Effects of Viscous Dissipation on the Thermal Boundary Layer of Pseudoplastic Power-Law Non-Newtonian Fluids Using DiscretizationMethod and the Boubaker Polynomials Expansion Scheme

Heat transfer of pseudoplastic power-law non-Newtonian fluids aligned with a semi-infinite plate is studied. Unlike in most classical works, the effects of viscous dissipation which is coupled with the temperature-dependent thermal diffusivity are considered in the energy equation. The discretization method is used to convert the governing partial differential equations into a set of nonlinear ordinary differential equations. The solutions are presented numerically by using the shooting technique coupled with the Newtonian method and the Boubaker polynomials expansion scheme. The effects of power-law index and the Zheng number on the dynamics are analyzed. The associated heat-transfer characteristics are also tabulated in some domains of the said parameters.


Introduction
Since 1960, a considerable attention has been devoted to predict the drag force behavior and energy transport characteristics of the non-Newtonian fluid flows.The main reason for this is probably that fluids (such as molten plastics, pulps, slurries, emulsions), which do not obey the assumption of Newtonian fluids that the tress tensor is directly proportional to the deformation tensor, are found in various engineering applications.A variety of constitutive equations have been proposed to describe the flow and heat transfer non-Newtonian characteristics, among them the empirical Ostwaald-de Waele model, which is known as the so-called power-law model, gained much acceptance.For an incompressible power law non-Newtonian fluid, its shear stress is characterized as τ xy = K|∂U/∂Y | n−1 ∂U/∂Y and the kinematics' viscosity is ν = γ|∂U/∂Y | n−1 (where γ = K/ρ, ρ is the density and K is the consistency index).The case n = 1 corresponds to a Newtonian fluid.0 < n < 1 is descriptive of pseudoplastic fluids while n > 1 describes dilatant's fluids.
Schowalter [1] and Acrivos et al. [2] successfully applied the power-law model to the boundary layer problems.The boundary-layer equations were formulated, and the conditions for the existence of similarity solutions were established.Following the pioneering works of Schowalter and Acrivos et al. [1,2], Wang [3,4], and Hady [5] analyzed the mixed convection heat transfer from a vertical or horizontal plate of non-Newtonian fluids with uniform surface heat flux.Kumari et al. [6] considered the freeconvection boundary-layer flow of a non-Newtonian fluid along a vertical wavy surface.Howell et al. [7] and Rao et al. [8] examined the momentum and heat transfer of the power-law fluids on a continuous moving surface.Hassanien et al. [9] investigated the heat transfer in power-law flow ISRN Thermodynamics over a nonisothermal stretching sheet.Luna et al. [10] considered the case of conjugated heat transfer of a power law laminar fluid in circular ducts.Pinarbasi and Imal [11] extended their work to the viscous heat effects on the linear stability of Poiseuille flow of an inelastic fluid.Chen [12] analyzed the effect of viscous dissipation on heat transfer in non-Newtonian power law liquids over an unsteady stretching sheet.For the non-Newtonian powerlaw fluids, the hydrodynamic problem of the MHD boundary layer flow over a continuously moving surface has been dealt with by several authors (e.g., Andersson et al. [13], Cortell [14], and M. A. A. Mahmoud and M. A. E. Mahmoud [15]).The effect of magnetic field is found to decrease the velocity distribution and thus to increase the skin-friction coefficient.
It seems that, in all the works cited above, the power law kinematic viscosity was applied only on the momentum equations and the thermal conductivity is still treated as constant.This is inconsistent with the known fact that the change of viscosity should affect both the momentum and heat equation of the power law flows.Pop et al. [16,17] proposed a model which states that the thermal conductivity of non-Newtonian fluids was power-law-dependent on the velocity gradient.Based on this consumption, Gorla et al. and Ece and Buyuk [18,19] performed a boundary-layer analysis for the free convection flow over a vertical flat plate embedded in a porous medium saturated by a power-law non-Newtonian fluid and gave the similarity solution to the classical boundary layer equations of the power-law wall plume problem.
Motivated by works mentioned above, this paper takes the effects of power law kinematic viscosity into account in the boundary layer heat transfer equations.In terms of the analogy between viscosity diffusion and thermal diffusion, we assumed that the thermal conductivity will be chosen as k(T) = λ|∂T/∂Y | n−1 and the thermal diffusivity α be defined as α = ω|∂T/∂Y | n−1 with ω = λ/ρc P as a positive constant (c P is the specific heat at constant pressure) [20,21].We first formulate a thermal boundary layer equation for the power-law non-Newtonian fluids with a new variable thermal conductivity and provide similarity solutions.

