Thermal Conductivity in the Boundary Layer of Non-Newtonian Fluid with Particle Suspension

Abstract

The present chapter is focused on studies concerned with three-dimensional flow and heat transfer analysis of Carreau fluid with nanoparticle suspension. The heat transfer analysis in the boundary was carried out with the fluid flow over a stretching surface under the influence of nonlinear thermal radiation, mixed convection and convective boundary condition. Suitable similarity transformations are employed to reduce the governing partial differential equations into coupled nonlinear ordinary differential equations. The equations in non-linear form are then solved numerically using Runge-Kutta-Fehlberg fourth fifth-order method with the help of symbolic algebraic software MAPLE. The results so extracted are well tabulated and adequate discussions on the parameters affecting flow and heat transfer analysis were carried out with the help of plotted graphs.

1. Introduction

Thermal radiation, the fundamental mechanism of heat transfer is an indispensable activity in rocket propulsion, plume dynamics, solar collector performance, materials processing, combustion systems, fire propagation and other industrial and technological processes at high temperatures. With the developments in computational dynamics, increasing attention has been diverted towards thermal convection flows with the significant radiative flux. Rayleigh initiated the theory of thermal convection, by deriving critical temperature gradient (Critical Rayleigh number). Importance of such radiations is intensified with absolute temperatures at higher level. Thus a substantial interest is driven towards thermal boundary layer flows with a strong radiation. Governing equation of radiative heat transfer with its integro-differential nature makes numerical solutions of coupled radiative-convective flows even more challenging. Multiple studies were conducted employing several models to investigate heat and mass transfer in boundary layer and fully-developed laminar convection flows. As a consequence several simultaneous multi-physical effects in addition to radiative heat transfer including gravity and pressure gradient effects [1], mhd flow of nanofluids [2], buoyancy effects [3, 4], ferrofluid dynamics [5], stretching surface flow [6, 7], time-dependent, wall injection and Soret/Dufour effects [8, 9, 10, 11].

These studies have however been confined to Newtonian flows. But industries related with fabrication of polymers and plastics at high temperatures show greater importance towards radiative flows of non-Newtonian fluids. The potential of non-Newtonian flows in ducts with radiative transfer were significantly developed after the studies on novel propellants for spacecraft [12]. The developments are extant and diversified the application of non-Newtonian fluid models. Most studies in this regard have employed the Rosseland model which is generally valid for optically-thick boundary layers. Recently, Kumar et al. [13] used such model to study melting heat transfer of hyperbolic tangent fluid over a stretching sheet with suspended dust particles. Cortell [14] and Batalle [15] have shown their earlier contribution towards radiative heat transfer of non-Newtonian fluids past stretching sheet under various circumstances. Relating to the studies Khan et al. [16] developed a numerical studies correlating MHD flow of Carreau fluid over a convectively heated surface with non-linear radiation. Appending to this studies Khan et al. [17] provided his results on hydromagnetic nonlinear thermally radiative nanoliquid flow with Newtonian heat along with mass conditions. Meanwhile, Rana and Bhargava [18] provided a numerical elucidation to study of heat transfer enhancement in mixed convection flow along a vertical plate with heat source/sink utilizing nanofluids. Hayat et al. [19] investigated the mixed convection stagnation-point flow of an incompressible non-Newtonian fluid over a stretching sheet under convective boundary conditions. Many diverse -physical simulations with and without convective and/or radiative heat transfer have been studied. Representative studies in this regard include [20, 21, 22, 23] with analogous to the property of radiation flow.

Endeavoring the complications in three dimensional flow analysis, Shehzad et al. [24] studied the effect of thermal radiation in Jeffrey nanofluid by considering the characteristics of thermophoresis and Brownian motion for a solar energy model. Hayat et al. [25] analyzed the effect non-linear thermal radiation over MHD three-dimensional flow of couple stress nanofluid in the presence of thermophoresis and Brownian motion. Rudraswamy et al. [26] observations on Soret and Dufour effects in three-dimensional flow of Jeffery nanofluid in the presence of nonlinear thermal radiation clearly showed that concentration and associated boundary layer thickness are enhanced by increasing Soret and Dufour numbers. Many such problems [27, 28, 29] were considered disclosing the feature of thermal radiation in three dimensional flow of non-Newtonian fluids.

