Mixed convection flow due to a moving vertical plate parallel to free stream under the influence of Newtonian heating and thermal radiation

The steady mixed convection flow from a moving vertical plate in a parallel free stream is considered to investigate the combined effects of buoyancy force and thermal diffusion in presence of thermal radiation as well as Newtonian heating effects. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar transformations. The resulting system of coupled non-linear partial differential equations is solved by an implicit finite difference scheme in conjunction with the quasi-linearization technique. Computations are performed and representative set is displayed graphically to illustrate the influence of the mixed convection parameter (  ), Prandtl number ( Pr ), the ratio of free stream velocity to the composite reference velocity (  ) and the radiation parameter ( R ) on the velocity and temperature profiles. The numerical results for the local skinfriction coefficient ( 1/2 Re L f C ) and surface temperature (   ,0 W G  ) are also presented. The results show that the streamwise co-ordinate  significantly influences the flow and thermal fields which indicate the importance of non-similar solutions. Also, it is observed that the increase of mixed convection parameter    causes the increase in the magnitude of velocity profile about 65% for lower Prandtl number fluids ( Pr =0.7), while it decreases in the temperature profile about 30%. Present results are compared with previously published work and are found to be in excellent agreement.


Introduction
The phenomenon of boundary layer behavior over a moving plate in a parallel free stream has more important practical applications such as the aerodynamic extrusion of plastic sheets, the cooling of an infinite metallic plate in a cooling bath, the boundary layer along material handling conveyers, the boundary layer along a liquid film in condensation processes, paper production etc. Tsou et al. [1] have showed experimentally that the flow and heat transfer problem from a continuously moving surface is a physically realizable one and studied its basic characteristics.Chappidi and Gunnerson [2]  for horizontal isothermal plate moving in parallel or reversibly to a free stream where similarity and non-similarity equations are used to obtain the flow and thermal fields.Sparrow and Abraham [5] used the relative velocity model where only one of the participating media is in motion.Karwe and Jaluria [6] presented numerical simulation of thermal transport associated with a continuously moving flat sheet in material processing.The steady laminar flow and heat transfer characteristics of a continuously moving vertical sheet of extruded material are studied close to and far downstream from the extrusion slot by Al-sanea [7].Soundalgekar and Murty [8] have discussed the effects of power law surface temperature variation on the heat transfer from a continuous moving surface with constant surface velocity.More recently, Cortell [9] extended the work of Afzal et al.
[3] by taking viscous dissipation effect in the energy balance.The effects of transpiration on the flow and heat transfer over a moving permeable surface in a parallel stream are analyzed by Ishak et al. [10].The development of the boundary layer on a fixed or moving surface parallel to a uniform free stream in presence of surface heat flux has been investigated by Ishak et al. [11].Patil et al. [12] have examined the role of internal heat generation or absorption effects on the flow and heat transfer over a moving vertical plate.In this study, authors have considered the steady flow and heat transfer characteristics.
The comprehensive analysis on steady mixed convection flow due to a moving semi-infinite vertical plate in a parallel free stream with thermal diffusion is also important to be investigated and such studies are yet to appear in the literature.Furthermore, steady mixed convection flows do not necessarily posses similarity solutions in many flow patterns.The nonsimilarity in such flows may be due to the free stream velocity or due to the curvature of the body or due to the surface mass transfer or even possibly due to all these effects.A solution is called self-similar if a system of partial differential equations can be reduced to a system of ordinary differential equations.If the similarity transformations are only able to reduce the number of independent variables, then the transformed equations are called as semi-similar and the corresponding solutions are the semi-similar solutions.Therefore, as a step towards the eventual development of studies on steady mixed convection flows it is important as well as useful to investigate the combined effects of buoyancy and thermal diffusion on a steady mixed convection flow along a semi-infinite vertical plate.
The objective of this study is to analyze the simultaneous effects of buoyancy and thermal diffusion on a mixed convection flow from a semi-infinite vertical plate in presence of thermal radiation and Newtonian heating effects.The plate is supposed to move parallel to the free stream velocity.The coupled non-linear partial differential equations governing the flow have been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique [13][14][15][16][17].The results are revealing that interesting features of flow and heat transfer phenomena.