Formalization of the Nonlinear Boundary Value Problem
Consider a semi-infinite plate at uniform wall temperature aligned with a power law fluid of constant speed U ∞ .In the presence of viscous dissipation, the laminar boundary layer equations are written as: where Uand V are the velocity components along X and Y directions, respectively.The boundary conditions are The following dimensionless variables are introduced: N Zh is defined as the Zheng number.The boundary layer equations then become with the boundary conditions: Equations ( 8)-( 11) are super-nonlinear.Since the problem is very complex to solve analytically, a discretized model may be a considerable approach.
The stream function ψ(x, y), similarity variable η, and dimensionless temperature function w(η) are defined as: where A, B, α, and β are constant to be determined, and f (η) denotes the dimensionless stream function.The u and v velocity component are Substituting u and v into ( 8)-( 11) combining with we arrive at the nonlinear boundary value problems of the form as:

Discretization Method Solution
In order to obtain numerical solutions, we transfer the problem ( 15)-( 18) to a system of four first-order equations by denoting the f , f , f , w, w using variables f , u, v, w, y, respectively: The corresponding boundary conditions are We introduce the parameters tand s as Then, the problem is to find parameters t and s such that the solution to ( 19)-( 24), (26) satisfies the boundary conditions (25).
The solution is consequently:

Numerical Results
The heat transfer of pseudoplastic power law fluids aligned with a semi-infinite plate is investigated in the presence of viscous dissipation.Numerical solutions are achieved using the discretization shooting method and the Boubaker polynomials expansion echeme (BPES).Figures 1-4 show the dimensionless temperature profiles for different power law index n, the Zheng number N Zh and Z * .It is observed that the length scale of thermal diffusion is significantly different for the values of parameters.On the whole, the thermal diffusion ratio increases with the increase of N Zh and Z * .Figure 1 depicts the effect of power-law index n on the temperature profiles using the two methods (Section 3, Section 4).It is a known fact that increasing values of n implies difference in temperature.The same is reiterated by Figure 1: the changing factor n arises some perturbation in temperature profiles and lead to the varying w for each η.
Figure 2 reveals the effect made by the viscous dissipation on the temperature.These profiles are well behaved and very little change occurs in the shapes of profiles with the varying parameter Z * .The different values of Z * contribute little to the thickness of the thermal boundary layer.
Figures 3-4 show temperature profiles for fixed Z * and n with different values of N Zh .For each N Zh (N Zh > 0), the thermal diffusion ratio increases with the increasing in N Zh as recorded earlier by Zheng et al. [38] and Ece and Buyuk [19].It is evident that small values of N Zh result in thinning of the thermal boundary layer.This is similar to the effects of Prandtl number Pron heat transfer.

Conclusions
An investigation on heat transfer of pseudoplastic non-Newtonian fluids aligned with a semi-infinite plate is made.The boundary layer equations for both momentum and energy boundary layers have been developed.Numerical solutions are presented and the associated transfer behavior for different power-law index, and the Zheng number N Zh and Z * are discussed.Two significant physical phenomena are observed, that is, large values of N Zh result in thicker thermal boundary layers, and, the different values of Z * contribute little to the thickness of the thermal boundary layer.

Nomenclature c P :
Fluid specific heat (J kg −1 C −1 ) f : Stream function (dimensionless) K: Consistency index k: Thermal conductivity (W kg −1 K −1 ) N Zh : Zheng number (dimensionless) n: Power-law index (dimensionless) T: Temperature(K) U, V : Velocity components along X and Y directions (m s −1 ) X, Y : Cartesian coordinates along the plate and normal to it (m).