# Lie group analysis of a Powell–Eyring nanofluid flow over a stretching surface with variable properties

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## Abstract

New applications of nanofluids that have enhanced thermo-physical properties have spurred new studies into the flow and heat transfer in nanofluids in the last decade. Most reported studies have considered the case where the fluid viscosity and thermal conductivity depend only on the size of nanoparticles. However, experimental data show that these properties may depend on the size of nanoparticles and the temperature. In this study, we investigate the flow and heat transfer in a Powell–Eyring nanofluid flow past a stretching surface using the nanofluid viscosity and thermal conductivity models derived from experimental data. Using Lie group analysis, the equations describing the flow and energy balance are reduced to a system of coupled differential equations. These equations are then solved using an efficient iterative spectral local linearization method. The computational results show that increasing the nanoparticle volume fraction and thermal radiation parameter enhances the temperature profiles, while an increase in the fluid parameter increased the velocity profiles. In addition, among other results, the Nusselt number increases with an increase in the temperature ratio parameter and thermal radiation. The results from this study may be useful to engineers in designing cooling devices for the enhancement of thermal systems.

## Keywords

Powell–Eyring model Nanofluid Thermal radiation Variable viscosity and thermal conductivity Spectral local linearization method## 1 Introduction

Efforts to improve the efficiency and performance of industrial and engineering processes have led to the replacement of traditional heat transfer fluids (e.g water, oil or ethylene glycol) with nanofluids. A nanofluid is a colloidal suspension containing metals or oxides, for example, copper oxide, alumina, zinc oxide or iron oxide having a diameter less than 100 nm. The applications of these fluids can be found in solar technology, where nanofluids are used to enhance the productivity and efficiency of a solar thermal system when used as a collector. In biomedical sciences, nanofluids find applications in cancer therapeutics and cryosurgery. A comprehensive review of other applications of nanofluids is given by Wong and De Leon [1], Robert et al. [2], Devendiran and Amirtham [3] and Munyalo and Zhang [4].

The superiority of nanofluids over traditional heat transfer fluids can be attributed to their stability and higher thermo-physical properties. Due to these remarkable characteristics and new applications of nanofluids, many studies have been carried out on the flow and heat transfer of nanofluids. Khan and Pop [5] studied the thermal boundary layer flow of a nanofluid past a stretching plate. In their study, emphasis was placed on the Brownian motion and thermophoresis effect. The implicit finite difference method was used to solve the flow equations. The exact solution to the thermal transport problem of different types of nanofluids was given by Turkyilmazoglu [6]. Sandeep and Gnaneswara [7] scrutinized the effects of nonlinear thermal radiation on the flow of a Cu–water nanofluid. The Runge–Kutta Newton–Raphson algorithm was used to solve the flow equations. Dhlamini et al. [8] discussed the second-grade nanofluid flow over a nonlinearly stretching sheet. Das et al. [9] solved the equations for the Casson nanofluid flow and heat transfer in a porous medium using a spectral quasi-linearization approximation. El-Aziz [10] studied the effect of variable viscosity on the flow and heat transfer of a power-law nanofluid. Recently, Das et al. [11] examined the influence of variable fluid properties on nanofluid flow over a wedge with surface slip.

The studies above considered the viscosity and thermal conductivity of the nanofluid to be a function of nanoparticle size. However, these properties may also change with temperature (Ogunseye et al. [12]). To accurately predict the heat transfer properties of nanofluids, it is important to consider a viscosity and thermal conductivity model that depends on the nanoparticle size and temperature. In recent years, scientists have proposed numerous viscosity and thermal conductivity models that are nanoparticle size and temperature dependent, and among these, the studies by Masoud et al. [13] and Hassani et al. [14] are worth mentioning.

