# A spectral relaxation method for thermal dispersion and radiation effects in a nanofluid flow

- Peri K Kameswaran
^{1}, - Precious Sibanda
^{1}Email author and - Sandile S Motsa
^{1}

**2013**:242

https://doi.org/10.1186/1687-2770-2013-242

© Kameswaran et al.; licensee Springer. 2013

**Received: **18 June 2013

**Accepted: **24 September 2013

**Published: **18 November 2013

## Abstract

In this study we use a new spectral relaxation method to investigate heat transfer in a nanofluid flow over an unsteady stretching sheet with thermal dispersion and radiation. Three water-based nanofluids containing copper oxide CuO, aluminium oxide Al_{2}O_{3} and titanium dioxide TiO_{2} nanoparticles are considered in this study. The transformed governing system of nonlinear differential equations was solved numerically using the spectral relaxation method that has been proposed for the solution of nonlinear boundary layer equations. Results were obtained for the skin friction coefficient, the local Nusselt number as well as the velocity, temperature and nanoparticle fraction profiles for some values of the governing physical and fluid parameters. Validation of the results was achieved by comparison with limiting cases from previous studies in the literature. We show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to common numerical methods for solving nonlinear boundary value problems. We show that the convergence rate of the spectral relaxation method is significantly improved by using the method in conjunction with the successive over-relaxation method.

### Keywords

spectral relaxation method heat transfer stretching sheet nanofluids## 1 Introduction

In recent years flow and heat transfer over a stretching surface has been extensively investigated due to its importance in industrial and engineering applications such as in the heat treatment of materials manufactured in extrusion processes and the casting of materials. Controlled cooling of stretching sheets is needed to assure quality products. Fiber technology, wire drawing, the manufacture of plastic and rubber sheets and polymer extrusion are some of the important processes that take place subject to stretching and heat transfer. The quality of the final product depends to a great extent on the heat controlling factors, and the knowledge of radiative heat transfer in the system can perhaps lead to a desired product with a sought characteristic.

The development of a boundary layer over a stretching sheet was first studied by Crane [1], who found an exact solution for the flow field. This problem was then extended by Gupta and Gupta [2] to a permeable surface. The flow problem due to a linearly stretching sheet belongs to a class of exact solutions of the Navier-Stokes equations. Since the pioneering work of Sakiadis [3], various aspects of the stretching problem involving Newtonian and non-Newtonian fluids have been extensively studied by several authors (see Cortell [4], Hayat and Sajid [5], Liao [6], Xu [7]).

The study of radiation effects has important applications in engineering. Thermal radiation effect plays a significant role in controlling heat transfer process in polymer processing industry. Many studies have been reported on flow and heat transfer over a stretched surface in the presence of radiation (see El-Aziz [8, 9], Raptis [10], Mahmoud [11]). El-Aziz [12] studied the radiation effect on the flow and heat transfer over an unsteady stretching sheet. He found that the heat transfer rate increases with increasing radiation and unsteadiness parameters and the Prandtl number. The effect of the radiation parameter on the heat transfer rate was found to be more noticeable at larger values of the unsteadiness parameter and the Prandtl number. In addition to radiation, it is important to consider the thermal dispersion effect on boundary layer flow since this has a direct impact on the heat transfer rate. Several studies on hydrodynamic dispersion have been reported in the literature. The double-dispersion phenomenon in a Darcian, free convection boundary layer adjacent to a vertical wall, using multi-scale analysis arguments, was investigated by Telles and Trevisan [13].

