Influence of nonlinear thermal radiation and Magnetic field on upper-convected Maxwell fluid flow due to a convectively heated stretching sheet in the presence of dust particles

A numerical investigation of two-dimensional MHD boundary layer flow and thermal characteristics of an electrically conducting dusty non-Newtonian fluid over a convectively heated stretching sheet has been considered. The effects of nonlinear thermal radiation, heat source or sink and viscous dissipation are also taken into the account. The Rosseland approximation is used to model the nonlinear thermal radiation. Suitable similarity transformations are used to transform the flow governing equations into a set of nonlinear differential equations of one independent variable. The Shooting method is adopted to solve transformed equations. The effects of various material parameters on the flow and heat transfer in terms of velocity and temperature distributions are drawn in the form of graphs and are briefly discussed. The numerical computations for the Nusselt number and skin friction drag are also carried out for the emerging parameters of interest in the problem. The obtained numerical results show the good agreement with the existing one for limiting case.


Introduction
The hydromagnetic flow and heat transfer over a stretching sheet have attracted many researchers, because of their applications in paper production, glass blowing, continuous casting of metals, hot rolling, wire drawing, cooling of an infinite metallic plate in a cooling bath, the boundary layer along a liquid film in condensation processes, metal spinning and the aerodynamic extrusion of plastic sheets.The study of heat http://www.ispacs.com/journals/cna/2016/cna-00254/International Scientific Publications and Consulting Services transfer has become an important factor in determining the quality of the final product, which greatly depends on the rate of cooling.Sakiadis [1,2] initiated the work on boundary layer flow on a continuously moving surface.Inspired by the work of Sakiadis, Crane [3] studied the problem with linear stretching sheet.Chiam [4] considered the hydromagnetic flow and heat transfer to a stretching surface.Liao and Pop [5] presented analytical solutions for similarity boundary layer equations governing flow and heat transfer from a stretching surface.The boundary layer flow of non-Newtonian fluid has wider applications, particularly in polymer depolarization, fermentation, composite processing, boiling, bubble columns and absorption, processing of plastic foam and many others.Its relevance is also seen in extrusion of molten polymers through a slit die for the production of plastic sheets, the processing of food stuffs, paper production and fibercoating.To understand the characteristics of non-Newtonian fluid, one can find several models.In the Maxwell fluid model, a subclass of rate type fluids, stress relaxation can be predicted.Therefore, several investigators have used the Maxwell fluid model because of its popularity.For instance, Hayat et al [6] studied the stagnation point flow of Maxwell fluid over a stretched surface with MHD effects by finding series solutions.Again, using the series solution Hayat and Qasim [7] discussed the Maxwell fluid with thermal radiation and Joule heating effects in the presence of thermophoresis.Aliakbar et al [8] have studied the influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets.The effect of magnetic field on stagnation point flow of an upper convected Maxwell fluid on a stretching surface were numerically analysed by Kumari and Nath [9].The MHD flow and heat transfer of an upper-convected Maxwell (UCM) fluid over a stretching sheet within a boundary layer is examined by Abel et al [10].They observed that, when the magnetic parameter increases the velocity decreases, also, for increase in Maxwell parameter, there is a decrease in velocity.Recently, Mukhopadhyay and Bhattacharyya [11] discussed the time-dependent mass transfer effect on Maxwell fluid flow passing through a stretching sheet.Very recently, Mushtaq et al [12] studied the response of the velocity and temperature characteristics of upper-convected Maxwell fluid due to the influence of thermal radiation.All the works mentioned previously are restricted to clean fluids, but in this work we have considered the suspension of dust particles in the fluid.In fact, a fluid flow with dust particles is a significant type of flow.Such flows find a wide range of applications such as, the paint spraying, gas cooling systems, powder technology, polymer technology, centrifugal separation of matter from fluid and fluid droplet sprays.Saffman [13] initially described the fluid-dust particle system, who derived the motion of gas equations carrying the dust particles.Later, Datta and Mishra [14] have investigated dusty fluid in boundary layer flow over a semi-infinite flat plate.The hydromagnetic flow and heat transfer of a dusty fluid over stretching sheet with suction effect was addressed by Vajravelu and Nayfeh [15].Makinde and Chinyoka [16] studied the flow and heat transfer of fluid between two parallel plates with variable fluid properties and uniform distribution of dust particles.Nandkeolyar et al [17] considered the natural convection flow of a dusty fluid past an impulsively moving vertical plate with ramped surface temperature, thermal radiation and MHD effets.Recently, the boundary layer flow and heat transfer of working fluids with suspension of dust particles with different physical conditions were investigated by Gireesha et al [18][19][20][21].In the above mentioned studies, some of (see [7], [8], [12], [17], [19] and [21]) were confined to the linear approximation for the radiative heat transfer effects which are valid for small temperature differences.It is difficult to construct a system in scientific and engineering applications in which the working fluids will have a small temperature differences.Therefore, the heat transfer with nonlinear radiation has been recently presented by some researchers.The Sakiadis flow with nonlinear Rosseland thermal radiation was considerd by Pantokratoras and Fang [22].Unlike, small temperature difference within the fluid, they have assumed large temperature differences within the fluid.Cortell [23] discussed the fluid flow and radiative nonlinear heat transfer over a stretching sheet.Recently, Mushtaq et al [24] studied a nonlinear radiative heat transfer in the flow of nanofluid due to solar energy.http://www.ispacs.com/journals/cna/2016/cna-00254/International Scientific Publications and Consulting Services Keeping all these specifics in mind, we intend to study the boundary layer flow and heat transfer of a dusty Maxwell fluid with a nonlinear thermal radiation.The effects of viscous dissipation, heat source/sink, convective boundary condition and magnetic field are also taken into account.Similarity solution is obtained for the governing basic equations.We employed an extensively validated, highly efficient fourth-fifth order Runge-Kutta-Fehlberg method coupled with shooting method to study the problem.A parametric study is conducted in order to understand the physics of the problem.

