1 Introduction

The heat transfer over a moving surface is one of the current research areas due to its ex- tensive applications in science and engineering disciplines,especially in chemical engineering processes. In such processes,the fluid mechanical properties of the end product mainly de- pend on two factors. One is the cooling liquid property,and the other is the stretching rate. Many practical situations demand for the physical properties with variable fluid characteristics. Thermal conductivity is one of such properties,which is assumed to vary linearly with the temperature ^[1] . Chiam ^[2] investigated the heat transfer with variable conductivity in the stag- nation point flow towards a stretching sheet. Vyas and Rai ^[3] examined the steady boundary layer flow with variable thermal conductivity over a non-isothermal stretching sheet placed at the bottom of a saturated porous medium. Misra et al. ^[4] investigated the unsteady boundary layer flow past a stretching plate and the heat transfer with variable thermal conductivity. Pal and Chatterjee ^[5] investigated the magnetohydrodynamic (MHD) mixed convection flow of a power law fluid with the combined action of Soret and Dufour. They considered the flow anal- ysis subjected to variable thermal conductivity,thermal radiation,chemical reaction,Ohmic dissipation,and suction/injection. Anselm and Koriko ^[6] analyzed the variable thermal conduc- tivity effects in the compressible boundary layer flow of a viscous fluid by a circular cylinder. They showed that the thermal radiation effects were especially important in the heat transfer processes at high temperatures. Many technological processes occur at high temperatures,and the knowledge of radiative heat transfer cooperates in designing the pertinent equipment. Su et al. ^[7] studied the MHD mixed convective flow and heat transfer of an incompressible fluid by a stretching permeable wedge. They analyzed the flow in the presence of thermal radiation and Ohmic heating. Bhattacharyya et al. ^[8] investigated the effects of thermal radiation on the flow of a micropolar fluid past a porous shrinking sheet. Prasad et al. ^[9] studied the thermal radiation effects in the MHD free convection heat and mass transfer from a sphere. Hayat et al. ^[10] examined the radiative flow of a Jeffrey fluid in a porous medium with power law heat flux and heat source. Rashad ^[11] studied the effects of radiation and variable viscosity on the unsteady MHD rotating flow of a viscous fluid over a stretching surface. Turkyilmazoglu and Pop ^[12] considered the heat and mass transfer effects in the flow of a viscous nanofluid past a vertical infinite flat plate with the radiation effect. Available information on the topic indicates that very little relevant to a stretching cylinder has been presented. Some representative studies in this direction can be refereed to Refs.^{[13, 14, 15, 16, 17]}.

Various experiments performed on blood with varying hematocrits,anticoagulants,temper- ature,etc. strongly suggest the behavior of blood as a Casson fluid. This article deals with the flow of a Casson fluid due to a stretching cylinder. The motivation of this study is to understand the flow of pigment suspensions in the lithography used for the preparation of printing inks and silicon suspensions. A Casson fluid describes the properties of many polymers over a wide range of shear rates. The main objective here is to analyze the boundary layer flow of a Casson fluid due to a stretching cylinder. The Casson fluid model has been used in Refs.^{[18, 19]}. In the present analysis,variable thermal conductivity is considered. In addition,the thermal radiation effects are present. The resulting problems for the velocity and temperature are first modeled and then solved by the homotopy analysis method ^{[20, 21, 22, 23, 24, 25]} ,and the skin friction and the local Nusselt number are examined carefully.

2 Mathematical formulation

We consider the steady and two-dimensional incompressible flow of a Casson fluid over an inclined stretching cylinder. The stretching cylinder makes an angle α with the horizontal axis, i.e.,the x-axis or r-axis,being normal to it. In addition,the energy equation is considered with combined effects of thermal radiation and viscous dissipation. The thermal conductivity depends on the temperature. The boundary layer equations comprising the balance laws of mass,linear momentum,and energy can be written as

In the above expressions,ν is the kinematic viscosity,ρ is the fluid density,T is the fluid temperature,c p is the specific heat,q r is the radiative heat flux expressed by

k^* is the mean absorption coefficient,g₀ is the acceleration due to gravity,β_T is the volumetric coefficient of thermal exponential,σ^* is the Stefan-Boltzmann constant,and k is the variable thermal conductivity of the fluid defined as in which ε is the small parameter,k ∞ is the thermal conductivity of the fluid far away from the surface,and

The boundary conditions are taken as follows:

Introduce the relations

Then,Eq.(1) is identically satisfied,and Eqs. (2)−(6) become where prime denotes the differentiation with respect to η,f is the dimensionless stream function, θ is the dimensionless temperature,and the dimensionless numbers are 

Here,β is the dimensionless material parameter,π_c is a critical value of the product of the component of the deformation rate with itself,τ_r is the yield stress of the fluid,R is the radiation parameter,γ is the curvature parameter,G is the mixed convection parameter,Pr is the Prandtl number,µ_B is the plastic dynamic viscosity,and Ec is the Eckert number.

The skin friction coefficient C_f is defined as

The local Nusselt number Nu_x is defined as 3 Homotopy analysis solutions

Initial approximations and auxiliary linear operators are taken as

subjected to the properties where C i (i = 1,2,3,4,5) are constant.

If p (p ∈ [0, 1]) indicates the embedding parameter,the zeroth-order deformation problems can be constructed as follows:

Here,h_f and h_θ are the non-zero auxiliary parameter and the nonlinear operator,respectively, and Then,we have the following equations:

It is noticed that when p varies from 0 to 1,f(η;p) and θ(η;p) approach from f 0 (η),θ 0 (η) to f(η),θ(η). The series of f and θ through Taylor’s expansion are chosen to be convergent for p = 1. Therefore,

The resulting problems at the mth-order can be presented as follows:

where

The general solutions (f_m ,θ_m ) comprising the special solutions (f^*_m ,θ^*_m ) are 4 Convergence of homotopy solutions

It is a well recognized fact that the convergence of the series solutions (26) and (27) depends upon the non-zero auxiliary parameter h. To find the pertinent values of the auxiliary parameters,we plot the h-curves for the velocity and temperature profiles (see Fig. 1 and Fig. 2).

