Effects of viscous dissipation on MHD boundary layer flow of Sisko fluid over a stretching cylinder

Mostly fluids which are very useful in our daily life and industry do not obey the Newtonian expression of viscosity, for examples paints, oil, lubricating greases, human blood, honey, biological fluids etc. These fluids are called non-Newtonian. Due to their importance in our daily life, in last few years many studies have been reported in which characteristics of non-Newtonian fluids are explored. Also, since the class of non-Newtonian fluids is diverse, so researchers were suggested different models to discuss physical properties of these fluids. In present investigation the Sisko fluid model is chosen to study the flow features of non-Newtonian fluids. This model is extension of power law model. As power law model failed to describe physical properties when shear rate is very high, this issue was resolved in this model. Sisko¹ the founder of this model investigated the lubricating greases. Nadeem et al.² studied the peristaltic flow in a uniform inclined tube with the help Sisko fluid model and found the analytical expressions of interested quantities with HAM. They discussed the peristaltic flow of both Newtonian and non-Newtonian fluids and suggested that Newtonian fluids have best peristaltic motion as compared to non-Newtonian fluids. Nadeem et al.³ extended the previous investigation and analyzed the peristaltic flow of a Sisko fluid in an endoscope and found the both analytical as well as numerical solution of the governing differential equations. The peristaltic flow of Sisko nano fluid in an asymmetric channel was investigated by Akber.⁴ She found numerical solution of governing equations by shooting method. She concluded that Sisko parameter rises pressure in peristaltic pumping region on the other pressure declines in augmented pumping region. Khan et al.⁵ described the flow of a magnetohydrodynamic Sisko fluid in annular pipe and calculated the numerical solution of the problem. They suggested that velocity of Sisko fluid is much less than Newtonian fluid. Malik et al.⁶ analyzed the boundary layer flow of Sisko fluid with convective heat transfer. They calculated analytical as well as numerical solution of problem. Recently, Malik et al.⁷ discussed the MHD flow of Sisko fluid over stretching cylinder and solved the flow equations by shooting method. They found that Sisko parameter enhances velocity profile and boundary layer thickness. The non-Newtonian Sisko fluid was also analyzed by Refs. 8–10 

The presence of magnetic field in fluid flow has great importance because their combination is used in many devices such as electromagnetic propulsion, MHD pump, nuclear reactors, MHD generators. Thus significant number of studies are available in literature which addressed MHD flows. The MHD flow of non-Newtonian Casson fluid over an exponentially shrinking sheet was investigated by Nadeem et al.¹¹ The governing equations of this problem were solved with Adomian decomposition method. Nadeem et al.¹² studied the MHD three dimensional flow of Casson fluid past a porous stretching sheet and determined the numerical solution. They recommended that fluid velocity decays when influence of MHD enhances. Ismail et al.¹³ studied the time dependent flow of viscous fluid between two parallel plates under the effects of applied magnetic field. The ordinary differential equations are solved numerically. Emad et al.¹⁴ discussed the flow of non-Newtonian power law fluid over moving cylinder under the impact of applied magnetic field. Akbar et al.¹⁵ discussed the two dimensional steady flow of Powell-Eyring fluid over stretching sheet. They calculated numerical solution via shooting method and concluded that transverse magnetic field declines velocity of the fluid. Malik et al.¹⁶ analyzed the stagnation point flow of Williamson fluid over stretching cylinder. The numerical solution was calculated through shooting method and they shown that magnetic field decelerates the motion. Recently, Malik et al.¹⁷ studied the effects of MHD on tangent hyperbolic fluid over stretching cylinder. The implicit finite difference scheme Kellor-Box was applied to solve governing equations. They found that magnetic field strength reduces the fluid velocity.

During the motion of fluid particles, viscosity of the fluid converts some kinetic energy into thermal energy. As this process is irreversible and caused due to viscosity, so this is called viscous dissipation. Initially the effect of viscous dissipation was considered by Brickman.¹⁸ He investigated the temperature distribution of Newtonian fluid in straight circular tube and interprets the result that the effects were produced in the close region. Ou et al.¹⁹ analyzed the viscous dissipation effects in the entrance region heat transfer in pipes with uniform heat flux. Chand et al.²⁰ studied the effects of viscous dissipation and radiation on unsteady flow of electrically conducting fluid through a porous stretching surface. The expressions of velocity and temperature were found with classical fourth order Runge-Kutta method. Van Rij et al.²¹ investigated the combined effects of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer. The solution of the problem was found analytically as well as numerically. Kishan et al.²² discussed the stagnation point flow of micropolar fluid through porous medium and heat transfer with viscous dissipation. The governing differential equations were solved numerically with implicit finite difference scheme. They found that Eckert number enhances the temperature. Singh²³ inspected the flow of MHD viscous fluid in porous medium through a moving vertical plate. And discussed heat transfer with viscous dissipation and variable viscosity. The problem was solved numerically. The combined influences of Joule heating and viscous dissipation on viscous fluid along a vertical plate were analyzed by Alim et al.²⁴ El-Amin,²⁵ Hossain²⁶ also studied the flow of Newtonian fluid under the combined effects of Joule heating and viscous dissipation.