Lie group symmetry analysis is a powerful technique for finding similarity solutions to a given set of partial differential equations. Using Lie group analysis, we can find similarity transformations that reduces *m* independent variables of a partial differential equation into *m* − 1 independent variables. Many authors have applied Lie group symmetry analysis to fluid flow models. Akgül and Pakdemirli [15] studied the transient flow of a power-law fluid using the Lie group symmetry analysis. Jalil and Asghar [16] analysed the boundary layer flow of a Powell–Eyring fluid using the Lie group symmetry analysis. The scaling group of transformations, a special form of the Lie group symmetry, was used by Rehman et al. [17] in studying the heat and mass transfer in a Powell–Eyring fluid flow past a stretching plate. Afify and El-Aziz [10] discussed the scaling group for the flow and heat transfer behaviour in a power-law nanofluid. Other studies using the Lie group analysis are reported in [18, 19, 20, 21].

The main focus of this study is the flow and heat transfer analysis in a non-Newtonian nanofluid with variable viscosity and thermal conductivity using the Lie group symmetry analysis. The Powell–Eyring model [22] is adopted due to its diverge advantages over other non-Newtonian fluid models. The model is derived from the molecular theory of fluids and not on empirical relations. Further, under low and high shear rates the Powell–Eyring fluid is reduced to a Newtonian fluid. A considerable number of studies on the Powell–Eyring fluid flow with constant viscosity have been reported by several authors such as Javed et al. [23], Jalil and Asghar [16], Hayat et al. [24], Mahanthesh et al. [25], Agbaje et al. [26] and Ramzan et al. [27]. However, the Powell–Eyring nanofluid flow with variable properties is yet to be considered. The second-order partial differential equation that models the thermal transportation problem is transformed into an ordinary differential equation using the classical Lie group symmetry approach. The equations are solved using an efficient iterative spectral local linearization method. The viscosity and thermal conductivity adopted here are derived from experimental data. The findings in this study may be useful for engineers in the design of heat exchangers and thermal solar collectors.

## 2 Formulation of the problem

A steady, two-dimensional, laminar flow of an incompressible Powell–Eyring nanofluid past a stretching surface is considered. The flow is restricted to the region \({ \bar{y}} > 0\), and the stretching velocity is assumed to be \(u_w({\bar{x}})\). The nanofluid is aluminium oxide Al_{2}O_{3}–water system. Further, the physical properties of the fluid are assumed to vary with the nanoparticle size and temperature.

*L*is the characteristic length, \(T_w\) is the wall temperature and \(T_\infty\) is the nanofluid temperature far away from the wall.

*bf*and

*p*represent the base fluid and nanoparticle, respectively, and \(\phi\) is the volume fraction of the nanoparticle.

*k*is the thermal conductivity,

*m*is a factor that depends on the nanoparticles, the base fluid and their interaction, \(\alpha ,\beta _1,\beta _2\) and \(\gamma _1\) are empirical parameters determined from experimental data, \(d_p\) is the diameter of the nanoparticle,

*r*is the capping layer thickness and \(\kappa\) is the Boltzmann constant. The values of these empirical parameters are given in Table 1, and the thermo-physical properties of water and aluminium oxide \(\text {Al}_2\text {O}_3\) are presented in Table 2.

| \(\alpha\) | \(\beta _1\) | \(\gamma _1\) | \(\beta _2\) | Volume fraction |
---|---|---|---|---|---|

0.72 | − 0.485 | 14.94 | 0.0105 | \(8.4407(100\phi )^{-1.07304}\) | \(1\%\le \phi \le 10\%\) |

Thermo-physical properties of water and aluminium oxide \(\text {Al}_2\text {O}_3\)

| \(c_p\) (J kg |
| |
---|---|---|---|

Water fluid | 997.1 | 4179 | 0.613 |

\(\text {Al}_2\text {O}_3\) | 3970 | 765 | 40 |

## 3 Lie symmetry analysis

*R*,

*S*represent either

*x*or

*y*depending on the component.

