Impact of Cattaneo-Christov heat flux in the flow over a stretching sheet with variable thickness

The present analysis concentrates on the boundary layer flow of Maxwell fluid over a stretching sheet with variable thickness. Cattaneo-Christov heat flux model is used instead of classical Fourier’s law to explore the heat transfer characteristics with variable thermal conductivity. Suitable transformations are employed to achieve the nonlinearordinarydi ff erentialequations.Convergentseriessolutionsofthemomentum and energy equations are obtained. Behavior of various pertinent parameters on the velocity and temperature distributions are analyzed and discussed. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http: // dx.doi.org / 10.1063 / 1.4929523]


INTRODUCTION
Transfer of heat is an important phenomenon in the nature which exists due to temperature difference between two bodies or within the same body. Characteristics of heat transfer have been explored by Fourier's law 1 of heat conduction in the last two centuries. However this model is not adequate in the sense that any initial disturbance is felt instantly throughout the whole substance. To overcome this difficulty Cattaneo 2 added a thermal relaxation time in the classical Fourier's of heat conduction which allows the transport of heat via propagation of thermal waves with finite speed. After that Christov 3  The analysis of boundary layer stretched flow accompanied with heat transfer has gained the interest of researchers and scientists due to their widespread applications in many areas of manufacturing, industrial, metallurgical and engineering phenomena. Moving surface into a cooling medium is a mathematical tool for the process of heat treatment. During this process metals are heated and cooled in a specific order so that to keep the metal away from molten state. The objective of heat treating process is to make a metal stronger, harder and resistant. Heat treating process is also used to make a metal softer and more ductile. Few applications here comprise of polymer extrusion, glass blowing, crystal growth, paper production, drawing plastic films, annealing and tinning of copper wires, manufacturing artificial fibers, boundary layer along a liquid film in condensation processes and aerodynamic extrusion of plastic sheets etc. In many manufacturing processes, the raw material passes through the die for the extrusion in liquefied state under high temperature. Unsteady boundary layer flow over a permeable stretching sheet with non-uniform heat source/sink is investigated by Zheng et al. 8 Rashidi et al. 9 examined the thermal magnetohydrodynamic (MHD) flow of nanofluid induced by stretching sheet. Su et al. 10 analyzed magnetohydrodynamic mixed convection flow past a permeable stretching wedge with Ohmic heating and thermal radiation. Turkyilmazoglu 11 investigated three-dimensional magnetohydrodynamic flow of viscoelastic fluid over a stretching sheet with different aspects. Magnetohydrodynamic flow past a shrinking surface with velocity slip and temperature jump is studied by Zheng et al. 12 Hayat et al. 13 19 Characteristics of heat transfer in the flow of nanofluid over a nonlinear stretching sheet with variable thickness are explored by Wahed et al. 20 The main objective of present analysis is to describe the rheological characteristics of upperconvected Maxwell fluid and Cattaneo-Christov heat flux model in flow over a stretching sheet with variable thickness. Thermal conductivity of the material is assumed variable. Non-similar analytic solutions are obtained by applying homotopic technique. [21][22][23][24][25][26] Impacts of various pertinent parameters are the velocity and temperature distributions are computed and discussed.

MATHEMATICAL FORMULATION
Consider the incompressible boundary layer flow of Maxwell fluid over a stretching sheet with variable thickness. Unlike the Fourier law of heat conduction Cattaneo-Christov heat flux model is employed to analyze the characteristics of heat transfer. Thermal conductivity of the fluid is assumed variable. Cartesian coordinates are chosen in such a way that x-axis is along the sheet while y-axis is normal to it (see Fig. 1). The conservation laws of mass, linear momentum and The new flux model is known as "Cattaneo-Christov heat flux model" and it has the following form: 3 Where λ 2 is the relaxation time of heat flux, k (T) is the variable thermal conductivity which depends linearly on temperature. It is noted that for λ 2 = 0 Eq. (4) reduces to classical Fourier's law.
As it is assumed that fluid is incompressible therefore Eq. (4) takes the form Comparing Eqs. (3) and (5) and then eliminating q, we get the energy conservation law corresponding to Cattaneo-Christov heat flux model as follows: The subjected boundary conditions are: 16 In the above expressions u and v denote the velocity components in the x− and y− directions respectively, λ 1 is the relaxation time, u w is the stretching velocity, ν is the kinematic viscosity, b is dimensional constant, n is the velocity power index, ρ is the density, c p is the specific heat, T is the fluid temperature and T ∞ is the ambient fluid temperature. Where k (T) is the variable thermal conductivity which is defined as follows: where k ∞ is the thermal conductivity of the ambient fluid, θ is the dimensionless temperature and ε is a small scalar parameter which shows the influence of temperature on variable thermal conductivity. Considering the following transformations 16 incompressibility condition is satisfied automatically and Eqs. (2), (6), and (7) are reduced to which reduces the nondimensionlized governing equations and associated boundary conditions as follows: where β is the Deborah number in terms of relaxation time, Pr is the Prandtl number and γ is the thermal relaxation parameter. These quantities are expressed as follows: It is noted that for n = 1, the problem reduces to flat plate with same thickness. Also it is noted that for n = 1 and ε = 0, the problem reduces as discussed by Han et al. 6

HOMOTOPIC SOLUTIONS
Homotopy analysis method was first proposed by Liao 17 in 1992 which is used for the construction of series solution of highly nonlinear problems. To proceed with such method, it is essential to define the initial guesses ( f 0 , θ 0 ) and linear operators (L f , L θ ) for the momentum and energy  equations which are expressed in the forms with where A i (i = 1, 2, . . . , 5) are the arbitrary constants. The zeroth and mth order deformation problems are:

ZEROTH-ORDER PROBLEM
Where p ∈ [0, 1] is embedding parameter and f and θ are the non-zero auxiliary parameters.