Inspired by the above works, we put forth the studies on the effect of non-linear thermal radiation on three dimensional flow of Carreau fluid with suspended nanoparticles. Present studies even include the phenomenon of mixed convection and convective boundary conditions. A numerical approach is provided for the above flow problem by employing Runge-Kutta-fourth-fifth order method.

2. Mathematical formulation

A steady three-dimensional flow of an incompressible Carreau fluid with suspended nano particles induced by bidirectional stretching surface at $z=0$ has been considered. The sheet is aligned with the $\mathit{xy}-$ plane $\left(z=0\right)$ and the flow takes place in the domain $z>0$. Let $u=u_w\left(x\right)=\mathit{ax}$ and $v=v_w\left(y\right)=\mathit{by}$ be the velocities of the stretching sheet along $x$ and $y$ directions respectively. A constant magnetic field of strength $B$ is applied in the $z-$ direction. Heat and mass transfer characteristics are taken in to account in the presence of Brownian motion and Thermophoresis effect. The thermo physical properties of fluid are taken to be constant.

Extra stress tensor for Carreau fluid is.

$${\overline{\tau }}_{\mathit{ij}}={\mu }_0\left[1+\frac{n-1}{2}{\left(\mathrm{\Gamma }\overline{\dot{\gamma }}\right)}^2\right]{\overline{\dot{\gamma }}}_{\mathit{ij}}$$

In which ${\overline{\tau }}_{\mathit{ij}}$ is the extra stress tensor, ${\mu }_0$ is the zero shear rate viscosity, $\mathrm{\Gamma }$ is the time constant, $\overline{\dot{\gamma }}$ is the power law index and is defined as.

$$\overline{\dot{\gamma }}=\sqrt{\frac{1}{2}\sum \sum {\overline{\dot{\gamma }}}_{\mathit{ij}}{\overline{\dot{\gamma }}}_{\mathit{ji}}}=\sqrt{\frac{1}{2}}\mathrm{\Pi }$$

Here $\mathrm{\Pi }$ is the second invariant strain tensor.

The governing boundary layer equations of momentum, energy and concentration for three-dimensional flow of Carreau nanofluid can be written as,

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,$$E1

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=\nu \frac{{\partial }^{2}u}{\partial z^2}+3\frac{\left(n-1\right)}{2}\mathrm{\Gamma }{\left(\frac{\partial u}{\partial z}\right)}^2\frac{{\partial }^{2}u}{\partial z^2}+g{\beta }_T\left(T-T_{\infty }\right)-\frac{\sigma B^2}{\rho }u,$$E2

$$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=\nu \frac{{\partial }^{2}v}{\partial z^2}+3\frac{\left(n-1\right)}{2}\mathrm{\Gamma }{\left(\frac{\partial v}{\partial z}\right)}^2\frac{{\partial }^{2}v}{\partial z^2}-\frac{\sigma B^2}{\rho }v,$$E3

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\alpha \frac{{\partial }^{2}T}{\partial z^2}+\tau \left[D_B\frac{\partial T}{\partial z}\frac{\partial C}{\partial z}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2\right]-\frac{1}{{\left(\mathit{\rho c}\right)}_f}\frac{\partial q_r}{\partial z},$$E4

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}=D_B\frac{{\partial }^{2}C}{\partial z^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial z^2}.$$E5

The boundary conditions for the present flow analysis are,

$$u=\mathit{ax},v=\mathit{by},w=0,k\frac{\partial T}{\partial z}=-h_f\left(T_f-T_{\infty }\right),C=C_w\mathrm{at}z=0$$E6

$$u\to 0,v\to 0,T\to T_{\infty },C\to C_{\infty }\mathrm{as}z\to \infty ,$$E7

where $\nu$ is the kinematic viscosity of the fluid, $\mu$ is the coefficient of fluid viscosity, $\rho$ is the fluid density, $B$ is the magnetic field, $\sigma$ is the electrical conductivity of the fluid, $T$ is the fluid temperature, $\alpha$ is the thermal diffusivity of the fluid, $k$ is the thermal conductivity. $\tau$ is the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid, $q_r$ is the radiative heat flux, $g$ is the gravitational acceleration, ${\beta }_T$ is thermal expansion coefficient of temperature, $D_B$ is the Brownian diffusion coefficient, $h_f$ is the heat transfer coefficient, $D_T$ is the thermophoretic diffusion coefficient, $c_p$ is the specific heat at constant pressure, $T_f$ is the temperature at the wall, $T_{\infty }$ is the temperatures far away from the surface. $C$ is the concentration and $C_{\infty }$ is the concentration far away from the surface. The subscript $w$ denotes the wall condition.

