Similarity treatment for MHD free convective boundary layer flow of a class of Non-Newtonian Fluids

Deductive group symmetry treatment is appied to derive the similarity transformations for the free convective boundary layer flow of a class of non-Newtonian fluids past over a two-dimensional surface and flowing under the influence of transverse magnetic field. Numerical solutions are obtained for particular Non-Newtonian fluid model namely Prandtl Eyring fluid, in a graphical form . The important physical quantities like velocity distribution, skin friction coefficent and temperature variations are discussed.


INTRODUCTION
The classical theory of fluid mechanics is based upon the hypothesis of a linear relationship between two tensor components, shearing stress and rate of strain as, u y The fluids with properties different from that described by equation (1), called Non-Newtonian fluids.
Flow of Non-Newtonian fluids has attained a great success in the theory of fluid mechanics due to its applications in biological sciences and industry.Problems involving Non-Newtonian fluid models have been studied by the many researchers since last decades.Several techniques are found to analyze and derive the solutions of governing equations.Some of these are cited in Refs.[1][2][3][4][5][6][7][8][9].The similarity technique plays an important role in problem analysis, especially in the boundary layer flows.The similarity method involves the determination of similarity variables which reduce the system of governing partial differential equations in to ordinary differential equations.Indeed similarity solution is the only class of more accurate solution for the governing differential equations.In the present work we have applied similarity treatment for the particular boundary layer problem.
In the literature several information are available on similarity solutions for the natural convective heat transfer of a Non-Newtonian fluids [ .Researchers have presented works on flow in an electrically conducting Non-Newtonian fluid over a stretching sheet [see [13][14][15][16][17][18][19].At this point it is worth to note that most of the work has been done for Non-Newtonian power-law fluids; this is because of its mathematical simplicity.However there are empirical Non-Newtonian fluid models based on functional relationship between shear stress and rate of the strain are available [20].In present work we concentrate our discussion on the similarity solution of steady laminar natural convection flows of generalized Non-Newtonian fluid.Such a class of fluids are severely omitted in the analysis due to mathematical complicity of its nonlinear stress-strain relationship.Further, from these charts, we noticed that all the similarity solutions presented there in are derived either by adopting or by ad-hoc assumption on similarity variables.In the context of this work it is necessary to develop systematical group transformation for similarity solution.Hence, present work focused on deductive group symmetry analysis based on general group of transformations.The analysis is applied to the particular problem of boundary layer theory.We investigate the MHD boundary layer flow of a class of Non-Newtonian fluids characterized by the property that its stress tensor component τ ij can be related to the strain rate component e ij by the arbitrary continuous function of the type ( ) The similarity equations obtained are more general and systematic along with auxiliary conditions.Recently this method has been successfully applied to various non-linear problems by Abd-el-Malek et al [21], Adnan et al [22], Darji and Timol [23,24].

GOVERNING EQUATIONS
Consider the steady laminar natural convection flow of a non-Newtonian fluid over a vertical permeable surface, in the presence of transverse magnetic field.Consider the vertical upward along the surface as positive x-direction, and the origin is fixed (Fig. 1).The transverse electrically conducting variable magnetic field of the strength B(x) is applied normal to the X -axis.It is assumed that the magnetic Reynolds number m Re is very small; i.e. µ is the magnetic permeability, L is the reference length of the plate and σ is the electric conductivity.We neglect the induced magnetic field, which is small in comparison with the applied magnetic field.Also for a class of Non-Newtonian fluids, the stress-strain relation, under the boundary layer assumption can be found in the form of arbitrary function with only non-vanishing component Using boundary layer approximations, the governing equations for a class of non-Newtonian fluids are given by [See 28, 29]: Continuity Equation: ' Energy Equation: Together with boundary conditions: ( ) , 0, at 0 , 0 as where ' α is thermal diffusivity, ' β is the volumetric thermal expansion coefficient.
Introducing the following dimensionless quantities, ' where L is the reference length of plate, 0 U is the reference velocity, U ∞ is the far velocity (near boundary layer), w θ and θ ∞ are the absolute temperatures of fluid near plate wall and near boundary layer respectively.
x y P y , together with boundary conditions:

