 Original Paper
 Open Access
Effect of nonlinear thermal radiation on doublediffusive mixed convection boundary layer flow of viscoelastic nanofluid over a stretching sheet
 K. Ganesh Kumar^{1},
 B. J. Gireesha^{1},
 S. Manjunatha^{2}Email author and
 N. G. Rudraswamy^{1}
https://doi.org/10.1186/s4071201700835
© The Author(s). 2017
 Received: 20 May 2017
 Accepted: 12 July 2017
 Published: 18 August 2017
Abstract
Background
The present exploration deliberates the effect of nonlinear thermal radiation on double diffusive free convective boundary layer flow of a viscoelastic nanofluid over a stretching sheet. Fluid is assumed to be electrically conducting in the presence of applied magnetic field. In this model, the Brownian motion and thermophoresis are classified as the main mechanisms which are responsible for the enhancement of convection features of the nanofluid. Entire different concept of nonlinear thermal radiation is utilized in the heat transfer process.
Methods
Appropriate similarity transformations reduce the nonlinear partial differential system to ordinary differential system which is then solved numerically by using the Runge–Kutta–Fehlberg method with the help of shooting technique. Validation of the current method is proved by having compared with the preexisting results with limiting solution.
Results
The effect of pertinent parameters on the velocity, temperature, solute concentration and nano particles concentration profiles are depicted graphically with some relevant discussion and tabulated result.
Conclusions
It is found that the effect of nanoparticle volume fraction and nonlinear thermal radiation stabilizes the thermal boundary layer growth. Also it was found that as the Brownian motion parameter increases, the local Nusselt number decreases, while the local friction factor coefficient and local Sherwood number increase.
Keywords
 Double diffusion
 Mixed convection
 Nanofluid
 Viscoelastic fluid
 Nonlinear thermal radiation
Background
In the modern day, a great deal of interest has been created on heat and mass transfer of the boundary layer flow over a stretching sheet, in view of its numerous applications in various fields such as polymer processing industry in manufacturing processes. Sakiadis (1961a; Sakiadis, 1961b) first studied the boundary layer problem assuming velocity of a bounding surface as constant. Crane (1970) computed an exact similarity solution for the boundary layer flow of a Newtonian fluid toward an elastic sheet which is stretched with velocity proportional to the distance from the origin. It is noteworthy to mention that both of these studies are regarding Newtonian fluid. Subsequently, the pioneering works of Sakiadis and Crane have been extended by including various effects such as suction/injection, porosity, magnetic field, variable material properties, thermal radiation, heat source/sink, and slip boundary, for either a Newtonian or nonNewtonian fluids. In recent years, researches on boundary layer flow and heat transfer in nanofluids have received amplified devotion due to their growing importance in numerous industrial and biomedical applications. Nanofluid is dilute suspension of nanoparticles with at least one of their principal dimensions smaller than 100 nm and the base fluid. Nanofluids are very stable and free from extra issues of sedimentation, erosion, additional pressure drops, and rheological characteristics. This is due to the tiny size and the lowvolume fraction of nanoelements. Thus, nanofluids have principal advantage about enhancement in thermal conductivity and the convective heat transfer coefficient when compared with the customary base fluids which comprise water, oil, and ethylene glycol. The nanomaterials are more operative in terms of heat exchange performance in micro/nanoelectro mechanical devices, for growing demands of modern technology, including power station, chemical production, and microelectronics. Especially, the magneto nanofluids are significant in applications like optical modulators, magnetooptical wavelength filters, tunable optical fiber filters, and optical switches. Choi (1995) was first to use the term nanofluids to refer to the fluid with suspended nanoparticles. Thermophysical properties of nanofluids such as thermal conductivity, diffusivity, and viscosity have been studied by Kang et al. (2006). The theoretical/experimental investigations on nanofluid flow and heat transfer were conducted by Rudyak et al. (2010), Wang et al. (1999), Eastman et al. (2001), and Rudraswamy et al. (2016; 2015) initiated an incompressible nanofluid over an impermeable stretching sheet and considered the uniform magnetic field. To make such investigations, the following studies are quite useful (KaiLong Hsiao (2016; Hsiao 2014; Hsiao 2017a; Hsiao 2017b; Hsiao 2017b) Buongiorno (2006), Khan and Pop (2010), and Kuznetsov and Nield (2010)). They concluded that the nanofluids possess the novel property of enhanced thermal conductivity of the working fluids.
During the past four decades, the investigators have devoted to the doublediffusive phenomena because of their various applications in chemical engineering, solidstate physics, oceanography, geophysics, liquid gas storage, production of pure medication, oceanography, highquality crystal production, solidification of molten alloys, and geothermally heated lakes and magmas, etc. Khan and Aziz (2011) analyzed the doublediffusive natural convective boundary layer flow in a porous medium saturated with a nanofluid over a vertical plate with prescribed surface heat, solute, and nanoparticle fluxes. Nield and Kuznetsov (2011) presented the doublediffusive nanofluid convection in a porous medium using analytical method. Numerous models and methods have been proposed by many researchers and academicians to advance their studies relating to problems involving various parameters. To make such investigations, the following studies are quite useful (Sharma et al. (2012), Hayat et al. (2014), Beg et al. (2014), Goyal and Bhargava (2014), Gaikwad et al. (2007), Wang (1989), and Gorla and Sidawi (1994)).
Motivated by the aforementioned researchers, we have considered the level of species concentration is moderately high so that the thermal diffusion (Soret) and diffusionthermo (Dufour) effects cannot be neglected. In view of the above discussion, the necessity of Soret and Dufour effects and inclusion of nonlinear thermal radiation in the heat transfer and firstorder chemical reaction not only enrich the present analysis but also complement the earlier studies. The inclusion of these phenomena gives rise to additional parameters such as viscoelastic parameter, Brownian motion parameter, thermophoresis parameter, modified Dufour parameter, regular doublediffusive buoyancy ratio, nanofluid buoyancy ratio, mixed convection parameter, radiation parameter, Dufour Lewis number, nanofluid Lewis number, and regular Lewis number. Thus, inclusion of additional parameters contributes to the complexity of the mathematical model representing the flow, heat, and mass transfer phenomena. To the best of author’s acquaintance, no work has been so far reported considering above phenomenal together.
Mathematical formulation
Methods
Comparison of the result for Nusselt number −θ ^{′}(0) when Nd = Le = Ld = α = λ = 0, Bi = 1 , R = 1 , θ _{ w } = 1, M = 1.5 , Nc = 0.1, Ln = 0, Nb = Nt → ∞
Comparison of the results for the Nusselt number (–θ ^{′}(0)) when Ln = Pr = 10 and Nd = Le = Ld = α = λ = 0, Bi = 1 , M = 1.5 , Nc = 0.1 , R = 1 , θ _{ w } = 1
Nt  −θ ^{′}(0)  