Mathematical Analysis
Consider a steady, laminar, two-dimensional boundary layer flow of an electrically conducting Maxwell fluid over stretching sheet with uniform distribution of dust particles.The -axis is taken along the stretching surface in the direction of motion and -axis is normal to it.The sheet coincides with the plane  = 0 and the flow is confined to  > 0. The flow is generated due to linear stretching of the sheet caused by the simultaneous application of two equal and opposite forces along the -axis.The stretching sheet temperature is maintained by convective heat transfer.The fluid conducts due to uniform transverse magnetic field of strength  0 along the  direction.The magnetic Reynolds number is assumed to be small, so that the induced magnetic field is neglected.The dust particles are non-deformable having constant number density and they are assumed to be spherical in shape.It is assumed that the fluid and the particle phases to be interacting continua and the interaction between two phases are according to Stokes drag law.Further, the particleparticle interaction is neglected, since the particle phase is to be relatively diluted in fluid.Under the usual boundary layer approximations subject to non-linear thermal radiation, heat source/sink and effect of viscous dissipation, the conservation equations of mass, momentum and heat transports for both fluid and particle phase as( [14], [16]); Fluid Phase: Particle Phase: ) ) The boundary conditions for the present problem are, Where ℎ  -heat transfer coefficient,   = -velocity of the stretching sheet and  > 0 is the stretching rate.
The radiative heat flux expression in equation ( 2.3) is given by the Rosseland approximation as; Here  * -Stefan Boltzmann constant and  * -mean absorption coefficient.In this model the optically thick radiation limit is considered.However, many researchers have solved this problem by assuming small temperature differences within the flow.In this circumstance, Rosseland formula can be linearized about ambient temperature  ∞ , this mean by replacing  3 in equation (2.8) with  ∞ 3 .That is; Where   =   / is local Reynolds number,  =  ∞ (1 + (  − 1)) and   =   / ∞ -temperature ratio parameter, the prime denote the differentiation with respect to .Apparently equation (2.12) has already satisfies the conservation of mass equations (2.1) and (2.4).Equations (2.2), (2.3), (2.6) and (2.11), produces the following nonlinear ordinary differential equations; (2.20) The dimensionless shear stress and rate of heat transfer at the stretching sheet  = 0, are characterized by skin friction coefficient and Nusselt number.They are defined as; Where   is the surface shear stress and   is the surface heat flux and they are given by; (2.23)

Numerical Solution and Validation
For solving a set of non-linear ordinary differential equations (2.13)-(2.16)with boundary conditions (2.17), we employed an efficient method called fourth-fifth order Runge-Kutta-Fehlberg method coupled with shooting technique.A brief explanation about the Shooting method on Maple functioning was found in Meade et al [25].In this method, we choose a finite value of  ∞ as  5 in a such a way that the boundary conditions are satisfied asymptotically.In addition, the relative error tolerance for convergence is considered to be 10 −6 and the step size is chosen as ∆ = 0.001throughout our numerical computation.To validate and to check the emplyed method used in the cureent work, the skin-friction coefficient  ′′(0)values are compared with the available results with limiting case.The comparison results are presented in Table 1 and are found to be excellent agreement.

 Influence of the Magnetic parameter (𝑴 𝟐 ):
Figures 1 and 2 show the fluid and particle phase velocity and temperature profiles for different values of magnetic parameter  2 .From these figures we observed that the increasing magnetic strength significantly reduces the thickness of the boundary layer thereby reducing the velocity components.Physically, the application of the transverse magnetic field, in the direction normal to the flow direction, has a tendency to create a drag force known as Lorentz force which dominates the motion of the fluid.Hence, one can observe damping effect on the flow velocity with increase of magnetic parameter  2 and this implies an accompanying reduction of the thickness of the momentum boundary layer for both phase.Figure 2 shows the influence of magnetic parameter  2 on temperature distribution.It is noticed that the temperature increases significantly with an increase in magnetic parameter for both fluid and particle phase.Thus the applied magnetic field tends to heat the fluid and thus reduces the heat transfer from the wall which in turn enlarges the thermal boundary layer thickness for both the phases. Influence of mass concentration dust particles parameter ( ) : Figures 3 and 4 show the effect of mass concentration of dust particles parameter () on velocity and temperature profiles for both fluid and particle phases.When the dust particles are suspended in the clean fluid, an internal friction is produced within the fluid, and also the dust particles absorb the heat from the fluid when they come into contact.As a result thinning of thermal boundary layer thickness takes place.This tends to reduce the velocity and temperature profiles for fluid and dust phase with increase in dust particle parameter. Influence of fluid-particle interaction parameters (  ,   ) : Figures 5 and 6 explore the effect of fluid-particle interaction parameters   and   on velocity and temperature profiles for both fluid and dust phases.It may be observed that the dust particle velocity and temperature distributions increase with an increase in the values of   and   in the flow region.This phenomenon is quite opposite for the fluid phase.When the fluid-particle interaction parameter for velocity is increased, the interaction between the fluid and dust particles also increases and then the particle phase velocity enhances until the relative velocity of the fluid and particle phase become identical.The conductive heat transfer in particle phase is intensified by increasing fluid-particle interaction parameter for temperature.As a result, the dust phase temperature amplifies and fluid phase temperature reduces in the boundary layer. Influence of Biot number and temperature ratio parameter (,   ): Figures 7 and 8 display the response of the fluid and particle phase velocity and temperature distributions for the variation of the Biot number and temperature ratio parameter   respectively.It is apparent from the figure 7 that, increasing the Biot number increases both the fluid and dust phase temperatures.This is because the amount of thermal energy in the body is directly proportional to its temperature, which in turn determines the rate of heat transfer into or out of it.Consequently, both the fluid and particle phase temperatures increase by increasing the Biot number.In this study, the influence of the Biot number on fluid temperature can be observed in three cases, namely uniform ( < 1), non-uniform ( > 1) and constant wall temperature ( ⟶ ∞).These cases can be easily verified through this figure.of the velocity gradient at the surface.An opposite behaviour is observed in the case of temperature profile for increasing values of Maxwell parameter.In figure10, thickening of the thermal boundary layer occurs due to an increase in the Maxwell parameter .Thus, the heat transfer rate at the surface increases with increase in .

 Influence of radiation parameter (𝑹):
Figures 11 and 12 display a comparison of the effect of linear and nonlinear thermal radiation on thermal boundary for varying values of the radiation parameter.It can be seen that the thermal boundary layer thickness increases as  increases for both fluid and particle phases.This decreases the absolute value of the temperature gradient at the surface.Thus, the heat transfer rate at the surface decreases with increase in, thereby causing the temperature to increase for fluid and dust phases respectively.Thus the thermal radiation parameter should beat its minimum in order to facilitate the cooling effect.It is also observed that, the thermal boundary layer is thicker for non-linear thermal radiation problem, when compared with that of the linear thermal radiation.We can conclude this by comparing figures11 and 12.

 Influence of Prandtl number and Eckert number (𝑷𝒓,𝑬𝒄):
The effect of Prandtl number on temperature distribution is illustrated in figure 13.The temperature and the thermal boundary layer thickness reduce for larger values of the Prandtl number.Prandtl number signifies the ratio of momentum diffusivity to thermal diffusivity.As the Prandtl number increases, the thermal diffusivity becomes lower.Consequently, the thermal boundary layer thickness of both the phase decreases as  increases.Large Prandtl number fluids can be used to increase the rate of cooling in conducting flows.In figure 14, we display the effect of the Eckert number on the temperature profiles for both phases.An increase in the values of the Eckert number is seen to increase the temperature of the fluid at any point above the stretching sheet.Physically, increasing the Eckert number allows energy to be stored in the fluid region as a consequence of dissipation due to viscosity and elastic deformation. Influence of heat source/sink parameter (): Figure 15 displays the temperature profile for different values of the heat source/sink parameter,Q.The temperature distribution for prescribed wall heat flux is different (less than unity) for different values of Q at the surface and reduces to zero in the free stream for both the phases.It is observed that the heat source Q generates energy, which causes the temperature to increase.However, for the case of heat, Q < 0, the fluid temperature decreases for both fluid and particle phases respectively.Finally, the variation of the skin friction coefficient and Nusselt number distributions for different values of pertinent parameters are recorded in Tables 2-4.From Table 2 it is noticed that the Nusselt number increases with the Biot number in both ordinary and UCM fluid cases.Also, the Nusselt number fluctuates notably by varying the Biot number from 0.1-100, but after  = 100 only slight variations can be observed.Further, we can conclude that, the Nusselt number is higher for ordinary fluid than that of Maxwell fluid.The response of the Nusselt number for different values of , , and was presented in Table 3.It depicts that, the Nusselt number is a decreasing function of , and, whereas it is increasing function of .It is also observed that, the Nusselt number is higher in the presence of linear thermal radiation effect, when we compare with that of non-linear thermal radiation effect.The skin friction coefficient increases for increasing values of and for both clean fluid and dusty fluid case, as can be shown in Table 4. Further, it is seen that, the drag force at surface is higher in dusty fluid ( ≠ 0) than that of clean fluid ( = 0) for all the values of and.

Conclusion
The present work includes the analysis of MHD flow with heat transfer in a boundary layer of dusty Maxwell fluid above a stretching sheet.The non-linear thermal radiation effect is considered under the influence of viscous dissipation, and magnetic field.Numerical results are presented in tabular/graphical form to elucidate the details of flow and heat transfer characteristics and their dependence on the various physical parameters.Some of the major findings of our analysis are listed below. The fluid velocity decreases when the magnetic parameter increases.Also an increase in the Maxwell parameter results in velocity decrements.However, the temperature is enhanced by increasing the values of the magnetic parameter as well as the Maxwell parameter. The velocity and temperature profile of both the phases decrease by strengthening the mass concentration of dust particles. The thermal radiation effect is favorable for the scientific and engineering processes involving the high thermal requirements. Due to viscous dissipation effect, the thermal boundary layer for both the phases thickens. In order to control the flow characteristics, the technique of including fine particles in a working fluid can be adaptable in many engineering processes. The prescribed thermal boundary can be recovered for increasing values of the Biot number from the convective thermal boundary. The Nusselt number is higher in a particle suspension viscous fluid than that of dusty Maxwell fluid. The skin friction drag is increased by embedding the dust particles in a clean Maxwell fluid.

Nomenclature
) here  and  denote the Cartesian co-ordinates, (, ) and (  ,   ) are velocity components along  and  directions of fluid and dust particle phase respectively,  and   are the density of the fluid and dust particles respectively,  -mass of dust particles per unit volume, -dust particles number density, -electrical conductivity of the fluid,  -dynamic viscosity of the fluid,  0 -uniform magnetic field,  = 6-Stokes drag coefficient and  -radius of dust particle, -relaxation time,  and   are the fluid and dust phase temperature respectively, -thermal conductivity of the fluid,   and   are the specific heat of the fluid and dust particles respectively,  0 -heat source/sink,   and   are relaxation time of the dust particles for velocity and temperature respectively and   radiative heat flux.http://www.ispacs.com/journals/cna/2016/cna-00254/International Scientific Publications and Consulting Services

Mathematically, the boundary
condition  ′ = −(1 − ) can be written as ( = −  ′  + 1), this implies the temperature  tends to 1 as  ⟶ ∞.The temperature distributions for both the fluid and dust phase get augmented with a rise in temperature ratio parameter as shown in figure 8.This is true for fluid and dust phases. Influence of Maxwell parameter (): Figures 9 and 10 exhibit the effect of the Maxwell parameter () on the velocity and temperature distributions for fluid and dust phases.The effect of increasing values of  is to reduce the velocity for both fluid and dust phase.Further, we can also be seen from figure 9 that the momentum boundary layer thickness decreases as Maxwell parameter increases, and hence induces an increase in the absolute value http://www.ispacs.com/journals/cna/2016/cna-00254/International Scientific Publications and Consulting Services

Figure 1 :
Figure 1: The response of velocity profile for variation of  2 .

Figure 2 :
Figure 2: The response of temperature profile for variation of  2 .

Figure 3 :
Figure 3: The response of velocity profile for variation of .

Figure 4 :
Figure 4: The response of temperature profile for variation of .

Figure 5 :
Figure 5: The response of velocity profile for variation of   .

Figure 6 :
Figure 6: The response of temperature profile for variation of   .

Figure 7 :
Figure 7: The response of temperature profile for variation of .

Figure 8 :
Figure 8: The response of temperature profile for variation of   .

Figure 9 :
Figure 9: The response of velocity profile for variation of .

Figure 10 :
Figure 10: The response of temperature profile for variation of .

Figure 11 :
Figure 11: The response of temperature profile for variation of .

Figure 12 :
Figure 12: The response of temperature profile for variation of .

Figure 13 :
Figure 13: The response of temperature profile for variation of .

Figure 14 :
Figure 14: The response of temperature profile for variation of .

Figure 15 :
Figure 15: The response of temperature profile for variation of .

Table 1 :
Comparison of the values of  ′′(0) for different values of Maxwell parameter when  2 =  = 0.The comprehensive numerical results are presented in Figs1-15.http://www.ispacs.com/journals/cna/2016/cna-00254/ 4 Results and DiscussionIn order to analyze results and physics of the present problem a numerical computation was performed.International Scientific Publications and Consulting Services

Table 3 :
http://www.ispacs.com/journals/cna/2016/cna-00254/International Scientific Publications and Consulting Services Table2: Numerical results of − ′ (0)for different values of  for ordinary and UCM fluid.Numerical results of − ′ (0)for different values of , ,  and  for linear and non-linear thermal radiation effect.

Table 4 :
Numerical results of′′ (0)for different values of and  for clean and dusty fluid.
International Scientific Publications and Consulting Services Velocity components of fluid phase along  and  directions ( −1 )  Coordinate along the plate ()  Coordinate normal to the plate () Greek symbols   Velocity of the fluid-particle interaction parameter   Temperature of the fluid-particle interaction parameter  Dynamic viscosity ( −1  −1 )  Electrical conductivity of the fluid  * Stefan-Boltzmann constant ( −2  −4 ) Fluid properties at ambient condition.http://www.ispacs.com/journals/cna/2016/cna-00254/ Fluid phase specific heat coefficient (/)   Dust phase specific heat coefficient (/) International Scientific Publications and Consulting Services