1. Introduction

The study of fluid and heat transfer over stretching surfaces have gained considerable attention in last decade due to its applications in wide range of industrial mechanism like polymer extrusion, crystal growing, wire drawing, cable coating, spinning of filaments and many more. Further an important phenomenon of suction or injection in the fluid through the bounding surfaces can significantly change the flow field and, as a result affects the skin friction coefficient and the heat transfer rate at the surface. It is well known that, suction helps to slow down the fluid motion which results in increasing of skin friction and heat transfer rate, whereas injection has totally different behavior ¹. In all these cases, controlling the rate of heat transfer on the stretching surface is responsible for the good quality of product. The study on this topic was first considered by Crane ², who investigated the boundary layer flow initiated due to stretching sheet and found the solution analytically. After the pioneering work of Crane ², investigators subsequently extended the work to Newtonian and non-Newtonian fluids under different physical conditions (see ^3-9).

The aforementioned references are related to heat and fluid flow characteristics over a stretching sheet and very less attention is given for the stretching cylinder. The study of fluid flow over stretching cylinder was first initiated by Wang¹⁰. In his study he converted the Navier Stoke equation in polar form. Following the Wang ¹⁰ some researcher have extended his work by incorporating different physical phenomena. Uniform suction/blowing effect on flow and heat transfer due to stretching cylinder is discussed by Ishak et al. ¹¹. Ishak and Nazar ¹² have studied the flow around stretching cylinder. They assumed that cylinder is stretching linearly in axial direction and found similarity solution. This work was extension of Grubka and Bobba ¹³ and Ali¹⁴, from a stretching sheet to a stretching cylinder. Ishak et al.¹⁵, Ganesan and Loganathan ¹⁶ discussed boundary layer flow and heat transfer over an unsteady stretching vertical surface and the magnetic effects on horizontal and vertical stretching cylinder, respectively. Abbas et al. ¹⁷ have studied the combine effects of MHD and porous medium over stretching cylinder. The thermal radiation through a porous medium was also incorporated by considering the fluid as transparent. Recently, Mukhopadhyay ^18-19 studied the effects of MHD and mixed convection boundary layer flow over a stretching cylinder.

In all aforementioned studies, the fluid physical properties were taken to be constant. The physical properties of the fluid may change with temperature variation. Heat generated in lubricants effect the physical properties, so that these properties can no longer be remained constant. The increase of temperature significantly increases the fluid carriage phenomena by the reduction of physical properties across the momentum boundary layer and the heat transfer rate is also effected near the wall. Therefore, to predict the flow behavior accurately it is essential to take physical properties as function of temperature. After examine the literature on variable fluid properties (see ^20-23), it is disclosed that lot of work can be done for flow and heat analysis over stretching cylinder with variable fluid properties. In present study, we analyze the effects of magnetic field and variable Prandtl number on flow and heat transfer over stretching cylinder. The governing non-linear ordinary differential equations have been solved numerically using the Chebyshev Spectral Newton’s Iterative Scheme (CSNIS) ²⁴. The accuracy of the results is verified with previous studies through tables. All the calculations are carried out with thermo-physical effects and variable Prandtl number.

2. Mathematical formulation

We considered the steady two dimensional flow over a hollow cylinder with radius a. The fluid is assume to be viscous, incompressible and electrically conducting. The axis of the cylinder is along x-axis and r-axis is measured along radial direction as shown in Fig. 1.

FIGURE 1 A sketch of physical plane of considered problem. 

It is assumed that the cylinder is stretched along axial direction with U_w = b(x/L) and surface temperature of the cylinder is taken as T_w = c(x/L) where L and b represents the reference length and stretching rate of the cylinder respectively, c represents dimensionless constant. The phenomenon of suction/blowing prevails all over the surface of the cylinder with constant radial velocity v_w . All the thermos-physical properties are considered as constant except that the fluid viscosity and thermal conductivity which are assumed as a function of temperature ²⁶ in the following form.