In recent years tremendous effort has been given to the study of nanofluids. The word nanofluid coined by Choi [14] describes a liquid suspension containing ultra fine particles (diameter less than 50 nm). Experimental studies (*e.g.*, Masuda *et al.* [15], Das *et al.* [16], Xuan and Li [17]) showed that even with a small volumetric fraction of nanoparticles (usually less than 5%), the thermal conductivity of the base liquid is enhanced by 10-50% with a remarkable improvement in the convective heat transfer coefficient. The literature on nanofluids was reviewed by Trisaksri and Wongwises [18], Wang and Mujumdar [19] among several others. Nanofluids thus provide an alternative to many common fluids for advanced thermal applications in micro and nano-heat transfer applications. Thermophysical properties of nanofluids such as thermal conductivity, diffusivity and viscosity were studied by Kang *et al.* [20], Velagapudi *et al.* [21], Rudyak *et al.* [22]. Hady *et al.* [23] studied the radiation effect on the viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet. They observed that the increase in the thermal radiation parameter and the nonlinear stretching sheet parameter yields a decrease in the nanofluid temperature leading to an increase in the heat transfer rate. The boundary layer flow of a nanofluid with radiation was studied by Olanrewaju *et al.* [24]. They observed that radiation has a significant influence on both the thermal boundary layer thickness and the nanoparticle volume fraction profiles.

Recently, Mahdy [25] studied the effects of unsteady mixed convection boundary layer flow and heat transfer of nanofluids due to a stretching sheet. He found that the heat transfer rate at the surface increased with the mixed convection parameter and the solid volume fraction of nanoparticles. Moreover, the skin friction increased with the mixed convection parameter and decreased with the unsteadiness parameter and the nanoparticle volume fraction. Narayana and Sibanda [26] studied the effects of laminar flow of a nanoliquid film over an unsteady stretching sheet. They found that the unsteadiness parameter has the effect of thickening the momentum boundary layer while thinning the thermal boundary layer for Cu-water and Al_{2}O_{3}-water nanoliquids. Kameswaran *et al.* [27] studied the effects of hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects. They observed that the velocity profile decreased with an increase in the nanoparticle volume fraction, while the opposite was true in the case of temperature and concentration profiles.

The objective of this study is to analyze the effects of fluid and physical parameters such as thermal dispersion, nanoparticle volume fraction and radiation parameters on the flow and heat transfer characteristics of three different water based nanofluids containing copper oxide CuO, aluminium oxide Al_{2}O_{3} and titanium dioxide TiO_{2} nanoparticles. The momentum and energy equations are coupled and nonlinear. By using suitable similarity variables, we convert these equations into coupled ordinary differential equations and solve them numerically via a novel iteration scheme called the spectral relaxation method (SRM) (see Motsa and Makukula [28], Mosta [29]). The SRM is an iterative algorithm for the solution of nonlinear boundary layer problems which are characterized by flow properties that decay exponentially to constant levels far from the boundary surface. The key features of the method are the decoupling of the governing nonlinear systems into a sequence of smaller sub-systems which are then discretized using spectral collocation methods. The method is very efficient in solving boundary layer equations of the type under investigation in this study. In cases where the SRM convergence is slow, it is demonstrated that successive over-relaxation can be used to accelerate convergence and improve the accuracy of the method. The current results were validated by comparison with published results in the literature and results obtained using the Matlab bvp4c routine. We further show that substantial improvement in the convergence rate of the SRM may be realized by using this method in conjunction with the successive over relaxation method.

## 2 Mathematical formulation

### Transient unsteady-state flow and heat transfer ($t>0$)

*x*and

*y*denoting coordinates along and normal to the sheet. The fluid is a water-based nanofluid containing either alumina, copper-oxide or titanium-oxide nanoparticles. The base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. The sheet is stretched with velocity

*x*-axis, where

*b*and

*α*are positive constants with dimensions (time)

^{−1}and $\alpha t<1$. The surface temperature distribution

where *u*, *v* are the velocity components in the *x* and *y* directions, respectively, *T* is the fluid temperature, ${C}_{p}$ is the specific heat at constant pressure, ${r}_{1}$ and *m* are constants, the expression for the effective thermal diffusivity taken as ${\alpha}_{y}={\alpha}_{m}+\gamma du$, ${\alpha}_{m}$ is the molecular thermal diffusivity, $\gamma du$ represent thermal diffusivity, *γ* is the mechanical thermal dispersion coefficient and *d* is the pore diameter.

