Flow of Sisko fluid over a stretching cylinder and heat transfer with viscous dissipation and variable thermal conductivity: A numerical study

In modern technology large number of complex fluids are used and these fluids have significantly different behavior from Newtonian fluids. Examples of these fluids are mixtures such as suspensions, paints, emulsions and lubricants etc. Also many products such as pharmaceutics, personal care products, toothpaste, paint, biological fluids etc. which are mostly encountered in our daily life do not satisfy Newton’s law of viscosity. These fluids are known as non-Newtonian. An interesting example of non-Newtonian fluid is blood flow in our arteries and veins. As non-Newtonian fluids have major contribution in our industry. Thus large number of studies have been attempted to discover the physical properties of non-Newtonian fluids. Different fluid models are proposed to analyze the properties of non-Newtonian fluids due to their diversity. The importance of non-Newtonian fluids is significant but one can never disregard the Newtonian fluids. So in current study the Sisko fluid model is considered which can predicts the properties of both Newtonian and non-Newtonian fluids. The most appropriate examples of Sisko fluid are lubricating greases, which were analyzed by Sisko.¹ Experimentally it is found that mostly real fluids such as human blood, paints, mud etc. obey the expression of viscosity proposed by Sisko. Recently, many researchers investigated Sisko fluid model with various physical assumptions and different geometries. Thin film flow of Sisko fluid over moving belt was analyzed by Akyildiz et al.² The modeled nonlinear differential equations were solved with HAM. Nadeem et al.³ studied peristaltic flow of non-Newtonian Sisko fluid in the uniform inclined tube. They presented contrast between different fluids and found that Newtonian fluids have better peristaltic pumping than all other fluids. Hayat et al.⁴ inspected the flow of MHD Sisko fluid over a porous wall with suction or blowing. Khan et al.⁵ deliberated the physical aspects of Sisko fluid through an annular pipe. They utilized homotopy analysis method to calculate solution of governing equations. The two dimensional axisymmetric, incompressible boundary layer flow of Sisko fluid over the radially stretching sheet was studied by Khan et al.⁶ They concluded that Sisko fluid are much faster than Newtonian and power-law fluids. Mekheimer et al.⁷ discussed the effects of chemical reactions on blood flow through tapered artery with time-variant overlapping stenoses. They utilized Sisko model for mathematical formulation. Khan et al.⁸ described the flow features of boundary layer flow of Sisko fluid over the stretching sheet. Variations in model were discussed for different values of pertinent parameters. Akber⁹ elaborated the peristaltic Sisko fluid with nano particles over asymmetric channel. She recommended that material parameter inclines pressure in peristaltic pumping regions, on the other hand it decays pressure in augmented pumping region. Moallemi et al.¹⁰ explored the physical properties of Sisko fluid through pipe and calculated the solution with He’s homotopy perturbation method. Mailk et al.¹¹ and Munir et al.¹² discussed the convective heat transfer of Sisko fluid. The influence of applied magnetic field on Sisko fluid over stretching cylinder was discussed by Mailk et al.¹³ They employed fifth order Runge-Kutta method to obtain numerical solution.

