Successive Linearization of Heat and Mass Transfer over an Unsteady Stretching Permeable Surface in the Presence of Thermal Radiation and a Variable Chemical Reaction Successive Linearization of Heat and Mass Transfer over an Unsteady Stretching Permeable Surface in the Presence of Thermal Radiation and a Variable Chemical Reaction

Theoretical studies of viscous incompressible flows over continuous stretching surfaces through a quiescent fluid have their origins in the pioneering work of Crane 1970. These types of flows occur in many industrial processes, such as in glass fibre production, food stuff processing reactor fluidization, and transpiration cooling. The prime aim in almost every extrusion is to maintain the surface quality of the extrudate. The pioneering works of Crane have been extended by many researchers to explore various aspects of the flow and heat and mass transfer occurring in infinite domains of the fluid surrounding the stretching sheet, [Liu and Andersson 2008; Abd EL-Aziz 2009; Abel and Mahesha 2008; Shateyi and Motsa 2009; Ziabakhsh et al. 2010; Motsa and Sibanda 2011], among others.


Introduction
Theoretical studies of viscous incompressible flows over continuous stretching surfaces through a quiescent fluid have their origins in the pioneering work of Crane 1970. These types of flows occur in many industrial processes, such as in glass fibre production, food stuff processing reactor fluidization, and transpiration cooling. The prime aim in almost every extrusion is to maintain the surface quality of the extrudate. The pioneering works of Many practical diffusive processes involve molecular diffusion of species in the presence of chemical reaction within and/or at the boundary. Chemical reaction can tremendously alter diffusion rates in convective heat and mass transfer processes. The effect of a chemical reaction depends on whether the reaction is heterogenous or homogeneous, as well as whether it occurs at an interface or a single phase volume reaction. We call a reaction of order n, if the reaction rate is proportional to the nth power of the concentration. The study of chemical reaction processes is useful for improving a number of chemical technologies such as polymer production and food processing. Various aspects of this problem have been studied by some researchers ( There has been much interest in the study of radiative heat transfer flows due to the effect of radiation on performance of many engineering systems applying electrically conducting fluids. Many engineering processes such as nuclear plants, gas turbines, satellites and space vehicles, take place at high temperatures and thus the effect of thermal radiation cannot be In this chapter, we explore the semi analytic solution of the non linear heat and mass transfer over an unsteady stretching permeable surface with prescribed wall conditions in the presence of thermal radiation and a non-uniform chemical reaction. The proposed method of solution employed in this work is based on an extension of the quasilinearization method (QLM) that was initially proposed in Bellman and Kalaba (1965). This method employs Taylor series linearization to convert a nonlinear two-point boundary value problem into an iterative scheme of solution which can be integrated using various numerical techniques. Mandelzweig  In this work we present new iteration schemes which are based on systematically extending the QLM. The objective of this work is to demonstrate that the convergence rate of the QLM can be significantly improved by using the proposed iterations schemes.

Mathematical formulation
The study investigates the unsteady laminar boundary layer in a quiescent viscous incompressible fluid on a horizontal sheet which comes through a slot at the origin. At t = 0, the sheet is stretched with velocity U w (x, t) along the x-axis, keeping the origin in the fluid of ambient temperature T ∞ and concentration C ∞ . The RosseLand approximation is used to describe the radiative heat flux in the energy equation. We also assume a variable chemical reaction.
The velocity, temperature and concentration fields in the boundary layer are governed by the two dimensional boundary layer equations for mass, and chemical species given by Where u, v are the velocity components in the x and y directions,respectively, ν is the kinematic viscosity,g is the acceleration due to gravity, ρ is the density of the fluid, T and T ∞ are the temperature of the fluid inside the thermal boundary layer and of the fluid in the free stream, respectively while C and C ∞ are the corresponding concentrations, α 0 is the thermal diffusivity, c p is the specific heat at constant pressure, D m is the mass diffusivity and q r is the radiative heat flux.
The boundary conditions are given as follows: The stretching velocity U w (x, t), the surface temperature T w (x, t) and the surface concentration are assumed to be of the form: The radiative heat flux q r is described by the Rosseland approximation such that where σ * and K are the Stefan-Boltzman constant and the mean absorption coefficient, respectively. Following Chamkha (1997), we assume that the temperature differences within the flow are sufficiently small so that the T 4 can be expressed as a linear function after using Taylor series to expand T 4 about the free stream temperature T ∞ and neglecting higher order terms. This results in the following approximation Using equations (7) and (8) in equation (3) we obtain

Similarity solutions
Now we introduce the following dimensionless functions of f , θ and φ and similarity variable η (Ishak et al., 2009).
where ψ(x, y, t) is a stream function defined as u = The governing equations are then transformed into a set of ordinary equations and associated boundary conditions as given below: where A=c/a is the component that measures the unsteadiness, Pr = ν/α is the Prandtl number, R = 16σT 3 ∞ /3Kk is the radiation parameter, Sc = ν/D m is the Schmidt number and The boundary conditions are: with f w < 0 and f w > 0 corresponding to injection and suction, respectively.