Sakiadis²⁷ started to investigate the boundary layer flow problems. He²⁷ analyzed the two dimensional axisymmetric flow over continuous solid surface. Crane²⁸ discussed the two dimensional boundary layer flow of Newtonian fluid over stretching plate. Gupta and Gupta²⁹ analyzed the heat and mass transfer on stretching sheet with suction or blowing. Malik et al.³⁰ studied the boundary layer flow of Casson nano fluid over a vertical exponentially stretching cylinder. The solution was calculated numerically with the help of fifth order Runge-Kutta Fehlberg method. Khan et al.³¹ discussed the two dimensional incompressible flow of non-Newtonian Casson fluid over a stretching sheet with nanoparticles. Makinde et al.³² investigated the boundary layer flow of nano fluid over stretching sheet with convective boundary conditions. Malik et al.³³ discussed the heat transfer analysis of Jaffery six-constant fluid between coaxial cylinder. The analytical expressions for velocity and temperature were calculated with homotopy analysis method. Nadeem et al.³⁴ studied the heat transfer of boundary layer flow of second grade fluid in a cylinder. Rangi et al.³⁵ discussed the boundary layer flow over stretching cylinder and heat transfer with variable thermal conductivity. The well known technique Kellor-Box was used to find the numerical solution.

In literature the combined effects of applied magnetic field and viscous dissipation over Newtonian and non-Newtonian fluids were reported by many researchers. But till now Sisko fluid model are not yet discussed with these effects. This motivates authors to investigate the boundary layer flow of MHD Sisko fluid model over the stretching cylinder and temperature with the effect of viscous dissipation. The nonlinear equations are solved numerically by shooting method. The effects of different physical parameters represent with the help of graphs and tables.

Let consider a two dimensional steady state boundary layer flow of a axisymmetric incompressible Sisko fluid along the continuously stretching cylinder. The cylinder is stretched in axial direction with the velocity U(x) which is defined as U(x) = cx where c is the positive constant. The transverse magnetic field is imposed orthogonal to x-direction with strength B₀. The influences of induced magnetic field and electric field are neglected. The effect of viscous dissipation is assumed in the heat transfer. With these suppositions and boundary layer approach the governing continuity, momentum and heat equations are

subject to the boundary conditions

Here u is the velocity component of fluid along x − axis while v is along r − direction. σ is the electrical conductivity of the fluid, n (power law index), a (high shear rate viscosity) and b (consistency index) are the material constants, B₀ is the magnetic field strength, ρ is the density, T is the temperature of the fluid, C_p is the specific heat, α is the thermal diffusivity, T_w is the temperature of fluid at wall and T_∞ is ambient temperature.

The stream function Ψ is defined such that

The proper similarity transformations which reduce the governing partial differential equations into ordinary differential equations are defined below

where Re_b is defined as $Re b = ρ x n U 2 − n b .$

Using above similarity transformations in Eqs. (1)-(3). The Eq. (1) is identically satisfied, on the other hand the Eqs. (2)-(3) will transformed to

Subject to boundary conditions

Where dimensionless quantities i.e. magnetic field parameter M, curvature parameter γ, Eckert number Ec, material parameter A and Prandtl number Pr are defined as

The quantities of practical attention like skin friction coefficient and local Nusselt number are computed from following relations

Where

After using similarity transformations in Eqs.(11)-(12) the dimensionless forms of skin friction coefficient and local Nusselt number are

The present problem deals with the analysis of boundary layer flow of MHD Sisko fluid under the influence of viscous dissipation. The governing equations i.e. Eqs. (7)-(8) are highly nonlinear simultaneous ordinary differential equations. The solution of governing equations is calculated with the help of shooting method in conjunction with Runge-Kutta-Fehlberg technique for different values of pertinent parameters i.e. curvature parameter γ, material parameter A , magnetic field parameter M, Prandtl number Pr and Eckert number Ec. As Runge-Kutta-Fehlberg method solves only first order ODE’s. Thus, initially governing equations are transformed to first order differential equations. For this purpose governing equations are re-arranged as

Above equations i.e. Eqs. (14)-(15) are of order three in f and order two in θ. Now five new variables are introduced to reduce Eqs. (14)-(15) into first order ordinary differential equations, which are defined in Eq. (16)

After imposing Eq. (16) in Eqs. (14)-(15) these are converted to system of ordinary differential equations Eqs. (17)-(21) which is defined below,

The subjected boundary conditions in new variables are

The shooting method consists the following steps;

1. First step is to choose the appropriate value for the limit η → ∞. In this problem η → ∞ is considered 5.

2. Second and most important step is to select good initial approximations for y₃(0) and y₅(0), initially assumption for these are taken −1 and −1. Then this system of first order differential equations with initial conditions is solved with the help Runge-Kutta-Fehlberg integration scheme. The solution will converge if absolute differences of given and computed values of y₂(∞) and y₄(∞) are less than error tolerance i.e. 10⁻⁶. If these difference are larger than error tolerance, the initial guesses are modified through Newton method. This procedure is repeated until the criterion is satisfied.