(1)

Where k _∞ and μ _∞ are thermal conductivity and viscosity of the fluid at the ambient temperature, ε = (k - k _∞)/ k _∞ is small parameter, γ is thermal property of the fluid, ΔT = T_w - T _∞ is the temperature difference, T_w and k_w are temperature and thermal conductivity at the surface. The temperature dependent viscosity in Eq. (1) can be simplified as

Here A and T_r are constant dependent on physical situations. In general, A > 0 represents gases and A < 0 represents liquid. Typical values of γ and A for air are γ = 0.026240 and A = -123.2 (see Weast ²⁹). Let fluid viscosity parameter θ_r is constant and defined by

which can be determined by the viscosity/temperature properties of the fluid under consideration. It is important to mention here that for γ → 0, i.e., μ(T) = μ _∞ (constant, from Eq. (3)), θ_r → ∞_. As the viscosity is decreasing function of temperature for liquids and gases, it is increasing function of temperature. Therefore θ_r is positive for gases and negative for liquids. The boundary layer equations for continuity, momentum and energy with variable fluid properties under the above assumptions are

(5)

(6)

(7)

where ρ _∞ be the density, B_o be magnetic field strength, T is the temperature of the fluid, c_p is the specific heat, T _∞ is the ambient temperature and Q represents temperature dependent volumetric rate. Where Q > 0 refers to internal heat generation and Q < 0 represents internal heat absorption. The relevant boundary conditions are

Following Ishak and Nazar ¹², we define the similarity transformations for Eqns. (6)-(8) as

Here η be the similarity variable, ƒ (η) and θ(η) are dimensionless quantities and v _∞ be the kinematic viscosity. ψ be the stream function for which we find the expressions of u = ∂ψ/r∂r and v = ∂ψ/r∂x. The expression of velocity takes the following forms

and Eqns. (6,7) becomes

(11)

(12)

With corresponding boundary conditions

Here

and prime denotes the differentiation with respect to η, K be the curvature parameter, M be the magnetic parameter, Pr _∞ be the ambient Prandtl number, β be the heat source sink and ƒ_w be suction/injection parameter, respectively. As by definition of Prandtl number, it is relation of viscosity, thermal conductivity and specific heat. It is well known that fluid viscosity and thermal conductivity varies across the boundary layer, therefore Prandtl number also varies. Moreover the assumption of constant Prandtl number inside the boundary layer with temperature dependent viscosity and variable thermal conductivity may leads to unrealistic results(see ^25-28). Due to said reason, the variable Prandtl number is used here. The Prandtl number under the assumption of variable fluid properties is defined as

Upon using Eq. (15), the non-dimensional energy Eq. (12) can be expressed as

(16)

The quantities of interest for engineers and scientists are skin friction coefficient C_ƒ and Nusselt number Nu respectively defined as

(17)

(18)

Utilizing the variables in (9), we get

Where Re_x is Local Reynolds number.

3. Numerical method

The solution of dimensionless boundary layer Eqs. (11,16) subject to the boundary conditions (13) is obtained by efficient numerical method named as Chebyshev Spectral Newton’s Iterative Scheme ²⁴ (CSNIS). For the validity of numerical scheme the results are equated with already published studies as a limiting case and found excellent agreement (see Table I and II). The outline of the employed scheme is as follows:

TABLE I The comparison of −ƒ´´(0) for different values of ƒ_ω when K=0, M=0, θᵣ→∞ and ɛ=0 

TABLE II The comparison of −θ´(0) for different values of Pr_∞ when K=0, M=0, ƒ_ω =0, ɛ=0, β=0 and θᵣ→∞ 

4. Results and discussion

The obtained boundary layer flow and energy equations are numerically solved by Chebyshev Spectral Newton’s Iterative Scheme. The numerical results are plotted graphically for different emerging parameters including curvature parameter K, magnetic parameter M, fluid viscosity parameter η, injection/suction parameter ƒ_w , variable thermal conductivity parameter, heat source/sink parameter β and the variable Prandtl number Pr _v .