Experimentally, it is observed that when fluid particles move the viscosity of fluid alters some part of kinetic energy into thermal energy, this means it converts energy from worthy to non-worthy form i.e. dissipate energy. Such type of dissipation is called viscous dissipation. In heat transfer of many practical problems it plays an important role. The effect of viscous dissipation in heat transfer of fluid flow was initially encountered by Brickman.¹⁴ He analyzed heat transfer of capillary flows. Lin et al.¹⁵ investigated the viscous dissipation effect in the entrance region of pipe with convective boundary conditions. Abel et al.¹⁶ formulated the problem on boundary layer flow of non-Newtonian viscoelastic fluid over the stretching sheet and discussed the heat transfer with the effects of viscous dissipation and non uniform heating. They studied the heat transfer for both prescribed surface temperature(PST) and prescribed heat flux(PHF). They recognized that fluid temperature rises for larger values of Eckert number in PST case while the influence of Eckert number is opposite on temperature in PHF case. The effects of viscous dissipation combined with radiation on thermal boundary layer over a nonlinear stretching sheet is discussed by Cortell.¹⁷ Thiagarajan et al.¹⁸ modeled the problem of MHD flow over a nonlinear stretching plate. They assumed the heat transfer with free stream pressure gradient and viscous dissipation in the presence of variable thermal diffusivity. They suggested that viscous dissipation rises fluid temperature slightly within the boundary layer. Singh¹⁹ described the effects of viscous dissipation and variable viscosity on the boundary layer flow of MHD viscous fluid past a porous stretching sheet with suction. Ragueb et al.²⁰ investigated the effect of viscous dissipation of non-Newtonian fluid inside elliptical duct. Chand et al.²¹ formulated the problem on MHD flow over the unsteady stretching surface in a porous medium and analyzed the effect of viscous dissipation together with radiation. They suggested that Eckert number enhances the temperature. El-Aziz²² investigated the influence of viscous dissipation and variable viscosity on unsteady mixed convection flow. Saleem et al.²³ studied the impact of viscous dissipation on slip flow of viscous fluid. They found that Eckert number escalates the temperature but it shows opposite behavior on wall temperature gradient. Mabood et al.²⁴ studied the viscous dissipation effects on MHD flow of nanofluid along with convective boundary and second order slip conditions. Malik et al.²⁵ examined the flow characteristics of MHD Sisko fluid over stretching cylinder under the impact of viscous dissipation. They found that Eckert number grows the momentum transport while it decays rate of heat transfer from the surface.

Heat transfer is an important feature in many metallurgical and engineering processes such as annealing and tinning of copper wires, polymer extrusion, crystal growth, drawing plastic films, artificial fibers and aerodynamic extrusion of plastic sheets etc. Heat transfer problem over stretching surface was firstly discussed by Gupta and Gupta.²⁶ They discussed the heat and mass transfer of Newtonian fluid over a stretching sheet with suction or blowing. It can be seen that in the above literature^1–26 thermal conductivity of fluid was considered constant. But it is recognized fact that thermo-physical property like thermal conductivity changes significantly when temperature rises. Especially, in lubricants the internal friction causes enhancement in temperature which changes fluid conductivity. Thus thermal conductivity is assumed to be temperature dependent. Chaim²⁷ assumed temperature dependent thermal conductivity in heat equation. They concluded that fluid temperature rises due to variation in thermal conductivity. Abel et al.²⁸ studied the flow of power law fluid over stretching sheet assuming temperature dependent thermal conductivity and non-uniform heating. Ahmad et al.²⁹ inspected the boundary layer flow of Newtonian fluid over the stretching plate and heat transfer with variable thermal conductivity. Mishra et al.³⁰ studied the unsteady flow of viscous fluid over stretching plate and heat transfer with variable thermal conductivity. They discussed both prescribed surface temperature (PST) and prescribed heat flux (PHF) cases. They show that the influence of variable thermal conductivity is insignificant. Rangi et al.³¹ considered the problem of boundary layer flow of viscous fluid over stretching cylinder and heat transfer problem with variable thermal conductivity. Recently, Miao et al.,³² Manjunatha et al.,³³ Si et al.³⁴ and Malik et al.³⁵ investigated the fluid flows assuming varying thermal conductivity.

After surveying the aforementioned works, the author seems that combined effects of temperature dependent thermal conductivity and viscous dissipation on Sisko fluid has not been yet examined. Thus the basic aim of present work is to investigate the boundary layer flow of Sisko fluid over stretching cylinder with viscous dissipation and variable thermal conductivity. The flow govern equations are highly nonlinear, to obtain solution of these equations fifth order Runge-Kutta method is applied. Characteristics of involving physical parameters on velocity and temperature profiles are demonstrated via graphs.

Let consider a two dimensional axisymmetric steady state boundary layer flow of non-Newtonian Sisko fluid along the continuously stretching cylinder. The disturbance in flow is produced due to stretching velocity U(x) which is defined as U(x) = cx. The surface of cylinder is r = r₀ and fluid occupies the region r > r₀. The motion of the fluid is along x-axis. In heat transfer the effect of viscous dissipation is considered, also thermal conductivity is assumed linearly varying. With all these suppositions and after using boundary layer approximations, the continuity, momentum and heat equations are given below in usual notations

with the boundary conditions

Here u , v denote axial and radial components of fluid velocity, n is power law index, a is high shear rate viscosity and b denotes power law region viscosity. Also n, a and b are known as material constants of Sisko fluid. Fluid particles density is denoted by ρ, T is the temperature of the fluid, T_w is the temperature of fluid at wall, T_∞ is the temperature away from surface, α is the thermal diffusivity and C_p is the specific heat.