Using the Rosseland approximation radiation heat flux $q_r$ is simplified as,

$$q_r=-\frac{4{\sigma }^{\ast }}{3k^{\ast }}\frac{\partial T^4}{\partial z}=-\frac{16{\sigma }^{\ast }}{3k^{\ast }}{T}^3\frac{\mathit{dT}}{\mathit{dz}},$$E8

where ${\sigma }^{\ast }$ and $k^{\ast }$ are the Stefan-Boltzmann constant and the mean absorption coefficient respectively.

In view to Eq. (8), Eq. (4) reduces to.

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\frac{\partial }{\partial z}\left[\left(\alpha +\frac{16{\sigma }^{\ast }{T}^3}{3k^{\ast }{\left(\mathit{\rho c}\right)}_f}\right)\frac{\mathit{dT}}{\mathit{dz}}\right]+\tau \left[D_B\frac{\partial T}{\partial z}\frac{\partial C}{\partial z}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial z}\right)}^2\right].$$E9

The momentum, energy and concentration equations can be transformed into the corresponding ordinary differential equations by the following similarity variables,

$$u=\mathit{ax}{f}^{\prime }\left(\eta \right),v=\mathit{by}{g}^{\prime }\left(\eta \right),w=-\sqrt{\mathit{a\nu }}\left(f\left(\eta \right)+g\left(\eta \right)\right),$$

$$\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\eta =\sqrt{\frac{a}{\nu }}$$E10

where $T=T_{\infty }\left(1+\left({\theta }_w-1\right)\theta \left(\eta \right)\right),$ ${\theta }_w=\frac{T_f}{T_{\infty }},$ $\left({\theta }_w>1\right)$ being the temperature ratio parameter.

Then, we can see that Eq. (1) is automatically satisfied, and Eqs. (2)–(7)are reduced to:

$$f^{‴}+\left(f+g\right)f^{″}-{f^{\prime }}^2+3\frac{n-1}{2}\mathit{\text{We}}{f^{″}}^2{f}^{‴}+\mathit{\lambda \theta }-M{f}^{\prime }=0$$E11

$$g^{‴}+\left(f+g\right)g^{″}-{g^{\prime }}^2+3\frac{n-1}{2}\mathit{\text{We}}{g^{″}}^2{g}^{‴}-M{g}^{\prime }=0$$E12

$$\frac{1}{\text{Pr}}\left({\left(1+R\left({\theta }_w-1\right)\theta \right)}^3{{\theta }^{\prime }}^{\prime }\right)+\left(f+g\right){\theta }^{\prime }+\mathit{Nb}{\theta }^{\prime }{\phi }^{\prime }+\mathit{Nt}{{\theta }^{\prime }}^2=0,$$E13

$${\varphi }^{″}+\mathit{Le}\text{Pr}\left(f+g\right){\varphi }^{\prime }+\frac{\mathit{Nt}}{\mathit{Nb}}{\theta }^{″}=0$$E14

With the boundary conditions,

$$f=0,g=0,f^{\prime }=1,g^{\prime }=c,{\theta }^{\prime }=-\mathit{Bi}\left(1-\theta \left(0\right)\right),\varphi =1,\mathrm{at}\eta =0,$$

$$f^{\prime }\to 0,g^{\prime }\to 0,\theta \to 0,\varphi \to 0\mathrm{as}\eta \to \infty .$$E15