Nb = 0.1  Nb = 0.2  Nb = 0.3  Nb = 0.4  
Khan and Pop  Goyal and Bhargava  Present result  Khan and Pop  Goyal and Bhargava  Present result  Khan and Pop  Goyal and Bhargava  Present result  Khan and Pop  Goyal and Bhargava  Present result  
0.1  0.9524  0.95244  0.95234  0.5056  0.50561  0.50556  0.2522  0.25218  0.25215  0.1194  0.11940  0.11940 
0.2  0.6932  0.69318  0.69317  0.3654  0.36536  0.36536  0.1816  0.18159  0.18159  0.0859  0.08588  0.08590 
0.3  0.5201  0.52025  0.52009  0.2731  0.27313  0.27311  0.1355  0.13564  0.13552  0.0641  0.06424  0.06408 
0.4  0.4026  0.40260  0.40261  0.2110  0.21100  0.21100  0.1046  0.10461  0.10462  0.0495  0.04962  0.04947 
0.5  0.3211  0.32105  0.32109  0.1681  0.16811  0.16810  0.0833  0.08342  0.08331  0.0394  0.03932  0.03939 
Comparison of the results for the nanoparticle Sherwood number (–ϕ ^{′}(0)) when Ln = Pr = 10 and Nd = Le = Ld = α = λ = 0, Bi = 1 , M = 1.5 , Nc = 0.1 , R = 1 , θ _{ w } = 1
Nt  −ϕ ^{′}(0)  