The stream function Ψ of fluid distribution is defined such that

The corresponding system of ordinary differential equations is obtained by applying following transformations

where α_∞ is the extreme thermal diffusivity, ϵ is the thermal conductivity parameter, Re_b is Reynolds number defined as $Re b = ρ x n U 2 − n b .$

The transformations which are defined in Eq. (6) are utilized in Eqs. (1)-(3). The continuity equation i.e. Eq. (1) is identically satisfied, while momentum and heat equations i.e. Eq. (2)-(3) are transformed to coupled nonlinear differential equations

along with the boundary conditions

In above system of equations γ, Ec, A and Pr denote curvature parameter, Eckert number, material parameter and Prandtl number respectively, which are defined below

The skin friction coefficient of flow field and local Nusselt number of temperature distribution are defined below

When transformations are used in Eqs. (11)-(12), the skin friction coefficient and local Nusselt number are converted into dimensionless form

The governing ordinary differential equations i.e. Eqs. (7)-(8) along with boundary conditions (9) are solved by applying fifth order Runge-Kutta method. In first step higher order differential equations are changed into first order differential equations. For this purpose first re-write Eqs. (7)-(8) as

define a new set of variables which is given below

After imposing Eq. (16) into Eqs. (14)-(15), they are converted into a system of five ordinary differential equations i.e. Eqs. (17)-(21)

The subjected boundary conditions reduce to

Since above system have five equations, its solution is computed with fifth order Runge-Kutta integration scheme unless five initial conditions are given. But in (22) only three initial conditions are defined. Thus before starting solution process, suitable initial approximations for y₃(0) and y₅(0) are selected. Also, upper limit of independent variable η should be chosen finite. After choosing some random values, η_∞ is selected 5 and both y₃(0) and y₅(0) are assumed −1. Now Runge-Kutta method is applied and solution is calculated. The computed solution will converge if boundary residuals i.e. difference between given and computed values of y₂(∞) and y₄(∞) is less than error tolerance i.e. 10⁻⁶. If computed solution does not meet the convergence criterion, then initial guesses are re-modified by Newton’s method. The procedure is continued unless solution meets the criterion.

In all above investigations of Sisko fluid, most of the authors ignored the influence of temperature dependent thermal conductivity and viscous dissipation on Sisko fluid. But it is observed that effects of viscous dissipation and thermal conductivity alters substantially against variations in temperature. Also, it is worth noticing that flow features are significantly changed by taking above mentioned effects into account. Thus to achieve more realistic results, the effect of viscous dissipation and thermal conductivity should be taken into account. Due to this fact, the main focus of present analysis is to explore the combined effects of viscous dissipation and thermal conductivity on Sisko fluid. The numerical solution of governing nonlinear coupled differential equations is found with fifth order Runge-Kutta method by varying pertinent parameters. The results show that fluid temperature and thermal boundary layer varies substantial against Eckert number while thermal conductivity affect them slightly. Also, to assess the accuracy of solution, values of local Nusselt number are compared with Chaim²⁷ and Abel et al.²⁸ results for different values of thermal conductivity parameter via Table. I. The present results have excellent agreement with previously computed results.

Figure. 1 displays the velocity f′(η) aptitude verses independent variable η for different values of material parameter A and power law index n. This figure shows that the fluid velocity is maximum at η = 0, but it decays and approaches to zero asymptotically at η = 5. Furthermore, Figure. 1 depicts that higher values of material parameter brings inciting in momentum for both shear thickening and shear thinning fluids. Because material parameter is the ratio between high shear rate viscosity and power law region viscosity. Finally, it can be seen from the figure that momentum boundary layer inclines as well.

Figure. 2 shows the impact of curvature parameter γ on velocity of fluid i.e. f′(η) for n = 1 and 2. It can be observed that velocity curves have decreasing manner near to surface but away from the surface velocity curves have opponent characteristics. From physical point of view, it is observed that increase in curvature corresponds to decrease in surface area by means of radius. As a result less resistance is offered to fluid motion.