$\mathit{\text{We}}=\frac{{\mathit{cU}}_w^2{\lambda }^2}{\nu }$ is the Weissenberg number, $M=\frac{\sigma B^2}{\mathit{\rho a}}$ is the magnetic parameter, $c=\frac{b}{a}$ is the ratio of stretching rates, $\text{Pr}=\frac{\nu }{\alpha }$ is Prandtl number, $R=\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3{\mathit{kk}}^{\ast }}$ is the radiation parameter, $\mathit{Nb}=\frac{\tau D_B\left(C_w-C_{\infty }\right)}{\nu }$ is the Brownian motion parameter, $\mathit{Nt}=\frac{\tau D_T\left(T_f-T_{\infty }\right)}{\nu T_{\infty }}$ is the Thermophoresis parameter, $\lambda =\frac{g{\beta }_T\left(T_f-T_{\infty }\right)}{{\text{Re}}_x}$ is the mixed convection parameter, $\mathit{Bi}=\frac{h_f}{k}\sqrt{\frac{\nu }{a}}$ is the Biot number, $\mathit{Le}=\frac{\alpha }{D_B}$ is the Lewis number.

The local skin friction $\left(C_f\right)$, local Nusselt number $\left({\mathit{Nu}}_x\right)$ and local number Sherwood $\left({\mathit{Sh}}_x\right)$ are defined as,

$$C_{\mathit{fx}}=\frac{{\tau }_w}{\rho u_w{\left(x\right)}^2},C_{\mathit{fy}}=\frac{{\tau }_w}{\rho v_w{\left(y\right)}^2},{\mathit{Nu}}_x=\frac{u_w{q}_w}{\mathit{ka}\left(T_f-T_{\infty }\right)}\text{and}{\mathit{Sh}}_x=\frac{u_w{q}_m}{D_{B}a\left(C_w-C_{\infty }\right)}$$

The local skin friction, local Nusselt number and Sherwood number is given by,

$$\sqrt{{\text{Re}}_x}{C}_{\mathit{fx}}=\left[f^{″}\left(0\right)+\frac{\left(n-1\right){\mathit{\text{We}}}^2}{2}{\left(f^{″}\left(0\right)\right)}^3\right],\sqrt{{\text{Re}}_x}{C}_{\mathit{fy}}=\left[g^{″}\left(0\right)+\frac{\left(n-1\right){\mathit{\text{We}}}^2}{2}{\left(g^{″}\left(0\right)\right)}^3\right],$$

$$\frac{{\mathit{Nu}}_x}{\sqrt{{\text{Re}}_x}}=-\left(1++R{\theta }_w^3\right){\theta }^{\prime }\left(0\right),\frac{{\mathit{Sh}}_x}{\sqrt{{\text{Re}}_x}}=-{\varphi }^{\prime }\left(0\right).$$

where ${\text{Re}}_x=\frac{u_{w}x}{\nu }$ is the local Reynolds number based on the stretching velocity. $u_w\left(x\right).$

3. Numerical method

The non-linear ordinary differential Eqs. (11)–(14) subjected to boundary conditions (15) has been solved using the Runge-Kutta-Fehlberg fourth-fifth order method with the help of symbolic algebraic software MAPLE. The boundary conditions for $\eta =\infty$ are replaced by $f^{\prime }\left({\eta }_{\max }\right)=1,\theta \left({\eta }_{\max }\right)=0$ and $\varphi \left({\eta }_{\max }\right)=0,$ where ${\eta }_{\max }$ is a sufficiently large value of $\eta$ at which the boundary conditions (15) are satisfied. Thus, the values of $\eta ={\eta }_{\max }$ are taken to be 6. To validate the employed method, the authors have compared the results of $f^{″}\left(0\right)$ and $g^{″}\left(0\right)$ with the that of published works by Wang [27] and Hayat [30] for the different values stretching parameter. These comparisons are given in Table 1 and it shows that the results are in very good agreement.

$c$	Wang [27]		Hayat et al. [30]		Present result
	$-f^{″}\left(0\right)$	$-g^{″}\left(0\right)$	$-f^{″}\left(0\right)$	$-g^{″}\left(0\right)$	$-f^{″}\left(0\right)$	$-g^{″}\left(0\right)$
0	1	0	1	0	1	0
0.25	1.0488	0.1945	1.048810	0.19457	1.04881	0.19457
0.5	1.0930	0.4652	1.093095	0.465205	1.09309	0.46522
0.75	1.1344	0.7946	1.134500	0.794620	1.13450	0.79462
1.0	1.1737	1.1737	1.173721	1.173721	1.17372	1.17372

Table 1.

Comparison of different values of c with Wang [27] and Hayat et al. [30].