Mathematical formulation
We consider the steady laminar viscous and incompressible mixed convection boundary layer flow along a semi-infinite vertical plate moving with velocity W U in the x-direction subjected to Newtonian heating effects.The x-axis is taken along the plate in the vertically upward direction and the y-axis is taken normal to it.A schematic representation of the physical model and co-ordinates system is depicted in Fig. 1.The buoyancy force arises due to the temperature difference in the fluid.The fluid is considered to be gray; absorbing-emitting radiation but non-scattering medium and the Rosseland approximation [18] is used to describe the radiative heat flux in the x-direction is considered negligible in comparison to the y-direction.All thermo-physical properties of the fluid in the flow model are assumed constant except the density variations causing a body force in the momentum equation.The Boussinesq approximation is invoked for the fluid properties to relate density changes, and to couple in this way the temperature field to the flow field (Schlichting and Gersten [19]).Under these assumptions, the dimensional equations of conservation of mass, momentum and energy governing the steady mixed convection boundary layer flow over a moving vertical plate are given by 0, uv xy ISSN 2347-3487 where all the physical quantities is mentioned in the Nomenclature.
The radiation heat flux r q under Rosseland approximation Brewster [18], we take where SB  is the Stefan-Boltzmann constant and R a is the Rosseland mean spectral absorption coefficient.
We assume that the temperature difference within the flow are sufficiently small such that 4 T can be expressed as a linear function of temperature.We now expand 4 T in a Taylor series about T  as follows: Neglecting higher order terms in the above equation beyond the first degree in (T -T ∞ ), we get, In view of Eqs. ( 5) and ( 7), the Eq.(3) becomes is the thermal diffusivity.
We now introducing the following transformations: where  is the stream function, which is defined as . Substituting transformations (9) into Eqs.( 1) -( 2) and ( 8), we find that Eq. ( 1) is identically satisfied and Eqs. ( 2) and ( 8) reduce to The corresponding non-dimensional Boundary conditions: ISSN 2347-3487 ), at 0, , 0, as . Here is the radiation parameter and we take Merkin [20]).Further,  is the mixed convection parameter and  is the ratio of free stream velocity to the composite reference velocity parameter, which are defined as We also notice that 1   corresponds to a fixed (static) plate, while 0   corresponds to a moving plate, respectively.It may be remarked that a single constant, "Pr/(1+R)" known as "Preffective" [21], can display the effects of radiation (R) and Prandtl number (Pr) when the sole interest is to study the effects of radiation (R) and Prandtl number (Pr) for a wide range of parameter values.In such situation a single parameter "Pr/(1+R)" known as "Preffective" [21] could serve the purpose for the whole range of parameter values.Since the present study concerned about the effect of Newtonian heating only for air (Pr = 0.7) and water (Pr = 7.0), numerical results are presented by indicating known Pr values of air (Pr = 0.7) and water (Pr = 7.0).
The main physical quantities of interest are the skin friction coefficient