MTH-ORDER DEFORMATION PROBLEMS
For p = 0 and p = 1 , we can writê and with the variation of p from 0 to 1, f (η; p) and ⌢ θ (η; p) vary from the initial solutions f 0 (η) and θ 0 (η) to the final solutions f (η) and θ(η) respectively. By Taylor's series we have The value of auxiliary parameter is chosen in such a way that the series (34) to (35) converge at p = 1 i.e.
The general solutions ( f m , θ m ) of Eqs. (26) and (27) in terms of special solutions ( f * m , θ * m ) are given by where the constants A i (i = 1, 2, . . . , 5) through the boundary conditions (28) have the values

CONVERGENCE ANALYSIS
To develop the series solutions by homotopy analysis technique it is also essential to check their convergence. Convergence region is the region parallel to -axis. Therefore we have plotted the -curves in the Fig. 2. It is found that the admissible ranges of the auxiliary parameters f and θ are −1.5 ≤ f ≤ −0.5 and −1.3 ≤ θ ≤ −0.3.

DISCUSSION
Description of various pertinent parameters on the velocity and temperature distributions is the main objective of this section. Impact of Deborah number (in terms of relaxation time) β on velocity distribution is illustrated in Fig. 3. It is noted that velocity distribution shows decreasing behavior corresponding to higher values of Deborah number. In fact Deborah number is the ratio of relaxation to observation times. So with increase in Deborah number relaxation time also increases which provides more resistance to the fluid motion. Therefore velocity profile decreases. Behavior of wall thickness parameter α on the velocity profile is sketched in Fig. 4. Higher values of wall thickness parameter result in the reduction of velocity profile and momentum boundary layer thickness for n < 1. As we increase the wall thickness parameter for n < 1, stretching velocity decreases which results in reduction of velocity profile. Analysis of power index non velocity distribution is sketched in Fig. 5. It is noted that velocity distribution shows increasing behavior corresponding to higher  values of n. In fact stretching velocity increases as we increase the values of n which produces more deformation in the fluid. Hence velocity distribution increases. Characteristics of Deborah number β on temperature distribution is displayed in Fig. 6. Temperature distribution increases for larger values of Deborah number. Because higher Deborah number corresponds to larger relaxation time which provides resistance to the fluid motion and as a result more heat is produced. Therefore temperature distribution increases. Behavior of wall thickness parameter α on temperature profile is shown in Fig. 7. It is concluded that temperature distribution as well as thermal boundary layer thickness decrease for higher wall thickness parameter. It is due the fact that as we increase the wall thickness parameter less heat is transferred from sheet to the fluid. Hence temperature profile decreases. Fig. 8 discloses the behavior of thermal conductivity parameter ε on temperature profile. Higher values of thermal conductivity parameter result in enhancement of temperature profile. Physically, as we increase the thermal conductivity parameter, thermal conductivity of the fluid increases. So more heat is transferred from sheet to the fluid and consequently temperature profile increases. Impact of thermal relaxation parameter γ on temperature distribution is illustrated in Fig. 9. Temperature distribution is a decreasing function of thermal relaxation parameter. Further it is also analyzed that thermal boundary layer thickness decreases. It is due to fact that as we increase the thermal relaxation parameter, particles of the material require more time to transfer heat to its neighboring particles. In other words we can say that for higher values of thermal relaxation parameter material shows a non-conducting behavior which is responsible in reduction of temperature distribution. Further it is also noted that for γ = 0 heat transfers promptly throughout the material. Therefore temperature distribution is higher for γ = 0 i.e., for Fourier's law as compared to Cattaneo-Christov heat flux model. Analysis of Prandtl number Pr on temperature profile is displayed in Fig. 10. Higher values of Prandtl number result in the reduction of temperature profile as well as thermal boundary layer thickness. Prandtl number is the ratio of momentum to thermal diffusivities. So higher prandtl number corresponds to lower thermal diffusivity which is responsible in reduction of temperature profile. Influence of power index n on temperature distribution is shown in Fig. 11. Temperature distribution increases for higher values of n. Further it is also noted that thermal boundary layer thickness increases. Table I shows the convergence of the series solutions for different order of approximations. It is noted that 9 th and 15 th order of approximations are sufficient for the convergence of momentum and energy equations respectively.

CLOSING REMARKS
We have investigated the characteristics of forced convection boundary layer flow of Maxwell fluid induced by stretching sheet with variable thickness. Cattaneo-Christov heat flux model is imposed to disclose the heat transfer characteristics of variable thermal conductivity vis-coelastic fluid. The main observations are summarized as follows: • Higher values of Deborah number result in the reduction of velocity and momentum boundary layer thickness. • Velocity distribution shows decreasing behavior for larger values of wall thickness parameter when n < 1. • Temperature distribution is higher in the case of Fourier's law as compared to Cattaneo-Christov heat flux model. • Variable conductivity parameter results in enhancement of temperature distribution.
• Velocity and temperature distributions enhance for larger values of n.