheet with convective boundary conditions. Nadeem et al.²⁷ formulated the problem of incompressible two dimensional boundary layer flow of viscous nanofluid over an exponentially stretching surface. The governing nonlinear ODEs were solved with HAM. The mixed convection boundary layer flow of non-Newtonian Casson nanofluid over a vertical exponentially stretching cylinder was investigated by Malik et al.²⁸ The solution was calculated with shooting method. They found that Brownian motion and thermopherosis both causes enhancement in fluid temperature while Reynolds number and Prandtl number causes fall down in temperature. Vajravelu et al.²⁹ formulated the problem of viscous flow over stretching cylinder with heat transfer analysis. The solution was calculated numerically using implicit finite difference scheme Kellor-Box method. Heat transfer of fluid flow are also dicussed by Hasnain et al.³⁰ and Andersson.³¹ Thermal conductivity is aptitude of any material to conduct heat. Thermal conductivity play vigorous role in heat transfer. Additionally according to Fourier law of heat conduction it is directly cognate to heat flux while it has inverse variation with temperature gradient. Thus thermal conductivity of fluid varies by incrementing or decrementing temperature. For fluids it is experimentally substantiated that thermal conductivity enhances when fluid temperature varies from 0 to 400 F. Chiam³² was primarily surmised thermal conductivity variable in his problem. In this analysis he considered boundary layer flow of two-dimensional viscous fluid over a porous stretching sheet and heat transfer with variable thermal conductivity. The dimensionless velocity and temperature profiles were computed numerically with shooting method. The main outcomes of this investigation are variable thermal conductivity enhances temperature while it declines wall gradient. Mishra et al.³³ analyzed the two dimensional unsteady boundary layer flow of Newtonian fluid past a stretching plate and heat transfer with variable thermal conductivity. In this study the heat transfer was discussed for presubscribed surface temperature (PST) and presubscribed heat flux (PHF). Rangi et al.³⁴ considered the boundary layer flow of incompressible two dimensional viscous fluid past over the stretching cylinder with variable thermal conductivity. The solution of governing simultaneous ODEs was computed with implicit finite difference scheme Keller-Box. They suggested that by varying curvature parameter and thermal conductivity fluid temperature elevates. In literature there are some other investigations related to variable thermal conductivity are available Singh,³⁵ Miaoa et al.,³⁶ Abel et al.,³⁷ Jhankal³⁸ and Miaoa et al.³⁹ The aim of present paper is to analyze the effects of variable thermal conductivity on MHD Sisko fluid over stretching cylinder which has not been discussed so for. The governing nonlinear momentum and energy equations are solved numerically with shooting method. Also the effects of curvature parameter, magnetic field parameter, material parameter, thermal conductivity parameter and Prandlt number on velocity and temperature distributions are analyzed via graphs.

Consider the two dimensional, axisymmetric, steady boundary layer flow of incompressible Sisko fluid model along the stretching cylinder. The stretching velocity U(x) is of the form U(x)  =  cx where c is the positive constant along x-axis i.e. in axial direction of cylinder. The upper half of the plane i.e. r > r₀ is filled with fluid. Magnetic field of strength B₀ is imposed on the fluid particles perpendicular to axial direction of cylinder. The influences of induced magnetic field and electric field are assumed zero. Also it can be observed that fluid changes its thermal conductivity as temperature varies. Thus thermal conductivity of the fluid is considered as temperature dependent (i.e. variable). By considering all these suppositions and after using boundary layer approximations the governing equations are

subject to the boundary conditions

In the above expressions u and v represent the velocity components along x and r axis respectively. Also U(x) is the stretching velocity, a called high shear rate viscosity, consistency index is denoted with b and n represents power law index, σ is the electrical conductivity, B₀ is magnetic field strength, ρ is the density of fluid particles. In last equation T denotes the temperature of fluid, wall temperature has symbol of T_w, T_∞ is the extreme temperature while α is the thermal diffusivity of fluid.

Define the stream function Ψ such that

which satisfies the continuity equation identically.

Modelled governing partial differential equations are reduced into ordinary differential equations by using the following similarity transformations

Here α_∞ is thermal diffusivity when r → ∞, ϵ is a positive number called thermal conductivity parameter.