Nb = 0.1  Nb = 0.2  Nb = 0.3  Nb = 0.4  
Khan and Pop  Goyal and Bhargava  Present result  Khan and Pop  Goyal and Bhargava  Present result  Khan and Pop  Goyal and Bhargava  Present result  Khan and Pop  Goyal and Bhargava  Present result  
0.1  2.1294  2.12949  2.1290  2.3819  2.38186  2.3816  2.4100  2.41009  2.4098  2.3997  2.39970  2.3994 
0.2  2.2740  2.27401  2.2735  2.5152  2.51537  2.5148  2.5150  2.51501  2.5147  2.4807  2.48066  2.4804 
0.3  2.5286  2.52855  2.5284  2.6555  2.65550  2.6550  2.6088  2.60876  2.6084  2.5486  2.54848  2.5483 
0.4  2.7952  2.79520  2.7949  2.7818  2.78181  2.7812  2.6876  2.68758  2.6871  2.6038  2.60380  2.6034 
0.5  3.0351  3.03511  3.0334  2.8883  2.88830  2.8876  2.7519  2.75190  2.7513  2.6483  2.64831  2.6478 
The numerical of skin friction coefficient with α = 0 and α = 0.5
Bi  R  Pr  θ _{ w }  Nd  Ln  Le  Ld  Nb  Nt  λ  M  Nc  Skin friction coefficient  

α = 0  α = 0.5  
0.2  1.43205  2.96707  
0.4  1.36583  2.85068  
0.6  1.32221  2.77284  
0  1.36251  2.85839  
0.5  1.34192  2.80810  
1  1.32089  2.75671  
3  1.32497  2.76526  
4  1.34192  2.80810  
5  1.35193  2.83360  
1  1.34876  2.82184  
1.4  1.33371  2.79157  
1.8  1.31274  2.74874  
0.1  1.26834  2.56823  
0.2  1.17796  2.29936  
0.3  1.07752  2.02579  
3  1.34801  2.81638  
4  1.34192  2.80810  
5  1.33795  2.80305  
5  1.34192  2.80810  
10  1.34160  2.80936  
15  1.33928  2.80541  
0.5  1.34192  2.80810  
1  1.34085  2.80503  
1.5  1.33978  2.80196  
0.2  1.36515  2.85138  
0.4  1.31797  2.76266  
0.6  1.26959  2.66933  
0.2  1.34910  2.82314  
0.4  1.33441  2.79227  
0.6  1.31846  2.75816  
0  1.58113  3.22750  
2  1.12634  2.44198  
4  0.73642  1.80525  
1  1.17104  2.47904  
2  1.49715  3.10837  
3  1.77371  3.64713  
1  1.09804  2.44336  
2  0.83753  2.06279  
3  0.58582  1.70306 
Numerical values of −f ^{′′}(0) , − γ ^{′}(0) , − θ ^{′}(0) and −ϕ ^{′}(0) for different physical parameters
Bi  R  Pr  θ _{ w }  Nd  Ln  Le  Ld  Nb  Nt  α  λ  M  Nc  −f ^{′′}(0)  −γ ^{′}(0)  −ϕ ^{′}(0)  −θ ^{′}(0) 