4. Result and discussion

The purpose of this section is to analyze the effects of various physical parameters on the velocities, temperature and concentration fields. Therefore, for such objective, Figures 1–11 has been plotted. Observations over these data with plotted graphs are discussed below.

Figure 1 characterizes the influence of Weissenberg number $\left(\mathit{\text{We}}\right)$ on velocity profiles for both $x$ and $y$ direction. It is found that increasing values of the Weissenberg number increases the momentum boundary layers in both directions. Physically, Weissenberg number is directly proportional to the time constant and reciprocally proportional to the body. The time constant to body magnitude relation is higher for larger values of Weissenberg number. Hence, higher Weissenberg number causes to enhance the momentum boundary layer thickness.

Figure 1.

Influence of $\mathit{\text{We}}$ on velocity profiles of both $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

The developments of a magnetic field $\left(M\right)$ on velocity profiles are circulated in Figure 2. We tend to discover depreciation within the velocity profile for ascent values of magnetic field parameter. Physically, the drag force will increase with a rise within the magnetic flux and as a result, depreciation happens within the velocity field.

Figure 2.

Influence of $M$ on velocity profiles $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 3 designed the velocity profiles of $f^{\prime }$ and $g^{\prime }$ for various values of stretching parameter $\left(c\right)$. The velocity profiles and associated momentum boundary layer thickness decrease, once the stretching parameter will increase whereas velocity profile g’, exhibits the opposite behavior of f’. Figure 4 shows the velocity profiles for different values of mixed convection parameter $\left(\lambda \right)$. It depicts that the velocity field and momentum boundary layer thickness increases in both $x$ and $y$ direction by increasing mixed convection parameter.

Figure 3.

Influence of $c$ on velocity profiles $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 4.

Influence of $\lambda$ on velocity profiles $f^{\prime }\left(\eta \right)$ and $g^{\prime }\left(\eta \right)$.

Figure 5 portraits the consequences of Brownian motion parameter on temperature and concentration profile. The Brownian motion parameter $\left(\mathit{Nb}\right)$ will increase the random motion of the fluid particles and thermal boundary layer thickness conjointly will increase which ends up in an additional heat to provide. Therefore, temperature profile will increase however concentration profiles show opposite behavior.

Figure 5.

Influence of $\mathit{Nb}$ on $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ profiles.

The development of the thermophoresis parameter $\left(\mathit{Nt}\right)$ on temperature and concentration profiles is inspecting in Figure 6. Form this figure we observed that, the higher values of thermophoresis parameter is to increases both $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ profiles. Further, the thermal boundary layer thickness is higher for larger values of thermophoresis parameter. This is because, it’s a mechanism within which little particles area unit force off from the new surface to a chilly one. As a result, it maximizes the temperature and concentration of the fluid.

Figure 6.

Influence of $\mathit{Nt}$ on $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ profiles.

Figure 7 describe the influences of Biot number $\left(\mathit{Bi}\right)$ on temperature profile. One can observe form the figure, the larger values of Biot number cause an enhancing the temperature profile. This is because, the stronger convection leads to the maximum surface temperatures which appreciably enhance the thermal boundary layer thickness.

Figure 7.

Influence of $\mathit{Bi}$ on temperature profile.

Figures 8 and 9 are sketched to analyze the effect of radiation parameter $\left(R\right)$ and temperature ratio $\left({\theta }_w\right)$ parameter on temperature profile. The above graphs elucidate that, the temperature profile and thermal boundary layer thickness area unit increased by ascent values of radiation parameter and temperature ratio. Larger values of thermal radiation parameter provide more heat to working fluid that shows an enhancement in the temperature and thermal boundary layer thickness.

Figure 8.

Influence of $R$ on temperature profile.

Figure 9.

Influence of ${\theta }_w$ on temperature profile.

The effect of the Prandtl number $\left(\text{Pr}\right)$ on $\theta \left(\eta \right)$ is seen in Figure 10. Since $\text{Pr}$ is that the magnitude relation of the viscous diffusion rate to the thermal diffusion rate, the upper worth of Prandtl number causes to scale back the thermal diffusivity. Consequently, for increasing values of $\text{Pr}$, the temperature profile gets decreases. The impact of Lewis number $\left(\mathit{Le}\right)$ on nanoparticle concentration is plotted in Figure 11. It is evident that the larger values of Lewis number cause a reduction in nanoparticles concentration distribution. Lewis number depends on the Brownian diffusion coefficient. Higher Lewis number leads to the lower Brownian diffusion coefficient, which shows a weaker nanoparticle concentration.