Figures 2(a), 2(b) and 2(c) demonstrate temperature profiles for different values of curvature and magnetic parameters by keeping ƒ_w = 0.0, -0.5, 0.5 respectively. The increasing values of curvature parameter leads to enhancement in thermal boundary layer thickness. This behavior is equally true for magnetic parameter, which in turn increases the temperature profile. It is noticed that in Figs. 2(b) and 2(c) thermal boundary layer thickness increases and decreases due to injection and suction respectively. Figures 3, 4 and 5 capture the effects of fluid viscosity parameter over a stretching cylinder for various curvature parameter K, keeping ƒ_w = 0.0, -0.5, 0.5 respectively. These lead to observation that with increasing values of θ _r temperature decreases slightly for all values of ƒ_w . The decreasing trend of temperature profile and thermal boundary layer is observed with increasing values of injection parameter ƒ_w . The temperature distribution for different values of the curvature parameter K and the heat source/sink parameter β for both blowing and suction are presented in Figs. 6(a,b). From these graphs, we observe that the temperature distribution is lower within the boundary layer for negative value of β (heat sink) and higher for positive values of β(heat source). It is observed that temperature and thermal boundary layer thickness increase with the increasing value of the heat source/sink parameter β. Same behavior is observed for all values of the transverse curvature parameter K. Figures 7(a,b) is drawn to show the effects of curvature parameter K on temperature profile against η for different value of variable thermal conductivity parameter ε for both blowing and suction cases. These figures clearly indicate that temperature and thermal boundary layer thickness

FIGURE 2 The impact of K and M on temperature profile with Pr _v = 0.71, ε = 0.2, θ_r = −3, β = −0.1 when (a) f_w = 0.0 (b) f_w = −0.5 (c) f_w = 0.5. 

FIGURE 3 Temperature profile for different values of K and θ_r with Pr _v = 0.71, ε = 0.2, M = 0.2, β = −0.1 when f_w = 0.0. 

FIGURE 4 Temperature profile for different values of K and θ_r with Pr _v = 0.71, ε = 0.2, M = 0.2, β = −0.1 when f_w = −0.5. 

FIGURE 5 Temperature profile for different values of K and θ_r with Pr _v = 0.71, ε = 0.2, M = 0.2, β = −0.1 when f_w = 0.5. 

FIGURE 6 Temperature profile for different values of K and β with Pr _v = 0.71, ε=0.2, M=0.2, θ_r =−5.0 when (a) f_w = −0.5 and (b) f_w = 0.5. 

FIGURE 7 Temperature profile for different values of K and ε with Pr _v = 0.71, β = −0.1, M = 0.2, θ_r = −5 when (a) f_w = −0.5 and (b) f_w = 0.5. 

increases with increase of ε and K where as in case of injection, temperature and thermal boundary layer thickness is larger as compared to suction. Figure 8(a,b) show the temperature profile for different value of variable Prandtl number Pr _v for both blowing and suction cases respectively. One can clearly see from these figures that an increase in variable Prandtl number leads to decrease in temperature profiles and hence thermal boundary layer thickness decreases.

FIGURE 8 Temperature profile for different values of K and Pr _v with ε = 0.2, β = −0.1, M = 0.2, θ_r = −5 when (a) f_w = −0.5 and (b) f_w = 0.5. 

Figures 9 and 10 exhibits the variations in variable Prandtl number in the domain of boundary layer for various values of the viscosity parameter θ_r with the fixed value of ambient Prandtl number Pr_∞ = 0.71. Both figures show that the variable Prandtl Pr _v asymptotically converges to the fix value of ambient Prandtl Pr_∞ as η→∞. It is noticed that at the surface of cylinder Pr _v approaches Pr_∞ for large values of θ_r . It is also observed that Pr _v decreases with increase in θ_r when it is positive while opposite behavior is observed for negative values of θ_r . Table III demonstrates the significance of variable viscosity parameter θ_r on the Nusselt number within the domain of boundary layer for both constant and variable Prandtl number.

FIGURE 9 Variation of variable Prandtl number Prv with ε = 0.2 for several values of θ_r (a) positive and (b) negative. 

FIGURE 10 Variation of variable Prandtl number Pr _v with ε = 0 for several values of θ_r (a) positive and (b) negative. 

Table III Values of NuRe _x ^−1/2 for different values of θ_r and ε = 0.1. 

Numerical data shows that values of Nusselt number are smaller in a fluid of constant Prandtl number than a fluid of variable Prandtl number when θ_r . is negative and opposite behavior is seen for positive values of θ_r . The relative error is increasing when θ_r is decreasing and decreasing when θ_r is increasing. These variations in rate of heat transfer clearly support the argument that constant Prandtl number produce unrealistic results.

5. Conclusions

Hydromagnetic flow and heat transfer is examined in the present article. The fluid properties and Prandtl number are considered as a function of temperature. The solution of present problem is achieved by Chebyshev Spectral method having excellent agreement with published studies and fast convergent. The results of detailed information concerning the effects of emerging parameters is discussed. Important deductions of the present theoretical study are