0.2  2.96707  1.50463  1.28818  0.26757  
0.4  2.85068  1.50341  1.27248  0.41719  
0.6  2.77284  1.50424  1.26552  0.50867  
0  2.85839  1.50345  1.27238  0.25574  
0.5  2.80810  1.50372  1.26835  0.46802  
1  2.75671  1.50957  1.27520  0.28380  
3  2.76526  1.50769  1.27218  0.45607  
4  2.80810  1.50372  1.26835  0.46802  
5  2.83360  1.50321  1.26976  0.47224  
1  2.82184  1.50071  1.26348  0.47644  
1.4  2.79157  1.50740  1.27434  0.45783  
1.8  2.74874  1.51675  1.28943  0.43212  
0.1  2.56823  1.57167  1.38537  0.30838  
0.2  2.29936  1.65304  1.52823  0.11343  
0.3  2.02579  1.74113  1.68545  −0.10123  
3  2.81638  1.49844  1.02955  0.47983  
4  2.80810  1.50372  1.26835  0.46802  
5  2.80305  1.50759  1.47537  0.45967  
5  2.80810  1.50372  1.26835  0.46802  
10  2.80936  2.26285  1.27719  0.45463  
15  2.80541  2.84139  1.28580  0.44357  
0.5  2.80810  1.50372  1.26835  0.46802  
1  2.80503  1.46854  1.26896  0.46828  
1.5  2.80196  1.43361  1.26957  0.46854  
0.2  2.85138  1.48668  1.18357  0.50788  
0.4  2.76266  1.52079  1.31270  0.42742  
0.6  2.66933  1.55364  1.35999  0.34719  
0.2  2.82314  1.49937  1.29030  0.47871  
0.4  2.79227  1.50818  1.25051  0.45699  
0.6  2.75816  1.51741  1.22730  0.43399  
1  3.99182  1.53308  1.30072  0.47469  
2  5.89049  1.57107  1.34290  0.48317  
3  7.43629  1.59502  1.36948  0.48839  
0  3.22750  1.46544  1.22729  0.45927  
2  2.44198  1.53371  1.29994  0.47455  
4  1.80525  1.58046  1.34835  0.48425  
1  2.47904  1.53351  1.30079  0.47471  
2  3.10837  1.47670  1.23908  0.46183  
3  3.64713  1.18758  1.18758  0.45060  
1  2.44336  1.52792  1.29302  0.47309  
2  2.06279  1.55163  1.31701  0.47793  
3  1.70306  1.57281  1.33831  0.48215 
Results and discussion
Figure 4a, b shows the velocity and concentration profile for different values of mixed convection parameter (λ). It depicts that the velocity field and momentum boundary layer thickness increase by increasing mixed convection parameter as shown in Fig. 4a. Figure 4b illustrates that the solutal concentration profile and its boundary layer thickness are a decreasing function of mixed convection parameter. As viewed from Fig. 5a, b, the temperature and nanoparticle concentration profiles decreased significantly with an increase in mixed convection parameter.
Figures 6a, b and 7a, b show the velocity, solutal concentration, temperature, and nanoparticle concentration profile for different values of viscoelastic parameter (α), respectively. From this plot, it is evident that increasing values of viscoelastic parameter oppose the motion of the liquid close to the stretching sheet and assist the motion of the liquid faraway from the stretching sheet. Increasing values of viscoelastic parameter enables the liquid to flow at a faster rate, due to which there is decline in the heat transfer. This is responsible for the increase in momentum boundary layer, whereas the thermal, solute concentration, and nanoparticle concentration boundary layers reduce when the viscoelastic effects intensify.
Figure 8a, b, respectively, shows the effect of magnetic parameter (M) on the velocity and solute concentration profile. In general, the application of transverse magnetic field will result a restrictive type of force (Lorenz’s force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity. It is clear that, as the magnetic parameter increases, it reduced the velocity profile and enhanced the solute concentration profile. Figure 9a, b reveals the temperature and nanoparticle concentration profile for different values of magnetic parameter. It is observed from the above figures that, for increasing the values of M, the temperature and nanoparticle concentration distributions increase and also the corresponding boundary layer thickness.
The effect of nanofluid buoyancy ratio parameter (Nr) on velocity and solute concentration profile are depicted in Fig. 10a, b. It is evident from this figure that the velocity profile increases and solute concentration profile decreases for increasing the values Nr. The temperature and nanoparticle concentration profiles for different values of nanofluid buoyancy ratio parameter are presented in Fig. 11a, b, respectively. Thermal and nanoparticle boundary layer thickness decrease by increasing the nanofluid buoyancy ratio parameter.
Figure 12a, b demonstrates the effect of temperature ratio and radiation parameter on temperature profile, respectively. Here, we observed that the temperature profile is an increasing function of temperature ratio and radiation parameter; as a result, thermal boundary layer thickness also increases. This is due to the fact that enhancement in the radiation parameter implies a decrease in the Rosseland radiation absorptive. Hence, the divergence of radiative heat flux q _{ r } increases as absorption coefficient decreases. Therefore, the rate of radiative heat transferred to the fluid increases and consequently the fluid temperature and simultaneously the velocity of the fluid also increases.
Figure 13a, b describes the influences of Biot number and modified Dufour parameter on temperature profiles, respectively. The influence of Biot number (Bi) on the temperature profile is shown in Fig. 13a. The stronger convection leads to the maximum surface temperatures which appreciably enhance the temperature and the thermal boundary layer thickness. From Fig. 13b, it is noticed that the thermal boundary layer thickness increases by increasing the modified Dufour parameter (Nd). The variation of dimensionless solute concentration with Dufour Lewis number and regular Lewis number is illustrated Fig. 14a, b. An increase Dufour Lewis number enhances the solute concentration in the boundary layer thickness. But in case of regular Lewis number, it reduces the solute concentration in the boundary layer thickness as shown in Fig. 14b.
The effect of Prandtl number and nanofluid Lewis number on temperature and nanoparticle concentration profiles are exhibited in Fig. 15a, b, respectively. From Fig. 15a, we observed that an increase in the Prandtl number is seen to decrease the fluid temperature above the sheet. Physically it means that, the thermal boundary layer becomes thinner for the larger Prandtl number. The Prandtl number signifies the ratio of momentum diffusivity to thermal diffusivity. Hence, the Prandtl number can be used to increase the rate of cooling in conducting flows. From another plot, it is evident that increasing values of Lewis number reduces nanoparticle concentration profile; physically, it means that Lewis number lessens the mass diffusivity which in turn lessens the penetration depth of the concentration boundary layer as shown in Fig. 15b.
Figure 16a, b portraits the consequences of Brownian motion parameter on temperature and nanoparticle concentration profile. The Brownian motion parameter (Nb) will increase the random motion of the fluid particles and boundary layer thickness conjointly, which ends up in an additional heat to provide. Therefore, temperature profile will increase. However, nanoparticle concentration profiles show an opposite behavior because increasing Brownian motion parameter enhances the nanoparticle volume fraction transfer rate. This can be shown in Fig. 16b. It is interesting that the variation of the Brownian motion parameter does not show a significant influence on the concentration and temperature profiles, but its effect on the velocity profiles is obvious in the vicinity of the wall.
Conclusions