Figure 10.

Influence of $\text{Pr}$ on temperature profile.

Figure 11.

Influence of $\mathit{Le}$ on concentration profile.

Table 2 presents the numerical values of skin friction for various physical values in the presence and absence $\left(\mathit{\text{We}}=n=0\right)$ of non-Newtonian fluid. It is observed that skin friction increase in both directions with increasing $c$ for both presence and absence of non-Newtonian fluid. In the other hand, the skin friction coefficient decreases in both directions by increasing $\mathit{Bi}$. The skin friction is higher in the presence of non-Newtonian fluid than in the absence of non-Newtonian fluid.

$\mathit{Bi}$	$\mathit{Le}$	$R$	$c$	$\lambda$	$M$	Absence		Presence
						$C_{\mathit{fx}}$	$C_{\mathit{fy}}$	$C_{\mathit{fx}}$	$C_{\mathit{fy}}$
0.2						1.2240	0.7261	1.3030	0.7836
0.4						1.1795	0.7280	1.2642	0.7847
0.6						1.1532	0.7289	1.2412	0.7854
	2					1.1719	0.7283	1.2575	0.7850
	3					1.1648	0.7285	1.2514	0.7851
	4					1.1619	0.7285	1.2489	0.7851
		1				1.1598	0.7289	1.2466	0.7855
		2				1.1420	0.7301	1.2300	0.7866
		3				1.1220	0.7314	1.2113	0.7878
			0.2			1.0852	0.2066	1.1657	0.2083
			0.4			1.1265	0.4509	1.2096	0.4657
			0.6			1.1648	0.7285	1.2514	0.7851
				0		1.3122	0.7242	1.3787	0.7816
				0.2		1.2523	0.7259	1.3143	0.7834
				0.4		1.1936	0.7276	1.2514	0.7851
					0	0.9611	0.5965	1.0281	0.6293
					0.5	1.1648	0.7285	1.2514	0.7851
					1	1.3441	0.8413	1.4521	0.9248

Table 2.

Numerical result of skin friction coefficient for different physical parameter values for present and absence non Newtonian fluid.

Table 3 also elucidates that, the wall temperature for different physical parameter for linear as well as nonlinear radiation. It reveals that, the wall temperature increases for increasing values of $\mathit{Bi},R$ and $c$ for both linear and nonlinear radiation but the wall temperature decreases by increasing $\mathit{Le},\mathit{Nb},\mathit{Nt}$ and $\text{Pr}$. Further, it is noticed that the wall temperature is higher for nonlinear radiation than that linear radiation.

$\mathit{Bi}$	$\mathit{Le}$	$M$	$\mathit{Nb}$	$R$	$\mathit{Nt}$	$\text{Pr}$	$c$	Linear	Nonlinear
								$-{\mathit{Nu}}_x{\left({\text{Re}}_x\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	$-{\mathit{Nu}}_x{\left({\text{Re}}_x\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$
0.2								0.1060	0.3289
0.4								0.1356	0.4683
0.6								0.1479	0.5421
	2							0.1825	0.5501
	3							0.1428	0.5102
	4							0.1200	0.4886
		0						0.1440	0.5206
		0.5						0.1428	0.5102
		1						0.1418	0.5011
			0.2					0.3354	0.8074
			0.4					0.2771	0.6974
			0.6					0.2091	0.5974
				1				0.1768	0.8621
				2				0.2059	1.5030
				3				0.2150	2.0523
					0			0.1834	0.5641
					0.5			0.1331	0.4971
					1			0.0913	0.4351
						2		0.2165	0.4958
						3		0.2059	0.5260
						4		0.1875	0.5307
							0.2	0.1268	0.4606
							0.4	0.1352	0.4870
							0.6	0.1428	0.5102

Table 3.

Numerical result of Nusselt number for different physical parameter values for linear and non nonlinear radiation.