Using above similarity transformations into Eqs. (2)–(4) following system of equations is obtained

the boundary conditions are reduces to

Here the curvature parameter γ, magnetic field parameter M, material parameter A and Prandtl number Pr are defined as

Physical quantities of practical interest i.e. skin friction coefficient and local Nusselt number are defined as

where τ_w is shear stress at the surface of cylinder while q_w called wall heat flux. These quantities are defined below

After applying the similarity transformations the skin friction coefficient and local Nusselt number are converted to

In this study the problem on MHD boundary layer flow of Sisko fluid with variable thermal conductivity is formulated. Since governing equations Eqs. (7)–(8) of this problem are highly nonlinear. So for computation of solution a numerical technique shooting method with Runge-Kutta Fehlberg method is applied. The variations in model are discussed for different values of curvature parameter γ, material parameter A, magnetic field parameter M, variable thermal conductivity parameter ϵ and Prandtl number Pr. Initially higher order equations are transformed to first order equations. So first governing equations are re-written as

And a new set of variables is defined to change Eqs. (14)–(15) into first order ordinary differential equations

When Eq. (16) is inserted into Eqs. (14)–(15), governing equations are transformed to system of first order ordinary differential equations Eqs. (17)–(21)

The subjected boundary conditions are reduce to

Now to compute the solution of above system i.e Eqs. (17)–(21) with Runge-Kutta Fehlberg method, it requires five initial approximations. But Eq. (22) have only three initial conditions, so it lacks two initial approximations for dependent variables y₃ and y₅. Thus before proceeding towards solution procedure firstly missing initial conditions must be chosen. y₃(0) = − 1 and y₅(0) = 1 are found good initial approximations. Additionally the domain of independent variable is semi-infinite, so finite upper limit of η must be chosen. The approximation for η → ∞ is selected 5. Now solution process is started for computation of fluid velocity and temperature. The process of solution is terminated if absolute difference between given and computed boundary conditions i.e. y₂(∞), y₄(∞) is less than error tolerance i.e. 10⁻⁶. But on the other hand if this differences are larger than error tolerance, then initial guesses are refine through Newton method.

The boundary layer flow of MHD Sisko fluid is discussed over stretching cylinder with variable thermal conductivity. The solution found numerically and accuracy of computed solution is certified by comparing it with available literature. In Table I numerical values of skin friction coefficient are compared with Hasnain et al.,³⁰ Cortell,¹⁸ Liao,¹⁹ Anderison³¹ and Abel et al.³⁷ It can be observed from the table that results agreed upto desired significant digits. Additionally comparison of wall temperature gradient i.e −θ′(0) is presented via Table II by varying Prandtl number. This graph reveals that present values are quite similar with existing literature i.e values computed by Yih,¹⁷ Ali²⁰ and Machireddy et al.²¹ 

Fig. 1 shows the fluid velocity f(η) curves for different values of Sisko fluid parameter A and n = 1, 2. This figure discloses that enhancement in Sisko fluid parameter A causes increase in fluid velocity and also boundary layer thickness. These results are physically valid as Sisko fluid parameter has inverse relation with consistency index b i.e. fluid viscosity of power law region. So when material parameter A increases the viscous force become weaker, thus less resistance is offered to fluid motion.

To figure out the effects of magnetic field parameter M on fluid velocity, Fig. 2 is constructed. This figure divulges that when magnetic field strength increases it decays fluid velocity. It is true because the Lorentz force i.e. resisting force become stronger when magnetic field parameter M increases. Thus fluid motion is decelerated and so flow field velocity. Many researchers illustrated such behavior of magnetic field on velocity and temperature profiles, some of the analysis is reported by Hsiao.^9,10

Effects of curvature parameter γ on velocity profile f′(η) are displayed through Fig. 3 for power law index n = 1 and 2. It can be perceived that when bending of cylinder enhances it reduces radius of cylinder and surface area. Hence the surface of the cylinder offers less resistance to flow which causes enhance in momentum transfer and velocity of the fluid.

Fig. 4 depicts the behavior of thermal conductivity parameter ϵ on heat equation for n = 1 and 2. This figure indicates that when thermal conductivity of fluid increases, it raises the fluid temperature. The reason behind this fact is that when thermal conductivity enhances, then fluid conduct heat rapidly and hence inclines the temperature.