Magnetic field reduces the velocity the profile and enhances the temperature, solute, and nanoparticle concentration profiles.

An increasing value of the regular buoyancy ratio, nanofluid buoyancy ratio, viscoelastic parameter, and mixed convection parameter is to increase the momentum boundary layer thickness and to decrease the thermal, solutal, and nanoparticle boundary layer thickness.

Brownian motion parameter that has an opposite effect on temperature and nanoparticle concentration profiles but similar effect on temperature and nanoparticle concentration profiles is observed in case of thermophoresis parameter.

Increasing values of temperature ratio parameter θ _{ w } extinguishes the rate of heat transfer θ ^{′}(0) for fixed Pr and R.

The temperature ratio parameter θ _{ w } and the thermal radiation parameter R have the same effect. From a qualitative point of view, temperature increases within creasing θ _{ w } and R. However, thermal boundary layer thickness decreases when the Prandtl number increases.

In coolant factor, nonlinear thermal radiation is a superior copier to linear thermal radiation (Table 6).

The temperature profile as well as thermal boundary layer thickness increases with an increase in both Biot and modified Dufour parameter.

Solutal concentration profile increases for higher values of Dufour Lewis number whereas it decreases with increase in values of regular Lewis number.
The numerical values of skin friction coefficient with different physical parameters with linear and nonlinear radiation
Bi  R  Pr  α  Nd  Ln  Le  Ld  Nb  Nt  λ  M  Nc  Nusselt number  

Linear radiation  Nonlinear radiation  
0.2  0.14372  0.26757  
0.4  0.22566  0.41719  
0.6  0.27708  0.50867  
0  0.25574  0.25574  
0.5  0.25407  0.46802  
1  0.24600  0.28380  
3  0.24744  0.45607  
4  0.25407  0.46802  
5  0.25664  0.47224  
1  0.27821  0.47469  
2  0.28283  0.48317  
3  0.28565  0.48839  
0.1  0.16878  0.30838  
0.2  0.05850  0.11343  
0.3  −0.07540  −0.10123  
3  0.26013  0.47983  
4  0.25407  0.46802  
5  0.24975  0.45967  
5  0.25407  0.46802  
10  0.24709  0.45463  
15  0.24131  0.44357  
0.5  0.25407  0.46802  
1  0.25425  0.46828  
1.5  0.25443  0.46854  
0.2  0.27452  0.50788  
0.4  0.23288  0.42742  
0.6  0.18971  0.34719  
0.2  0.27936  0.47871  
0.4  0.26948  0.45699  
0.6  0.25883  0.43399  
0  0.27034  0.45927  
2  0.27770  0.47455  
4  0.28247  0.48425  
1  0.27811  0.47471  
2  0.27120  0.46183  
3  0.26518  0.45060  
1  0.27720  0.47309  
2  0.27975  0.47793  
3  0.28196  0.48215 
Declarations
Authors’ contributions
GK has involved in conception and design of the problem. NGR has involved in analysis and interpretation of data. MS has involved in drafting the manuscript or revising it critically for important intellectual content. BJG has given final approval of the version to be published. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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