*ϕ*is the solid volume fraction of spherical nanoparticles. The effective density of each nanofluid is given as

*et al.*[32]):

*nf*,

*f*and

*s*represent the thermophysical properties of the nanofluid, base fluid and nanoparticles, respectively. The continuity equation (1) is satisfied by introducing a stream function $\psi (x,y,t)$, where

*S*, the thermal dispersion parameter

*D*, the radiation parameter ${N}_{R}$ and the Prandtl number

*Pr*. They are respectively defined as

**Thermophysical properties of water and nanoparticles, Oztop and Abu-Nada** [33]

Properties → | ρ (kg/m | ${\mathbf{C}}_{p}$ (J/kgK) | k (W/mK) |
---|---|---|---|

Pure water | 997.1 | 4,179 | 0.613 |

CuO | 3,620 | 531.8 | 76.5 |

Al | 3,970 | 765 | 40 |

TiO | 4,250 | 686.2 | 8.9538 |

## 3 Skin friction and heat transfer coefficients

where $R{e}_{x}$ is the local Reynolds number defined by $R{e}_{x}=x{U}_{w}/{\nu}_{f}$.

## 4 Initial steady state flow and heat transfer ($t\le 0$)

*i.e.*, $S\to 0$, equations (16) and (17) along with boundary conditions (18) and (19) are replaced by

*s*is a parameter associated with the nanoparticle volume fraction. This satisfies the equation

where ${\lambda}_{1}=(\frac{{k}_{f}}{{k}_{nf}})\frac{{\varphi}_{2}}{{k}_{R}}$ and $P{r}^{\ast}=Pr/{s}^{2}$ is the modified Prandtl number.

*η*, we get

## 5 The spectral relaxation method

- 1.
Reduce the order of the momentum equation for $f(\eta )$ by introducing the transformation ${f}^{\prime}(\eta )=p(\eta )$ and express the original equation in terms of $p(\eta )$.

- 2.
Assuming that $f(\eta )$ is known from a previous iteration (denoted by ${f}_{r}$), construct an iteration scheme for $p(\eta )$ by assuming that only linear terms in $p(\eta )$ are to be evaluated at the current iteration level (denoted by ${p}_{r+1}$) and all other terms (linear and nonlinear) are assumed to be known from the previous iteration. In addition, nonlinear terms in

*p*are evaluated at the previous iteration. - 3.
The iteration schemes for the other governing dependent variables are developed in a similar manner but now using the updated solutions of the variables determined in the previous equation.

*L*is chosen to be large enough to numerically approximate the conditions at infinity. The basic idea behind the spectral collocation method is the introduction of a differentiation matrix which is used to approximate the derivatives of the unknown variables at the collocation points as the matrix vector product of the form

**D**, that is,

where *p* is the order of the derivative.

In equations (52)-(54), **I** is an identity matrix. The size of the matrix **O** is $(N+1)\times 1$ and $diag[\phantom{\rule{0.2em}{0ex}}]$ is a diagonal matrix, all of size $(\overline{N}+1)\times (\overline{N}+1)$ where $\overline{N}$ is the number of grid points, **f**, **g** and **p** are the values of the functions *f*, *g* and *p*, respectively, when evaluated at the grid points and the subscript *r* denotes the iteration number.

*ω*is introduced and the SRM scheme for finding, say

*X*, is modified to

The results in the next section show that for $\omega <1$, applying the SOR method improves the efficiency and accuracy of the SRM.

## 6 Results and discussion

We have studied heat transfer in a nanofluid flow due to an unsteady stretching sheet. We considered three different nanoparticles, copper oxide (CuO), aluminium oxide (Al_{2}O_{3}) and titanium oxide (TiO_{2}), with water as the base fluid.