Re
Lf C and the wall temperature These quantities are defined respectively, as

Method of solution
The set of coupled non-linear partial differential Equations ( 10) - (11) under the boundary conditions (12), represent a non-linear two point boundary value problem for partial differential equations which is solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique [13][14][15][16][17].An iterative sequence of linear equations is carefully constructed to approximate the non-linear Eqs. ( 10) - (11) under the boundary conditions (12) achieving quadratic convergence and monotonicity of the sequence of solutions.
Applying the quasi-linearization technique [13][14][15][16][17], the non-linear coupled partial differential equations ( 10) -( 11) are replaced by the following sequence of linear partial differential equations: The coefficient functions with iterative index i are known and the functions with iterative index (i+1) are to be determined.The corresponding boundary conditions are given by: ), at 0, , 0, as .
The resulting sequence of linear partial differential equations ( 15) and ( 16) were discretisized using second order central difference formula in  -direction and backward difference formula in  -direction.In each iteration step, the equations were then reduced to a system of linear algebraic equations, with a block tri-diagonal structure which is solved using Varga's algorithm [22].To ensure the convergence of the numerical solution to the exact solution, step sizes   and   are optimized and the results presented here are independent of the step sizes at least up to the fourth decimal place.A convergence criterion based on the relative difference between the current and previous iteration values is employed.The solution is assumed to have converged and the iteration process is terminated when the difference reaches i.e.

Results and discussion
The comprehensive numerical parametric study was conducted and results are reported in terms of graphs.This is done to illustrate special features of solutions.[3] have discussed only self-similar solutions where all solutions along streamwise direction were made congruent using similarity transformations.Governing equations were finally reduced to a set of ordinary differential equations.In such cases, single asymptotic solution will not be able to represent all physical solutions.Further, it is reported by many other investigators also that there are dual solutions with different asymptotic nature.In contrast, authors have captured nonsimilar solutions at each streamwise location by solving coupled set of partial differential equations.In this investigation, authors have observed a single non-similar solution at each streamwise location.It may be noted that  = 0 is a singular point while obtaining the non-similar solutions and results are valid only for  > 0. Further,  = 0 results are obtained separately and compared with the existing results in literature as displayed in Table 1.Also, results show an excellent agreement with the results reported by earlier investigators.
In order to verify the correctness of the present investigation, steady state results of heat transfer rate compared with the results previously reported by Tsou et al. [1], Soundalgekar and Murty [8], and Patil et al. [12].Some of the comparisons are presented in Table 1 and are found to be in excellent agreement.
The effects of Prandtl number (Pr) and buoyancy (mixed convection) parameter    on velocity and temperature profiles FG     as well as skin friction coefficient and wall temperature

Re
Lf C increases approximately about 281% and 299% as  increases from 1.0 to 5.0 for Pr = 0.7 and for Pr = 7.0 (see Fig. 4), respectively.Furthermore, Figure 5 shows that wall temperature   ,0 W G  decreases approximately about 5% and 11%, respectively, for Pr = 0.7 and for Pr = 7.0 when  increases from 1.0 to 5.0. Figure 6 displays the effect of  (the ratio of free stream velocity to the composite reference velocity) and the thermal radiation parameter R on the velocity profile   , F  with  = 1.0,  = 1 and Pr = 0.7.The velocity is strongly depending on  because it occurs in the governing momentum equation as well as in the boundary condition for the velocity profile   , F  .It has been observed that the magnitude of the velocity profile within the moving boundary layer increases with the increase of  , while it decreases as R increases from R = 0.0 to R = 1.0.The physical reason is that the assisting buoyancy force due to thermal gradients as well as increase in  act likes a combined favorable pressure gradient which accelerates the fluid for uniform motion.The effect of radiation is to decrease the rate of energy transport to the fluid, thereby decreasing the temperature and velocity of the fluid.  , R = 0.2 and Pr = 0.7 at 6.0   , the temperature profile   , G  is reduced approximately by 46% as  increases from 0.0 to 1.0.