Method of solution
To solve the governing system of equations (12 -14) we observe that equation (10) depends on f (η) only. Thus, it can be solved independently of the other equations in the system. The solution for f (η) is then substituted in equations (11) and (12) which can also be solved for θ and φ separately. We begin by obtaining the solution for f (η). We assume that an estimate of the the solution of (10) is f γ . For convenience, we introduce the following notation In terms of the new variables (15), equation (10) can be written as where We introduce the following coupled system, Note that when equations (18) and (19) are added, we obtain equation (16). Separating the known and unknown variables, equation (18) can be written as where We use the quasilinearization method (QLM) of Bellman and Kalaba (1965) to solve equation (20). The QLM determines the (i + 1)th iterative approximation f s,i+1 as the solution of the differential equation Separating the unknowns f s,i+1 from the known functions f s,i yields We assume that f s,0 is obtained as a solution of the linear part of equation (20) given by which yields the iteration scheme It can easily be shown that equation (26) is the standard QLM iteration scheme for solving (16).
When i = 0 in (23) we can approximate f s as Thus, setting i = 0 in (23) we obtain which yields the iteration scheme where f The general iteration scheme obtained by setting i = m (m ≥ 2) in equation (23), hereinafter referred to as scheme-m is where f (m−1) s,r+1 is obtained as the solution of The iterative schemes (26) and (31) The derivatives of f at the collocation points are represented as where D = 2D/Le, with D being the Chebyshev spectral differentiation matrix (see for example, Canuto et al. 1988, Trefethen 2000. Thus, applying the CSC on the functions f s we obtain where Thus, applying the spectral method, with derivative matrices on equation (26) and the corresponding boundary conditions gives the following matrix system with the boundary conditions where The vector H r corresponds to the function H when evaluated at the collocation points and a s,r (s = 0, 1, 2) is a diagonal matrix corresponding to the vector of a i,r which is defined as and η d is an (M + 1) × (M + 1) diagonal matrix of η. The boundary conditions (38) are imposed on the first, Mth and (M + 1)th rows of C r and H r to obtain Starting from a suitable initial guess f 0,0 (η), the iteration scheme (41) can be used to iteratively give approximate solutions of the governing equation (10). The application of the CSC on the general iteration schemes (31) and (32) can be done in a similar manner for any value of m.

Results and discussion
In this section we present the results for the governing physical parameters of interest. In applying the Chebyshev spectral method described in the previous section M = 100 collocation points were used. The value of L e for numerically approximating infinity was chosen to be Le = 20. In order to assess the accuracy of the proposed iteration methods, the present numerical results were compared against results generated using the MATLAB routine bvp4c. For illustration purposes, results are presented for the first three iterations schemes obtained by setting m = 0, 1, 2.
In Table 1 we give a comparison between the results of scheme-0 against results generated using bvp4c. We observe that the QLM results converge very rapidly to the bvp4c results. It takes only three or four iterations to achieve an exact match that is accurate to order 10 −8 for the selected parameters of A. We also observe in this table that stretching increases the absolute values of the skin friction.
Scheme-0 iter.   Tables 2 and 3 give the results of the comparison of the values of f ′′ (0) between three levels of the iteration schemes and their corresponding errors. In computing the errors it was assumed that the result corresponding to the 6th iteration is the converged solution. From numerical experimentation it was found that all the iteration schemes would have completely converged to a fixed value by the time the 5th or 6th iteration is used. The results from Table 2 and 3 clearly indicate that the the convergence to the solution progressively improves when you use the higher level iteration schemes. For instance, from Table 3 we note that it takes only 2 iterations to achieve full convergence in scheme-1 and scheme-2 compare to four iterations in scheme-0. This results demonstrates the improvement offered by the proposed new iteration scheme on the original quasilinearization method which corresponds to scheme-0.
In Figs. 1 -7 we give illustrations showing the effect of the governing parameters on the flow properties. Unless otherwise specified, the sample illustrations were generated using R = 1, Pr = 0.7, Sc = 1, K = 1, A = 1. In Figs. 1 -3 we show the velocity, temperature and concentration profiles for different values of A. In Fig. 1 we observe that the velocity f ′ (η) is a monotonically decreasing function of the stretching parameter A.  Fig. 4 depicts the effects of the Prandtl number on the temperature distributions. We observe that as Pr increases, the temperature profiles and the thermal boundary layer thickness become smaller. This is because when Pr increases, the thermal diffusivity decreases, leading to the decrease of the energy transfer ability that decreases the thermal boundary layer. The effect of thermal radiation R on the temperature field is shown in Fig. 5. From this figure we see that the effect of increasing the thermal radiation parameter R is to reduce the temperature profiles.   Fig. 6 shows the dimensionless concentration profiles for different values of the Schmidt number Sc. We clearly see from this figure that the concentration boundary layer thickness decreases as the Schmidt number Sc increases. This phenomenon occurs because when Sc, the mass diffusivity decreases and the fluid becomes heavier. The effects of chemical reaction K on the concentration distributions is displayed in Fig. 7. It should be noted here that physically positive values of K implies the destructive reaction. We observe in this figure that an increase in the chemical reaction leads to the decrease in the concentration can profiles. This shows that diffusion rate can be tremendously altered by chemical reaction.

Conclusion
In this chapter we explored the semi-analytic solution of the non linear heat and mass transfer flow over an unsteady stretching permeable surface with prescribed wall conditions in the presence of thermal radiation and non-uniform chemical reaction. From the present investigation we may conclude the following: