Double-diffusive convection flow of Casson fluid with nonlinear thermal radiation and convective condition

This study focused on the effects of nonlinear thermal radiation and magnetic field on a double-diffusive convection flow. Double-diffusive convection is being utilized in many bio-medical applications like laser treatment in cornea, destruction of tumours and radiofrequency heating for the elimination of cardiac arrhythmias. We considered Casson liquid model as a working liquid. Convective type boundary condition is imposed. Similarity solution is obtained numerically using RKF 45 method. Results of the emerging parameters for velocity and temperature fields are plotted. Also skin friction, Nusselt number and Sherwood number are presented in tables. Analysis shows that presence of nonlinear radiation gives the more impact on thermal layer when contrasted with linear radiation.


Introduction
Flows due to convection heat transport become important in several industrial applications such as cooling of electronic systems, float glass, solar ponds and metal solidification processes.Aldoss et al. [1] examined the convectional flow past a flat vertical plate due to magnetic field.Hossain et al. [2] studied the heat transport phenomenon in a viscous liquid with temperature oscillation.Hayat et al. [3] addressed the twodimensional stagnation point flow past a vertical porous surface.Lok et al. [4] examined the mixed convectional flow of micropolar liquid in a double infinite flat plate.Chan [5] explore the heat and mass transfer mechanism in a vertical stretching surface.Hossain [6] reported the influence of Joule heating and viscous dissipation on conducting liquid generated by semi infinite plate.Ishak et al. [7] investigated the steady case of mixed convection flow viscous fluid through stretching vertical sheet.Significant analyses on mixed convectional flow have been made by the researchers [8][9][10][11].http://www.ispacs.com/journals/cna/2018/cna-00350/International Scientific Publications and Consulting Services In heat transfer analysis more attention can be found in thermal radiation because to get the finest quality products in industry.Several engineering processes include space vehicles, hypersonic fights, gas turbines, nuclear power plants etc. involve the phenomenon of radiation.Recently, radiative heat transport has also role in the techniques of renewable energy.Various researches [12][13][14][15][16][17][18][19] have been done in literature to describe the mechanism of radiation.Recently Ramesh et al. [20] analyzed the behaviour of nonlinear radiation effect on three dimensional Oldroyd-B nanoliquid.Double-diffusion concept may be found when the fluid moving with temperature and concentration gradients Double-diffusive convection is very essential for better understanding of engineering and industrial technological problems like migration of moisture in fibrous insulation, oceanography, crystal growth process and geophysics.Due to these applications Mohamed [21] addressed the double-diffusive convection and radiation performance in unsteady moving porous plate.Reddy et al. [22] analyzed the chemical reaction effect on micropolar fluid flow through porous medium.Unsteady behaviour of Kuvshinski fluid past a moving porous plate have discussed by Sharma et al. [23].Dual solution natures of MHD convection flow over a vertical plate have been examined by Subhashini et al [24].Numerical treatment for boundary layer flow over a moving surface with the influence of unsteadiness and Hall effect are obtained by Shateyi and Motsa [25].Recently Gireesha et al. ( [26], [27]) studied the dusty fluid over a vertical surface with various conditions.Here Casson liquid model is considered as a non-Newtonian fluid.This model is very popular amongst the recent investigators.Inspired by the above analyses, we investigated the behaviour of nonlinear thermal radiation on doublediffusive convection flow of Casson fluid in the presence of magnetic field.The Casson fluid model is considered to fit the rheological information for several ingredients such as soup, jelly, honey, and tomato sauce.This model can also be chosen for human blood flow investigations because human blood has many constituents like fibrinogen, protein, red blood cells, etc.This model gives a very popular amongst the recent researchers [28][29][30][31][32][33][34][35][36][37].The Robin's condition is accounted for convective heat transport at boundary of sheet.Numerical computation is made to find the solution of non-linear governed expressions.The results of dimensionless quantities have been visualized for various values of emerging physical constraints.

Mathematical formulation
We consider laminar, mixed convection flow of non-Newtonian Casson fluid by stretching surface.The x-axis is along axis of the surface and yaxis normal it.Assumed that stretching surface of the velocity u w (x) = b√x, surface temperature T f and magnetic field of strength B 0 is exposed on y − direction (see Fig. 1).Also assumed that surface is heated by convection from a hot fluid at temperature T f it gives a heat transfer coefficient h f .The rheological equation of state for an isotropic flow of a Casson fluid [33] can be expressed as: boundary conditions of the problem is (2.7) In the equation (2.4) q r is denoted by Rosseland approximation, i.e. (2.9) Following dimensionless variables are defined, with , θ w > 1 being the temperature ratio parameter.
Substituting (2.11) into (2.3)-(2.5) are reduced to the following set of form: ) The transformed boundary conditions (2.6) and (2.7) becomes (2.15)Here λ for mixed convection parameter and N for buoyancy force parameter, which are defined by Biot number.
The skin friction coefficient C f , local Nusselt number Nu x and the Sherwood number Sh x , are given by; The dimensionless form of the skin friction coefficient, Nusselt number and Sherwood number are:

Numerical method
The system of coupled extremely nonlinear ordinary differential equations (2.11)-(2.13)subject to the boundary conditions (2.14) and (2.15) are resolved numerically using Runge-Kutta-Feldberg 45 technique.The equations are first reduced into system of seven first order simultaneous equations having seven unknowns as follows:

Result and discussions
The systems of coupled equations (2.11)-(2.13)are highly non-linear in nature so analytical solution may not be feasible.Therefore we intend to obtain the numerical solution by applying Runge-Kutta-Feldberg 45 method.The present problem has some special case, it may be noted that in the absence of Casson parameter, magnetic field, radiation parameter and convective type boundary condition our equations (2.11)-(2.13)http://www.ispacs.com/journals/cna/2018/cna-00350/International Scientific Publications and Consulting Services along with boundary conditions (2.14),(2.15)reduces to Subhashini et al [24].Further in the absence of concentration equation our problem reduces to Ishak et al [7].We examined the impact of emerging parameters on velocity, temperature and concentration curves as well as friction factor, local Nusselt and Sherwood number profiles.Figure 2 drawn to visualize the effects of Casson parameter (β) on velocity field.Noted that velocity curve decrease with an development of Casson parameter, this happens because of increment in Casson parameter, the yield stress diminishes and consequently the thickness in the momentum boundary layer decreases.On contrast, an opposite response is observed for both temperature and concentration curves and its relevant layer of thickness (see figure 3).Influence of buoyancy parameter (N) and magnetic parameter (M) on velocity, temperature and concentration profiles are presented in figures 4-7. Figure 4 illustrated that buoyancy effect increases the velocity curve as well as the momentum thickness of boundary layer.Also in this graphs we examined N for two cases i.e., β = 0 and β = 1.5, it reveals that presence of β, the velocity distribution higher than that of absence of β.In figure 5 opposite behaviour can be found for temperature curve and concentration curve.The increasing values of parameter of magnetic lead to a lower velocity profile.The central reason for this is an applied magnetic force normal to electrically-conducting liquid which has the ability to generate draglike force named as Lorentz force.This force acts in direction opposite to that of flow which tends to impede its motion can be seen in figure 6.From figure 7 opposite trend can be found in temperature and concentration distribution.11 is plotted to visualize the effects of radiation on temperature profile.The above graphs elucidate that, the temperature profile and corresponding thermal boundary layer thickness increases by increasing the parameter of radiation.The presence of parameter of radiation provides more heat to liquid that give rise to temperature.Figure 12 presents the impacts of parameter of temperature ratio (θ w ) on liquid temperature.As excepted, the increment in ratio parameter corresponds to higher temperature of the liquid.Figure 13 is http://www.ispacs.com/journals/cna/2018/cna-00350/International Scientific Publications and Consulting Services sketched to determine the Sc on concentration profile.One can note that larger values of Sc create to reduction in the concentration profiles and associated boundary layer thickness.Physical reason is larger values of Sc have a stronger viscous diffusion which increases the molecular motions hence the temperature increases.Figure 14 elaborates the variations of temperature for several values of Biot number.We can observe from this plot, an increase in the Biot number gives rise to the enhancement in temperature profile.This result qualitatively agrees with expectations, because the coefficient of heat transport enhanced for higher Biot number which is responsible for an increase in temperature.Biot numbers relate inward and outside protections from heat transitions and are valuable to recognize controlling instruments and the reference is [31].

Conclusion
In this model we examined the behaviour of non-linear thermal radiation on double-diffusive convection flow of Casson fluid past a vertical stretching surface.The numerical calculation is gain by RKF-45 http://www.ispacs.com/journals/cna/2018/cna-00350/International Scientific Publications and Consulting Services technique.From table 2, we observed that increasing values of Bi, N, R, θ w , β and λ is to decreases the friction factor and increases the Nusselt and Sherwood numbers.The opposite trend can be found for M, Pr and Sc.Also from table 3 we noted that friction factor values are higher for Casson parameter (β = 1.5) comparable with Newtonian fluid(β = 0) and nonlinear radiation has more influence than linear radiation in the thermal boundary layer (table 4).

Figure 2 :
Figure 2: Influence of β on velocity profile.

Figure 4 :
Figure 4: Influence of N on velocity profile.

Figure 8
Figures 8-10 are depicted to study the behaviour λ and Pr. Figure 8 illustrate the effect of mixed convection parameter (λ) on f ′ (η) .It is noted that with an enhancement of λ, the velocity field and associated boundary layer thickness increases.Here λ = 0 indicates the non-existing mixed convection parameter.An increase of λ shows a reduction in the temperature and concentration curves and its related boundary layer thickness which are displayed in figures 9 and 10.The ratio of momentum and thermal diffusivities is the definition of Prandtl number.The reason is weaker thermal-diffusion is obtained in the case of higher Prandtl number that produces a reduction in temperature of the liquid.

Figure 6 :
Figure 6: Influence of M on velocity profile.

Figure 8 :
Figure 8: Influence of λ on velocity profile.

Figure 10 :
Figure 10: Influence of Pr on temperature profile.

Figure 11 :
Figure 11: Influence of R on temperature profile.

Figure 12 :
Figure 12: Influence of θ w on temperature profile.

Figure 13 :
Figure 13: Influence of Sc on concentration profile.

Figure 14 :
Figure 14: Influence of Bi on temperature profile.

Figure 15 :Figure 16 :
Figure 15: Influence of λ with the various values of β on skin friction coefficient

Figure 17 :
Figure 17: Influence of θ w with the various values of Bi on Nusselt number.

Figure
Figure 11  is plotted to visualize the effects of radiation on temperature profile.The above graphs elucidate that, the temperature profile and corresponding thermal boundary layer thickness increases by increasing the parameter of radiation.The presence of parameter of radiation provides more heat to liquid that give rise to temperature.Figure12presents the impacts of parameter of temperature ratio (θ w ) on liquid temperature.As excepted, the increment in ratio parameter corresponds to higher temperature of the liquid.Figure13 is

Figure 18 :
Figure 18: Influence of λ with the various values of R on Nusselt number.

Figure 19 :
Figure 19: Influence of Pr with the various values of R on Nusselt number.

Figures 15 -
Figures 15-20 are drawn to examine the friction factor and Nusselt number on important key parameters.Figure 15 displayed the relation of λ and β on skin-friction coefficient.It is observed that increase of λ and β decreases the skin-friction coefficient.The same observation also found the in the variation between λ and M which is depicted in figure 16.Figures 17-19 are drawn to observe the effects of Bi, λ, Pr, R and θ w on temperature gradient.These figures indicates that increase of Bi, λ, Pr, R and θ w , the results of Nusselt number also increases.Finally figure 20 describe the variation of the Sherwood number with Sc for distinct values of λ.The local Sherwood number linearly enhanced with an increase Sc and λ.

Figure 15
Figures 15-20 are drawn to examine the friction factor and Nusselt number on important key parameters.Figure 15 displayed the relation of λ and β on skin-friction coefficient.It is observed that increase of λ and β decreases the skin-friction coefficient.The same observation also found the in the variation between λ and M which is depicted in figure 16.Figures 17-19 are drawn to observe the effects of Bi, λ, Pr, R and θ w on temperature gradient.These figures indicates that increase of Bi, λ, Pr, R and θ w , the results of Nusselt number also increases.Finally figure 20 describe the variation of the Sherwood number with Sc for distinct values of λ.The local Sherwood number linearly enhanced with an increase Sc and λ.

Table 1 :
1.855 seconds.http://www.ispacs.com/journals/cna/2018/cna-00350/InternationalScientificPublicationsand Consulting ServicesTo assess the accuracy of aforementioned numerical method, comparison of local Nusselt number values between the present results and existing results for various values presented in the Table1.Comparison of −θ ′ (0) for different values of Prandtl number Pr when M = 0, θ w = 1 and R = 0.

Table 4 :
Computational values of Nusselt number for different parameter.