The spectral relaxation method algorithm (49)-(51) has been used to solve the nonlinear coupled boundary value problem due to flow over a steady stretching sheet in a nanofluid. We established the accuracy of the spectral relaxation method by comparing the SRM results with those obtained using the Matlab bvp4c solver. The comparison in Tables 3-8 shows a good agrement between the two methods. This comparison provides a benchmark to measure the accuracy and efficiency of the method.

**Effect of the unsteadiness parameter and a comparison of wall temperature gradient**
$\mathbf{-}{\mathit{g}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**for different values of the Prandtl number**

$\mathit{P}\mathit{r}$ | S | $\mathbf{-}{\mathbf{g}}^{\prime}\mathbf{(}\mathbf{0}\mathbf{)}$ | ||
---|---|---|---|---|

El-Aziz [12] | bvp4c | SRM | ||

0.1 | 0.8 | 0.4517 | 0.45148717 | 0.45148717 |

1.0 | 1.6728 | 1.67284531 | 1.67284531 | |

10 | 5.70503 | 5.70597574 | 5.70597574 | |

0.1 | 1.2 | 0.5087 | 0.50850265 | 0.50850265 |

1.0 | 1.818 | 1.81800501 | 1.81800501 | |

10 | 6.12067 | 6.12102434 | 6.12102434 | |

0.1 | 2 | 0.604013 | 0.60351763 | 0.60351763 |

1.0 | 2.07841 | 2.07841323 | 2.07841323 | |

10 | 6.88506 | 6.88615127 | 6.88615127 |

*i.e.*, $\varphi =0$). Firstly, we observe that both the skin friction and the heat transfer coefficients increase with unsteadiness. The skin friction increases as the unsteadiness parameter increases. This is a consequence of the inverted boundary layer that is formed. The negative values of ${f}^{\u2033}(0)$ are an indication that the solid surface exerts a drag force on the fluid. This is due to the fact that the development of the boundary layer is caused solely by the stretching sheet. Secondly, we note that the number of iterations required for the two methods to give a consistent solution decreases as the unsteadiness parameter increases. We further note that the number of iterations to convergence of the SRM decreases by more than a factor of 2 if the SRM is used in conjunction with the SOR method. The fact that the SRM solutions are in good agreement with the bvp4c results is indicative of the accuracy and robustness of the spectral relaxation method.

**Effect of fluid unsteadiness on the skin friction and a comparison of bvp4c with SRM when**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{2}$
**,**
$\mathit{D}\mathbf{=}\mathbf{1}$
**,**
$\mathit{\varphi}\mathbf{=}\mathbf{0}$
**,**
${\mathit{r}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$
**,**
$\mathit{m}\mathbf{=}\mathbf{1.5}$
**,**
$\mathit{\omega}\mathbf{=}\mathbf{0.75}$
**and**
${\mathit{N}}_{\mathit{R}}\mathbf{=}\mathbf{1}$

S | Basic SRM | SRM with SOR | bvp4c | ||
---|---|---|---|---|---|

Iter | $\mathbf{-}{\mathbf{f}}^{\u2033}\mathbf{(}\mathbf{0}\mathbf{)}$ | Iter | $\mathbf{-}{\mathbf{f}}^{\u2033}\mathbf{(}\mathbf{0}\mathbf{)}$ | ||

0.5 | 38 | 1.16721152 | 14 | 1.16721152 | 1.16721152 |

1 | 23 | 1.32052206 | 14 | 1.32052206 | 1.32052206 |

1.5 | 18 | 1.45966589 | 13 | 1.45966589 | 1.45966589 |

**Effect of**
S
**,**
Pr
**,**
D
**and**
${\mathit{N}}_{\mathit{R}}$
**on the heat transfer coefficient and a comparison of bvp4c with SRM for fixed values of**
$\mathit{\varphi}\mathbf{=}\mathbf{0}$
**,**
${\mathit{r}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$
**,**
$\mathit{m}\mathbf{=}\mathbf{1.5}$