Figure. 3 presents the effects of Eckert number Ec and power law index n on the temperature profile θ(η) verses independent variable η. Since Eckert number defined as the ratio of advective heat transfer to heat dissipation potential. The work done against the viscous fluid stresses i.e. the transformation of kinetic energy into internal energy was manifested by it. It is observed from the graph that an appreciable growth has been noticed in temperature due to dissipative heat. This provides evidence to the well known fact that thermal energy is accumulated in fluid as a consequence of dissipation due to fractional heating.

Figure. 4 demonstrates the influence of thermal conductivity parameter ϵ on temperature θ(η) for n = 1, 2. The temperature profile increases against larger values of thermal conductivity parameter ϵ. Because thermal conductivity is the aptitude of any material to conduct heat, so the fluid possess less thermal conductivity have low temperature and vice versa. But it is obligatory to mention here that the effects of viscous dissipation are more prominent on fluid temperature than variable thermal conductivity.

The variation of Prandtl number Pr on temperature profile θ(η) for n = 1 and 2 is displayed via Figure. 5. Since Prandtl number is ratio between convective to conductive mode of heat transfer, so when Pr increases it decelerates conductive mode of heat transfer which yields decline in fluid temperature.

Figure. 6 portrays the influence of curvature parameter γ on the temperature θ(η) verses η for n = 1 and 2. It is observed that in the dynamic region i.e. [0,0.9] the temperature of fluid decays, whereas fluid temperature have opponent features outside the region. Fluid temperature rises considerably for higher values of curvature parameter γ, this result is true because greater values of curvature parameter accelerates motion which results to increase in kinetic energy. And temperature is average kinetic energy of particles, so ultimately temperature increases.

Figure. 7 shows the curves of skin friction coefficient verses curvature parameter γ for variations in material parameter A and power law index n. It can be observed that γ brings enhancement in skin friction coefficient. Further, when values of n changes from 1 to 2 the wall shear stress reduces while A has opposite affect on it. This holds physically, because increment in both parameters A and γ diminishes the Reynolds number. Consequently, viscous forces near the surface become strong and increases the skin friction coefficient.

The graph of wall temperature gradient i.e. −θ′(0) verses ϵ is displayed in Figure. 8 for variations in Eckert number Ec and power law index n. Local Nusselt number decreases for each increment thermal conductivity parameter ϵ, because ϵ provides strength to conductive mode of heat transfer which has converse relation with Nusselt number. Also larger values of Eckert number Ec reduces the heat dissipation potential i.e. less heat transfer from the surface, as a consequence wall temperature gradient decays.

Figure. 9 depicts the influence of curvature parameter γ on Nusselt number for variations in ϵ and n. Curvature parameter γ enhances fluid velocity which accelerates convective heat transfer and so local Nusselt number which can be analyzed from the figure. Also this figure indicates that ϵ decelerates heat transfer from the surface while n accelerates it.

Figure. 10 stands for variation in −θ′(0) against different values of power law index n, Prandtl number Pr and thermal conductivity parameter ϵ. Since enhancement in Prandtl number causes increase in momentum which eventually accelerates the convection of heat and so local Nusselt number.

Table. II reflects the impact of flow parameters A and γ on the skin friction coefficient for material parameter n = 1 and 2. This table interprets that curvature parameter γ and material parameter A increases skin friction coefficient but n reduces it absolutely.

Table. III shows the influence of curvature parameter γ, Eckert number Ec, thermal conductivity parameter ϵ and Prandtl number Pr on local Nusselt number for material parameter n = 1 and 2. It is concluded form the table that the local Nusselt number decreases for flow parameters Ec and ϵ whereas it shows opposite behavior towards curvature parameter γ, power law index n and Prandtl number Pr.

In the present work, flow of Sisko fluid is investigated with viscous dissipation and variable thermal conductivity effects over stretching cylinder. The resulting nonlinear governing velocity and temperature equations are solved with shooting method. The main findings of this analysis are summarized as

• The increment in curvature parameter γ and Sisko parameter A enhances the velocity profile f′(η) while motion of the fluid decelerates by varying n.

• The temperature profile θ(η) increases when curvature parameter γ, Eckert number Ec and thermal conductivity parameter ϵ increases, while fluid temperature θ(η) falls down by increasing Prandtl number Pr.

• The skin friction coefficient enlarges verses both parameters γ and A while it shows decays against power law index n.

• Prandtl number Pr and curvature parameter γ increases the values of local Nusselt number. On the other hand Nusselt number declines for larger values of thermal conductivity parameter ϵ and Eckert number Ec.