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

## 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.

### Keywords

- Carreau nano fluid
- nonlinear thermal radiation
- mixed convection
- stretching sheet
- convective boundary condition
- numerical method

## 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

Extra stress tensor for Carreau fluid is.

In which

Here

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

The boundary conditions for the present flow analysis are,

where

Using the Rosseland approximation radiation heat flux

where

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

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

where

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

With the boundary conditions,

The local skin friction

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

where

## 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

## 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

The developments of a magnetic field

Figure 3 designed the velocity profiles of

Figure 5 portraits the consequences of Brownian motion parameter on temperature and concentration profile. The Brownian motion parameter

The development of the thermophoresis parameter

Figure 7 describe the influences of Biot number

Figures 8 and 9 are sketched to analyze the effect of radiation parameter

The effect of the Prandtl number

Table 2 presents the numerical values of skin friction for various physical values in the presence and absence

Absence | Presence | ||||||||
---|---|---|---|---|---|---|---|---|---|

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 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

Linear | Nonlinear | ||||||||
---|---|---|---|---|---|---|---|---|---|

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 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

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 |

## 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

Concentration profile increase by increasing the values

Effects of

Magnetic parameter reduces the velocity profiles in both

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

We also noticed that the velocity profile and its associated boundary layer thickness are increases by increasing the values of