The solution of the problem is computed by applying numerical technique shooting method. The variations in velocity and temperature profiles are computed and displayed for different values of pertinent parameters. Also accuracy of computed numerical solution is certified by comparing it with existing literature. In Table I the comparison between present and previously computed Rangi et al.³⁵ numerical values of skin friction coefficient for different values of curvature parameter γ is displayed. It can be seen that both values are agreed upto significant digits. Table II represents contrast of present result with recently published work i.e. Akbar et al.¹⁵ and Malik et al.¹⁶ This table shows that results are quite similar with each other.

Figure 1 is drawn to discuss the effects of applied magnetic field on the velocity profile for different values of power law index. As enhancement in magnetic field strength the Lorentz force produces. Since Lorentz force is opponent force, hence increase in Lorentz force causes decrease in the velocity, it can be observed from this figure.

Figure 2 demonstrates the impact of curvature parameter γ on the velocity profile for n = 1 and 2. As radius of the cylinder diminishes when curvature parameter γ enhances. Decrease in radius causes reduction in surface area, hence it provides less resistance to fluid motion. Thus motion of the fluid accelerates, which can be indicated from this figure that the momentum transfer of the flow distribution increases by increasing the curvature parameter γ.

Figure 3 shows the characteristics of the material parameter A on the velocity profile for n = 1 and 2. The viscous forces become weaker when material parameter A increases and offers less resistance which causes increase in the velocity of the fluid. It can be observed from figure that velocity increases by increasing the material parameter.

Temperature distribution demonstrated in Figure 4 for the different values of Eckert number Ec and power law index n. As Eckert number Ec is the relation between flow kinetic energy to heat enthalpy difference. So increase in Eckert number causes enhancement in the kinetic energy. Additionally it is well known fact that temperature is defined as average kinetic energy. Thus alternatively temperature of the fluid rises. It could be analyzed form this graph that fluid temperature increases when Eckert number Ec increases.

Figure 5 is created to discuss the behavior of curvature parameter γ on temperature distribution for n = 1 and 2. Since temperature is average kinetic energy and increase in curvature parameter γ enhances the velocity as well as kinetic energy, hence it increases the temperature. The above results is verified from this figure.

Figure 6 describes the impact of Prandtl number Pr on temperature profile. The thermal conductivity of the fluid declines by enhancing the Prandtl number Pr. Thus transfer of the heat slows which fall down the temperature of flow distribution. This figure validates the above result i.e. the temperature of the flow distribution falls when Prandtl number Pr increases.

Figure 7 displays the combined effects of magnetic field parameter M, material parameter A and power law index n on skin friction coefficient. This figure shows that magnetic field parameter and material parameter increase the shear stress on the wall of cylinder. But this figure indicates that power law index declines the skin friction coefficient.

The behavior of wall shear stress under the influences of magnetic field parameter M, curvature parameter γ and power law index n exhibits via Figure 8. Since by increasing curvature of cylinder boundary layer thickness and viscosity enhances near the surface, which increase the shear stress on the wall of cylinder. This fact can be validated from this figure.

Effects of Eckert number Ec and curvature parameter γ on local Nusselt number are depicted via Figure 9 by considering n = 1, 2. Since Eckert number varies inverse as boundary layer enthalpy. Thus when Eckert number inclines it declines difference between surface and ambient temperatures. Ultimately, it decreases the rate of heat transfer from the surface. Also rate of heat transfer enlarges when curvature parameter and power law index enhances.

Figure 10 demonstrates the variations in local Nusselt number by varying Eckert number Ec, Prandtl number Pr and power law index n. The convective mode of heat transfer accelerates when high Prandtl number is assumed. And since Nusselt number is directly proportional to convective heat transfer, thus alternatively it increases. The fact can be verified from this figure.

Table III is constructed to discuss the impact of physical parameters n, γ, M and A on skin friction coefficient. It can be observed form table that when all parameters increases the values of skin friction coefficient also increases.

Table IV is created to investigate the influence of all physical parameters γ, Ec and Pr on Nusselt number for n = 1, 2. This table shows that the values of Nusselt number increases by increasing flow parameters γ and Pr, while it diminishes by increasing the Eckert number Ec.

In the present analysis boundary layer flow of Sisko fluid over the stretching cylinder with combined influences of applied magnetic field and viscous dissipation is studied numerically. The following results are acquired

• The velocity of the fluid enhances by increasing both material parameter A and curvature parameter γ, on the other hand magnetic field parameter M decelerates the velocity.

• The temperature of the fluid rises when curvature parameter γ, Eckert number Ec are increases while effect of Prandtl number Pr on fluid temperature is opposite.

• Skin friction coefficient increases for larger values of A, M and γ while it decays when power law index varies from n = 1 to n = 2.

• Numerical values of local Nusselt number enhances by increasing γ, Pr and n while it declines for larger values of Eckert number Ec.