Conclusions
A numerical study is performed for the problem of steady mixed convection flow over a moving vertical plate in a parallel free stream in the presence of Newtonian heating applied at the wall and the thermal radiation.We draw the following conclusions in the investigation:  Mixed convection parameter  enhances the skin friction coefficient while the surface temperature is reduced.
 From the computed results, it is found that in presence of buoyancy force   0   , the velocity profile exhibits velocity overshoot 65% more for lower Prandtl number (Pr = 0.7) as compared to the magnitude of the velocity overshoot for higher Prandtl number (Pr = 7.0) fluid.
 The effect of thermal radiation leads to a fall in the momentum and thermal boundary layers.Further, it is noted that the thermal radiation parameter to increase the surface temperature.
 The magnitude of the velocity profile within the moving boundary layer increases about 45% with the increase of  (the ratio of free stream velocity to the composite reference velocity) from 0.1 to 0.9, while it decreases about 30% as R increases from R = 0.0 to R = 1.0.
have studied the laminar boundary layer in two cases w UU   and w UU   separately and formulated two sets of boundary value problems.Afzal et al. [3] formulated a single set of boundary conditions by employing the composite reference velocity w U U U  , where w U is the moving plate velocity and U  is the free stream velocity, instead of considering w U and U  separately, irrespective of whether w UU   or w UU   .Lin and Haung [4] were analyzed number referring to the wall temperature and Re / the forced convection flow case.
The computations have been carried out for various values of The edge of the boundary layer     has been taken between 8.0 and 18.0 depending on the values of the parameters.Afzal et al.

5 and 1 ,FF. 0 WGRe 1 
 are displayed in Figs.2 -5.The influence of buoyancy assisting force   0   shows the significant overshoot in the velocity profiles    near the plate for lower Prandtl number (Pr) fluids (air, Pr = 0.7) while for higher Prandtl number (Pr) fluids (water, Pr = 7.0), the velocity overshoot is not observed as shown in Fig.2.The magnitudes of the velocity overshoot increases with the buoyancy parameter   0   while it decreases as Prandtl number (Pr) increases.The physical reason is that the buoyancy force    effect is more in lower Prandtl number (Pr) fluid (air, Pr = 0.7) due to the lower viscosity of the fluid enhances the velocity profile within the stretching boundary layer as the assisting buoyancy force    acts like a favourable pressure gradient and hence, the velocity overshoot occurs.For higher Prandtl number (Pr) fluids (water, Pr = 7.0), the overshoot is not present because higher Prandtl number (Pr) fluids implies more viscous fluid which have less impact on the buoyancy parameter    .For example,  = 0.5, R = 1.0 and Pr = 0.7 when 5   , the velocity profile   ,  is enhanced approximately by 65% for lower Prandtl number fluids as compared to the higher Prandtl number fluids.The effect of buoyancy opposing force on velocity and temperature profiles is also presented in Figs. 2 and 3 by a representative negative value of    0.25   to limit the number of lines in the figures.It is worthy to note from the Fig. 2 that for opposing buoyancy flow (  < 0), the buoyancy opposing force reduces the magnitude of the velocity significantly within the moving boundary layer for low Prandtl number fluid (Pr = 0.7, air) as well as for high Prandtl number fluid (Pr = 7.0, water).The influence of buoyancy parameter    has relatively less influence on the temperature profiles   , G  , as shown in Fig. 3. Furthermore, Fig. 3 also show that the effect of higher Prandtl number fluids (water, Pr = 7.0) results into a thinner thermal boundary layer since the higher Prandtl number (Pr) fluids (water, Pr = 7.0) have lower thermal conductivity.Figures 4 and 5 represent the influence of mixed convection parameter  and Prandtl number on the skin friction coefficient and wall temperature Skin friction coefficient increases with the increase of buoyancy parameter    while wall temperature   , decreases.The physical reason is that the buoyancy force   0   implies favorable pressure gradient, and the fluid gets accelerated, which results in thinner momentum and ISSN 2347-3487 864|P a g e O c t o b e r 1 4 , 2 0 1 4 thermal boundary layers.Further, when Prandtl number Pr increases from 0.7 to 7.0, the wall temperature Lf C decrease, as shown in Figs. 4 and 5.For example, when  = 0.5, R = 0.5 and

Figure 7 ,G
Figure 7 illustrates the role of stream wise coordinate  and thermal radiation parameter R on the temperature profile   , G  when  = 0.5,