Table 4 clearly shows the numerical values of skin friction, Nusselt number and Sherwood number for various physical parameters values. It reveals that, numerical values of wall temperature $\theta \left(0\right)$ increase by increasing $\mathit{Bi},{\theta }_w,R$ and $c$. In the other hand Nusselt number decreases by increasing. $\mathit{Le},M,\mathit{Nb},\mathit{Nt}$ and $\text{Pr}$. From this table, the skin friction coefficient increases by increasing $\mathit{Bi}$ and $m$. Further, the Sherwood number increases by increasing $\mathit{Bi},{\theta }_w,R,\text{Pr}$ and $\mathit{\text{We}}$.

$\mathit{Bi}$	${\theta }_w$	$\mathit{Le}$	$M$	$\mathit{Nb}$	$R$	$\mathit{Nt}$	$\text{Pr}$	$\mathit{\text{We}}$	$c$	$\lambda$	$-C_{\mathit{fx}}$	$-C_{\mathit{fy}}$	$-{\mathit{Sh}}_x{\left({\text{Re}}_x\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$	$-{\mathit{Nu}}_x{\left({\text{Re}}_x\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$
0.2											1.3030	0.7836	1.4859	0.3289
0.4											1.2642	0.7847	1.4877	0.4683
0.6											1.2412	0.7854	1.4886	0.5421
	1.8										1.2434	0.7856	1.4825	0.7398
	2										1.2353	0.7860	1.4812	0.9218
	2.2										1.2250	0.7865	1.4814	1.1226
		2									1.2575	0.7850	1.1427	0.5501
		3									1.2514	0.7851	1.4882	0.5102
		4									1.2489	0.7851	1.7744	0.4886
			0								1.0281	0.6293	1.5340	0.5206
			0.5								1.2514	0.7851	1.4882	0.5102
			1								1.4521	0.9248	1.4493	0.5011
				0.2							1.3140	0.7831	1.2408	0.8074
				0.4							1.2926	0.7837	1.4118	0.6974
				0.6							1.2713	0.7844	1.4647	0.5974
					1						1.2466	0.7855	1.4781	0.8621
					2						1.2300	0.7866	1.4727	1.5030
					3						1.2113	0.7878	1.4740	2.0523
						0					1.2674	0.7845	1.4869	0.5641
						0.5					1.2472	0.7853	1.4901	0.4971
						1					1.2255	0.7863	1.5060	0.4351
							2				1.2113	0.7878	1.4740	0.4958
							3				1.2345	0.7863	1.4731	0.5260
							4				1.2449	0.7857	1.4767	0.5307
								0			1.1936	0.7276	1.4746	0.5070
								1			1.2974	0.8329	1.4985	0.5127
								2			1.3712	0.9109	1.5141	0.5164
									0.2		1.1657	0.2083	1.2770	0.4606
									0.4		1.2096	0.4657	1.3862	0.4870
									0.6		1.2514	0.7851	1.4882	0.5102
										0	1.3787	0.7816	1.4738	0.5068
										0.2	1.3143	0.7834	1.4811	0.5085
										0.4	1.2514	0.7851	1.4882	0.5102

Table 4.

Numerical result of local skin friction coefficient, Sherwood number and Nusselt number for different physical parameter.

5. Conclusions

In the present study, influence of nonlinear radiation on three dimensional flow of an incompressible non-Newtonian Carreau nanofluid has been obtained. The obtained results are presented in tabulated and graphical form with relevant discussion and the Major findings from this study are:

The velocity profiles increase in $x-$ directions and decrease in the $y-$ direction by increasing the stretching parameter.

Concentration profile increase by increasing the values $\mathit{Nb}$ but in case of $\mathit{Nt}$ concentration profile decreases.

$\mathit{Nb}$ and $\mathit{Nt}$ parameter shows the increasing behavior for temperature profile.

Effects of $\mathit{Le}$ nanoparticle fraction $\varphi \left(\eta \right)$ show the decreasing behavior.

Magnetic parameter reduces the velocity profiles in both $x$ and $y-$ directions.

Temperature and thermal boundary layer thickness are decreased when the Pr and tl number increases.

Nonlinear thermal radiation should be kept low to use it as a coolant factor.

The rate of heat transfer increases with the increases in parameters $\mathit{Rd}$ and ${\theta }_w$.

We also noticed that the velocity profile and its associated boundary layer thickness are increases by increasing the values of $\mathit{\text{We}}$.