S | $\mathit{P}\mathit{r}$ | D | ${\mathbf{N}}_{R}$ | $\mathbf{-}{\mathbf{g}}^{\prime}\mathbf{(}\mathbf{0}\mathbf{)}$ | |
---|---|---|---|---|---|

bvp4c | SRM | ||||

0 | 7 | 1 | 1 | 1.73404471 | 1.73404471 |

0.5 | 1.98512673 | 1.98512673 | |||

1 | 2.21161765 | 2.21161765 | |||

1.5 | 2.41898107 | 2.41898107 | |||

1.5 | 3 | 1 | 1 | 1.52380810 | 1.52380810 |

4 | 1.78458969 | 1.78458969 | |||

5 | 2.01570707 | 2.01570707 | |||

6 | 2.22549552 | 2.22549552 | |||

7 | 2.41898107 | 2.41898107 | |||

1.5 | 2 | 0 | 1 | 1.45831884 | 1.45831884 |

1 | 1.21745970 | 1.21745970 | |||

5 | 0.77377295 | 0.77377295 | |||

1.5 | 2 | 1 | 1 | 1.21745970 | 1.21745970 |

5 | 1.40695145 | 1.40695145 | |||

10 | 1.43699000 | 1.43699000 |

Table 4 shows the effect of flow unsteadiness, the Prandtl number, thermal dispersion and the radiation parameter on the heat transfer coefficient in the absence of nanoparticles. The heat transfer coefficient decreases with an increase in thermal dispersion and increases with thermal radiation. The results show that the rate of unsteady heat transfer can be accelerated by the thermal dispersion. The thermal dispersion may be regarded as the effect of mixing to enhance heat transfer in the medium. Table 4 further gives a comparison of the SRM and the bvp4c results in the case $\varphi =0$. The spectral relaxation method converges to the numerical solutions for all parameter values matching the bvp4c results up to nine significant digits.

_{2}O

_{3}, TiO

_{2}nanofluids. We observe that the skin friction coefficient decreases with increasing nanoparticle volume fraction, and the same trend is observed in the case of the heat transfer coefficient. Table 5 further shows that for a particular nanoparticle volume fraction, the skin friction coefficient increases with an increase in the unsteadiness parameter. A similar trend is observed in the case of the heat transfer coefficient.

**Effect of unsteadiness and the solid volume fraction on the skin friction coefficient and a comparison of bvp4c with SRM for different values of nanoparticle volume fraction with CuO-water nanofluid for fixed values of**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{6.7850}$
**,**
$\mathit{D}\mathbf{=}\mathbf{1}$
**,**
${\mathit{r}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$
**,**
$\mathit{m}\mathbf{=}\mathbf{1.5}$
**,**
${\mathit{N}}_{\mathit{R}}\mathbf{=}\mathbf{1}$
**and**
$\mathit{\omega}\mathbf{=}\mathbf{0.75}$

S | ϕ | Basic SRM | SRM with SOR | bvp4c | ||
---|---|---|---|---|---|---|

Iter | $\mathbf{-}{\mathbf{f}}^{\u2033}\mathbf{(}\mathbf{0}\mathbf{)}$ | Iter | $\mathbf{-}{\mathbf{f}}^{\u2033}\mathbf{(}\mathbf{0}\mathbf{)}$ | |||