Temperature distribution is shown for different values of Prandtl number Pr in Fig. 5 and n = 1 and 2. It can be seen that temperature profile is declines for larger value of Prandtl number Pr. It holds practically since Pr is the ratio of momentum to thermal diffusivity. Thus by increasing Pr, momentum transport accelerates which enhances convective heat transfer and declines conductive heat transfer. Hence it causes fall down in fluid temperature.

The influence of curvature parameter γ on temperature profile for n = 1 and 2 is shown via Fig. 6. This graph illustrates that curvature parameter γ raises fluid temperature and it enriches the thermal boundary layer. This is physically true because when curvature of cylinder increases it reduces radius of the cylinder. It accelerates the heat transfer rate which causes increase in fluid temperature.

The effects of magnetic field parameter M on temperature profile for different values of power law index n are display in Fig. 7. This figure reflects that fluid temperature enhances by increasing the strength of magnetic field. The fact behind that when magnetic field strength increases, it produced heat in the fluid which causes enhancement in fluid temperature.

To reflect the influence of magnetic field parameter M, material parameter A and power law index n on wall shear stress Fig. 8 is developed. This graph demonstrates that shear stress near the wall of cylinder enhances by increasing material parameter A and magnetic field parameter M. The results hold practically because enhancement in A and M causes increase in momentum boundary layer, thus skin friction coefficient increases. Also this figure proves that wall shear stress of pseudoplastic fluid in much higher than both Newtonian and dilatant fluids.

Fluctuation in skin friction coefficient by varying curvature parameter γ and material parameter A is displayed in Fig. 9 for different values of n. It can be seen that larger values of curvature parameter γ increase the skin friction coefficient. It is true because inclination in curvature parameter γ decrease Reynolds number which causes enhancement in wall shear stress.

Fig. 10 exemplifies the combined inspiration of variable thermal conductivity parameter ϵ and curvature parameter γ on local Nusselt number. This figure shows that curvature parameter γ accelerate heat transfer from the wall while the influence of thermal conductivity parameter ϵ is opposite on it. These results are true because Nusselt number varies inversely to conductive heat transfer, thus when thermal conductivity increases it causes decrease in Nusselt number. Also it can be analyzed that power law index n causes inclination in wall temperature gradient.

Fig. 11 expresses variations in local Nusselt number for different values of Prandtl number Pr, variable thermal conductivity parameter ϵ and power law index n. This figure illustrates that Prandtl number enhances local Nusselt number. As Prandtl number is ratio of momentum to thermal diffusivity, so increment in Prandtl number causes reduction in thermal conductivity which increase the local Nusselt number.

Table III reflects the influence of physical parameters A, M, γ and n on skin friction coefficient. In this table a comparison is presented between power law fluid i.e (A = 0) with Sisko fluid i.e (A = 1). It can analyze that Sisko fluid has larger skin friction coefficient than power law fluid for all values of n. Also wall shear stress declines for larger values of power law index while it inclines when physical parameters M and γ increases.

The impact of curvature parameter γ, thermal conductivity parameter ϵ, Prandtl number Pr and power law index n on Nusselt number is discussed in Table IV. It can be observed from the table that fluid with constant thermal conductivity (ϵ = 0) has larger local Nusselt number as compared to fluid with variable thermal conductivity (ϵ = 0.5). Also and Pr enhances the values of Nusselt number.

In this investigation the MHD flow of incompressible Sisko fluid is analyzed over stretching cylinder. Thermal conductivity of fluid is considered temperature dependent. The numerical solution is computed with Runge-Kutta-Fehlberg method. It can be observed that

• Material parameter A and curvature parameter γ enhances fluid velocity while magnetic field parameter M decayed velocity profile.

• By increasing thermal conductivity and curvature parameter γ fluid temperature rises while Prandtl number Pr causes reduction in temperature.

• Skin friction of power law fluid is much less than Siko fluid and also pseudoplastic fluids have larger skin friction coefficient than both Newtonian and dilatant fluids.

• Magnetic field parameter M, Sisko fluid parameter A and curvature parameter γ increases skin friction coefficient absolutely.

• Wall temperature gradient increases for larger values of curvature parameter γ and Prandtl number Pr.

• Nusselt number is larger of constant thermal conductivity as compared to variable thermal conductivity.