It is clear from Fig. 1 and Fig. 2 that the admissible values of h_f and h_θ are

These series solutions converge for the whole region of η when h_f = −0.7 and h_θ = −0.7.

Table 1 shows the convergence of the homotopy solutions. It is obvious that the 25th-order approximations are enough for the convergent series solutions.

5 Discussion

Our intention in this section is to analyze the velocity and temperature profiles for different physical parameters,e.g.,the Casson fluid parameter,the curvature parameter angle of inclina- tion,the mixed convection parameter,the Eckert number,the Prandtl number,the radiation parameter,and the variable thermal conductivity parameter. The features of the Casson fluid and curvature parameters are displayed in Fig. 3 and Fig. 4. It is observed that the velocity field is a decreasing function of β. The velocity increases with the increase in the curvature parameter γ. The rate of transport decreases with the increase in the distance from the cylinder,and vanishes at some large distance from the cylinder.

Figure 5 exhibits the variation of the angle of inclination. It is seen that α increases with a decrease in the velocity. This is due to the fact that the angle of inclination increases the effect of the buoyancy force due to the thermal decrease by the factor of cosα. Figure 6 presents the effects of the mixed convection parameter G on the velocity profile. When the mixed convection parameter increases,the velocity decays very slowly to the ambient,while the fluid motion can infiltrate quite deeply into the ambient fluids. Physically speaking,positive G means heating the fluid or cooling the boundary surface,while negative G means cooling the fluid or heating the boundary surface. When G = 0,the free convection current is absent. When η > 9,the temperature field vanishes asymptotically.

The effects of the curvature parameter γ on the temperature profile are displayed in Fig. 7. When γ = 0,the outer surface of the cylinder behaves like a flat plate. The temperature profile increases the most rapidly when the curvature parameter γ increases. When 0 ≤ η ≤ 1,no variation can be observed for various γ. The temperature and thermal boundary layer thickness decrease for large Pr. The fluid with higher Pr is more viscous,and that with lower Pr is less viscous. The fluid with a higher viscosity has a lower temperature,while the fluid with a lower viscosity has a higher temperature. This leads to the decreases in the temperature and boundary layer thickness (see Fig. 8).

Figure 9 illustrates the effects of ε on the temperature profile. It can be seen that the temperature increases with the increase in the thermal conductivity,whereas the temperature profile decays when the thermal conductivity is constant (e.g.,ε = 0). As we have assumed that the thermal conductivity varies linearly with the temperature,which causes the reduction in the magnitude of the transverse velocity,and the temperature profile increases with an increase in the variable thermal conductivity parameter. The enhancement in R accelerates the temperature and the boundary layer thickness (see Fig. 10). Because larger R implies higher surface heat flux and thereby makes the fluid become warmer,the temperature profile enhances with the increase in R. Here,we note that the temperature increases the most rapidly in the region 0 ≤ η ≤ 4.

Figure 11 plots the temperature distribution versus the Eckert number Ec. It is seen that the fluid temperature rises when the Eckert number becomes pronounced. Physically,the Eckert number depends on the kinetic energy. When we increase the values of the Eckert number, the kinetic energy enhances. This enhancement in the kinetic energy leads to the increases in the temperature and thermal boundary layer thickness. In all the cases,the velocity is the maximum at the surface of the cylinder,and it starts decreasing when η → ∞.

Fig. 12,Fig. 13,Fig. 14,Fig. 15 show the effects of ε,R,Ec,and Pr on the temperature distribution for the flat plate (i.e.,γ = 0) and the stretching cylinder (i.e.,γ = 0.7),respectively. It is noticed that the temperature profile increases remarkably when γ = 0.7 in comparison with that when γ = 0 (see Fig. 12,Fig. 13,Fig. 14). The magnitude of the boundary layer thickness in the case of the stretching cylinder is larger than that of the flat plate. Figure 15 illustrates that for larger Pr, the temperature profile decays,and the thermal boundary layer thickness reduces. It is also noted that the effect of Pr is much more prominent for the stretching flat plate than for the stretching cylinder. The fluid with lower Pr possesses higher thermal conductivity than that with higher Pr,resulting that the heat in it can diffuse faster from the cylinder.

The magnitude of the skin friction coefficient increases when the curvature parameter γ and the angle of inclination α increase (see Table 2). Table 3 shows the local Nusselt number for different sets of the involved parameters. The effects of β and γ on the local Nusselt number are similar. The magnitude of the local Nusselt number decreases for larger Ec,R,and ε.

6 Conclusions

An analysis is carried out for the flow and heat transfer in the presence of variable thermal conductivity and thermal radiation effects by a stretching cylinder. The present study indicates that due to the increase in the curvature parameter (γ),the velocity and temperature increase. Variable thermal conductivity enhances the temperature profile and the associated boundary layer thickness. The magnitude of the temperature in the case of a stretching cylinder rises when the Eckert number (Ec),the radiation parameter (R),and the variable thermal conduc- tivity parameter (ε) increase. However,it is smaller than that for a flat plate. The distance of the boundary layers from the leading edge is greater when we take a stretching cylinder instead of a flat plate. The local Nusselt number decreases when the radiation parameter (R) and the variable thermal conductivity parameter (ε) increase.

Acknowledgements We are grateful to the reviewers for their useful suggestions.