0.5 | 0.05 | 38 | 1.16449176 | 14 | 1.16449176 | 1.16449176 |

0.1 | 37 | 1.14990897 | 14 | 1.14990897 | 1.14990897 | |

0.15 | 38 | 1.12498006 | 13 | 1.12498006 | 1.12498006 | |

0.2 | 35 | 1.09094877 | 13 | 1.09094877 | 1.09094877 | |

1 | 0.05 | 23 | 1.31744508 | 14 | 1.31744508 | 1.31744508 |

0.1 | 23 | 1.30094687 | 13 | 1.30094687 | 1.30094687 | |

0.15 | 23 | 1.27274360 | 14 | 1.27274360 | 1.27274360 | |

0.2 | 23 | 1.23424238 | 14 | 1.23424238 | 1.23424238 | |

1.5 | 0.05 | 18 | 1.45626469 | 13 | 1.45626469 | 1.45626469 |

0.1 | 18 | 1.43802806 | 13 | 1.43802806 | 1.43802806 | |

0.15 | 18 | 1.40685300 | 13 | 1.40685300 | 1.40685300 | |

0.2 | 18 | 1.36429489 | 14 | 1.36429489 | 1.36429489 |

**Comparison of**
$\mathbf{-}{\mathit{g}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**obtained by bvp4c and SRM for different values of nanoparticle volume fraction with CuO-water nanofluid for fixed values of**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{6.7850}$
**,**
$\mathit{D}\mathbf{=}\mathbf{1}$
**,**
${\mathit{r}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$
**,**
$\mathit{m}\mathbf{=}\mathbf{1.5}$
**,**
$\mathit{\omega}\mathbf{=}\mathbf{0.75}$
**and**
${\mathit{N}}_{\mathit{R}}\mathbf{=}\mathbf{1}$

S | ϕ | Basic SRM | SRM with SOR | bvp4c | ||
---|---|---|---|---|---|---|

Iter | $\mathbf{-}{\mathbf{g}}^{\prime}\mathbf{(}\mathbf{0}\mathbf{)}$ | Iter | $\mathbf{-}{\mathbf{g}}^{\prime}\mathbf{(}\mathbf{0}\mathbf{)}$ | |||

0.5 | 0.05 | 38 | 1.85367196 | 14 | 1.85367196 | 1.85367196 |

0.1 | 37 | 1.75916556 | 14 | 1.75916556 | 1.75916556 | |

0.15 | 37 | 1.66693860 | 13 | 1.66693860 | 1.66693860 | |

0.2 | 35 | 1.57677868 | 13 | 1.57677868 | 1.57677868 | |

1 | 0.05 | 23 | 2.06861805 | 14 | 2.06861805 | 2.06861805 |

0.1 | 23 | 1.96536212 | 13 | 1.96536212 | 1.96536212 | |

0.15 | 23 | 1.86412754 | 14 | 1.86412754 | 1.86412754 | |

0.2 | 23 | 1.76472535 | 14 | 1.76472535 | 1.76472535 | |

1.5 | 0.05 | 18 | 2.26481693 | 13 | 2.26481693 | 2.26481693 |

0.1 | 18 | 2.15317826 | 13 | 2.15317826 | 2.15317826 | |

0.15 | 18 | 2.04340762 | 13 | 2.04340762 | 2.04340762 | |

0.2 | 18 | 1.93533542 | 14 | 1.93533542 | 1.93533542 |

*D*have no direct effect on the skin friction coefficient, but that the skin friction decreases with an increase in the unsteadiness parameter. From Table 8 we observe that an increase in the thermal radiation parameter produces significant increases in the heat transfer coefficient. The skin friction and heat transfer rates decrease with an increase in the nanoparticle volume fraction.