*X*on Eqs. (10) and (11), we obtain the following infinitesimals after some algebraic simplification:

*X*to the boundary conditions, Eq. (12) yields:

## 4 Method of solution

*D*, at the collocation points as a matrix vector product, that is:

*f*and \(\theta\) are evaluated as powers of \({\mathbf {D}}\), that is

## 5 Numerical validation

Comparison of the SLLM results for \(-f^{\prime \prime }(0)\) with Jalil and Asghar [16] for distinct values of \(F_w\) when \(\lambda = A_1 = A_4 = A_5 = 0\), \(\theta _w = A_2 = 1\) and \(A_3 = 1\)

\(F_w\) | \(f^{\prime \prime }(0)\) | ||
---|---|---|---|

Jalil and Asghar [16] | SLLM | Relative error | |

0.75 | 0.984436 | 0.98443940 | 0 |

0.50 | 0.873640 | 0.87364290 | 0 |

0 | 0.677647 | 0.67764824 | 0 |

− 0.50 | 0.518869 | 0.51887049 | 0 |

− 0.75 | 0.453523 | 0.45352500 | 0 |

## 6 Results and discussion

The computed results for skin friction coefficient, \(C_f,\) and Nusselt number, Nu, for \(\hbox {Al}_2\hbox {O}_3\)/water nanofluid for varying the parameters: \(\phi ,\lambda ,\delta ,fw,\theta _w\) and Nr

\(\phi\) | \(\lambda\) | \(\delta\) |
| \(\theta _w\) | Nr | \(C_f\) | Nu |
---|---|---|---|---|---|---|---|

0.01 | 1 | 1 | 1 | 1.5 | 1 | − 1.31566819 | 4.04218773 |

0.04 | − 1.53640389 | 2.69425801 | |||||

0.07 | − 1.83309969 | 1.82571968 | |||||

0.1 | − 2.23626267 | 1.25732143 | |||||

0 | − 2.05499683 | 1.24567815 | |||||

1 | − 2.23626267 | 1.25732143 | |||||

3 | − 2.55833356 | 1.27519931 | |||||

5 | − 2.84185954 | 1.28832024 | |||||

1 | − 2.23626267 | 1.25732143 | |||||

1.5 | − 2.23360279 | 1.25715057 | |||||

2 | − 2.23091741 | 1.25697829 | |||||

2.5 | − 2.22820586 | 1.25680457 | |||||

− 1 | − 1.47135446 | 0.51499994 | |||||

− 0.5 | − 1.64056501 | 0.67145981 | |||||

0.5 | − 2.02355151 | 1.04510263 | |||||

1 | − 2.23626267 | 1.25732143 | |||||

1.2 | − 2.03895232 | 1.11810439 | |||||

1.5 | − 2.23626267 | 1.25732143 | |||||

1.8 | − 2.43248202 | 1.41661565 | |||||

2.1 | − 2.62786066 | 1.59104731 | |||||

0 | − 2.27747031 | 0.97952469 | |||||

0.5 | − 2.25163761 | 1.14019626 | |||||

1 | − 2.23626267 | 1.25732143 | |||||

1.5 | − 2.22583993 | 1.34955641 |

## 7 Conclusions

- 1.
The velocity profiles are enhanced with an increase in the fluid parameter,

*λ*and nanoparticle volume fraction. - 2.
The velocity and temperature profiles are decreasing functions of the suction parameter, while injection shows an opposite trend.

- 3.
An increase in the nanoparticle volume fraction, thermal radiation parameter and temperature ratio parameter enhances the thermal boundary layer thickness as well as the temperature profiles.

- 4.
The skin friction coefficient increases with an increase in the fluid parameter,

*δ*. - 5.
Increasing the temperature ratio parameter and thermal radiation parameter increases the Nusselt number.

## Notes

### Funding

The authors are grateful to the University of KwaZulu-Natal, South Africa, for financial support.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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