SIMILARITY ANALYSIS
We now seek some sort of transformation, namely, similarity transformation which transforms the partial differential Eqs ( 8) to (10) into the ordinary differential equations along with appropriate auxiliary conditions (11).
To search this transformation, the one-parameter general deductive group of transformations is introduced as: ( ) ( ) where Q stands for , , , , ,
Finally, we get the one-parameter group G, which transforms invariantly the system of equations ( 8)- (10) along with the auxiliary conditions (11). (

b. The complete set of absolute invariants
Now we want to develop a complete set of absolute invariants so that the original problem ( 8)-(10) will be transformed into similarity equations under the derived deductive group (21).If ( ) is the absolute invariant of the independent variables, then variables of the four absolute invariants for dependent variables ψ, θ, τ yx , B are given by ( ) ( )   , , , , , , and can be obtained by the following first-order linear partial differential equation: (see Morgan [25], Moran and Gaggioli [26]) where, ( ) 0 0 and 1,..6 and 'a 0 ' denotes the value of 'a' which yields the identity element of the group G. Since 0 ( ) ( ) The corresponding characteristic equation of ( 25) is ( ) / 3 Applying the variable separable method, the absolute invariants of independent and dependent variables owing the equation ( 25) are given by As ( )

c. Reduction to ordinary differential equations
Substituting the values of derivatives from (28) in equations ( 8)-( 10), yields the following system of nonlinear ordinary differential equations.
( ) where ( ) H η is similarity variable related to non-dimensional strain-stress relation ( ) Together with boundary conditions, subject to ( ) ( )

PRANDTL EYRING FLUID MODEL
Non-Newtonian fluid models based on functional relationship between shear-stress and rate of the strain, shown by equation ( 3), are defined by various empirical explicit or implicit functional relations See [20,27].Among these models most research work has been so far carried out on power-law fluid model, this is because of its mathematical simplicity.On the other hand fluid models other than the Power-law model presented in Table 1 are mathematically more complex and the nature of partial differential equations governing these flows is too non-liner boundary value type and hence their analytical or numerical solution is a bit difficult.For the present study the Prendtl Eyring model, although mathematically more complex, is chosen mainly due to two reasons.Firstly, it can be deduced from kinetic theory of liquids rather than the empirical relation as in the powerlaw model.Secondly, it correctly reduces to Newtonian behavior for both low and high shear rate.This reason is somewhat opposite to the pseudo plastic system, whereas the power-law model has infinite effective viscosity for the low shear rate, thus limiting its range of applicability.Mathematically, the Prandtl-Eyring model can be written as (Bird et al [20], Skelland [27]) where A and C are flow consistency indices.
Introducing the dimensionless quantities, Substituting it into the equation ( 29), we get ( ) Further, the expression of local skin-friction coefficient f C is:

RESULTS AND DISCUSSIONS
• The numerical solutions in a graphical form of nonlinear system (34) subject to the boundary conditions (31) are obtained using bvp4c solver in Matlab (Figs. [2][3][4][5][6][7][8][9][10].This is second order accurate and allows uniform and non-uniform grid size.[31].This comparision gurranteed the validity of present analysus.• It is worth noting that all solutions have derived for non-dimensional quantities and hence these results are applicable for all types of considered non-Newtonian fluids.

CONCLUSION
The deductive group symmetry method is applied to search similarity transformations to transform the partial differential system to ordinary differential system for the class of Non-Newtonian fluids.Numerical solutions are presented in a graphical form for Prandtl-Eyring fluids using Matlab.Effects of rheological parameters on the boundary layers are discussed in detail.It is found that change in all the dimensionless parameters and rheological parameters causes the boundary layers thickness.The analysis is made for generalized Non-Newtonian fluid and work of Na and Hansen [12] and T. Hayat et al. [30] are particular cases of he present work.
the relation (2) can be given by [see 20]

Figure 1 .
Figure 1.Schematic diagram of MHD natural convection Substitute the values in (3) to(7) along with nondimensional stream function ℵ and ' s ℜ are real-valued and at least differential in the real argument .a To transform the differential equation, transformations of the derivatives of ψ can be obtained from G via chain-rule operations.
Figures 2-4 are graphical representation of the profiles similarity variables F', G and F''(0) are related to velocity along X-axis, local shear stress and temperature respectively under the influence of magnetic field M.These figures mean that increase in M causes the boundary layers to thicken.• At this point it is worth noting that a similar kind of effects have been observed in the work of Na and Hansen [12] and T. Hayat et al. [30] for Power-law and Powel-Eyring Non-Newtonian fluids respectively.• This warrants that the present work is consistent with earlier work, indeed we have analyze the most general case and one can studied any Non-Newtonian fluid model using present analysis.