**Comparison of**
$\mathbf{-}{\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**obtained by bvp4c and SRM for different values of nanoparticle volume fraction with Al**
_{
2
}
**O**
_{
3
}
**-water nanofluid for fixed values of**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{6.7850}$
**,**
${\mathit{r}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$
**,**
$\mathit{m}\mathbf{=}\mathbf{1.5}$
**and**
$\mathit{\omega}\mathbf{=}\mathbf{0.75}$

Quantity | ϕ | Basic SRM | SRM with SOR | bvp4c | ||
---|---|---|---|---|---|---|

Iter | $\mathbf{-}{\mathbf{f}}^{\u2033}\mathbf{(}\mathbf{0}\mathbf{)}$ | Iter | $\mathbf{-}{\mathbf{f}}^{\u2033}\mathbf{(}\mathbf{0}\mathbf{)}$ | |||

${N}_{R}=1$ | 0.05 | 23 | 1.32762309 | 14 | 1.32762309 | 1.32762309 |

0.1 | 23 | 1.31890045 | 14 | 1.31890045 | 1.31890045 | |

0.15 | 23 | 1.29654739 | 13 | 1.29654739 | 1.29654739 | |

0.2 | 23 | 1.26231187 | 14 | 1.26231187 | 1.26231187 | |

${N}_{R}=5$ | 0.05 | 23 | 1.32762309 | 14 | 1.32762309 | 1.32762309 |

0.1 | 23 | 1.31890045 | 14 | 1.31890045 | 1.31890045 | |

0.15 | 23 | 1.29654739 | 14 | 1.29654739 | 1.29654739 | |

0.2 | 23 | 1.26231187 | 13 | 1.26231187 | 1.26231187 | |

${N}_{R}=10$ | 0.05 | 38 | 1.17348812 | 14 | 1.17348812 | 1.17348812 |

0.1 | 38 | 1.16577817 | 14 | 1.16577817 | 1.16577817 | |

0.15 | 37 | 1.14602026 | 14 | 1.14602026 | 1.14602026 | |

0.2 | 37 | 1.11575944 | 13 | 1.11575944 | 1.11575944 |

**Comparison of**
$\mathbf{-}{\mathit{g}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**obtained by bvp4c and SRM for different values of nanoparticle volume fraction with Al**
_{
2
}
**O**
_{
3
}
**-water nanofluid for fixed values of**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{6.7850}$
**,**
${\mathit{r}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$
**,**
$\mathit{m}\mathbf{=}\mathbf{1.5}$
**and**
$\mathit{\omega}\mathbf{=}\mathbf{0.75}$

Quantity | ϕ | Basic SRM | SRM with SOR | bvp4c | ||
---|---|---|---|---|---|---|

Iter | $\mathbf{-}{\mathbf{g}}^{\prime}\mathbf{(}\mathbf{0}\mathbf{)}$ | Iter | $\mathbf{-}{\mathbf{g}}^{\prime}\mathbf{(}\mathbf{0}\mathbf{)}$ | |||

${N}_{R}=1$ | 0.05 | 23 | 2.07754148 | 14 | 2.07754148 | 2.07754148 |

0.1 | 23 | 1.98429513 | 14 | 1.98429513 | 1.98429513 | |

0.15 | 23 | 1.89386696 | 13 | 1.89386696 | 1.89386696 | |

0.2 | 23 | 1.80581775 | 14 | 1.80581775 | 1.80581775 | |

${N}_{R}=5$ | 0.05 | 23 | 2.43674192 | 14 | 2.43674192 | 2.43674192 |

0.1 | 23 | 2.35621359 | 14 | 2.35621359 | 2.35621359 | |

0.15 | 23 | 2.27587972 | 14 | 2.27587972 | 2.27587972 | |

0.2 | 23 | 2.19532143 | 13 | 2.19532143 | 2.19532143 | |

${N}_{R}=10$ | 0.05 | 23 | 2.49509121 | 13 | 2.49509121 | 2.49509121 |

0.1 | 23 | 2.41787850 | 14 | 2.41787850 | 2.41787850 | |

0.15 | 23 | 2.34049758 | 13 | 2.34049758 | 2.34049758 | |

0.2 | 23 | 2.26250267 | 14 | 2.26250267 | 2.26250267 |

Figure 1 shows the effect of the unsteadiness parameter on the velocity and temperature profiles, respectively, in the case of a CuO-water nanofluid. It is observed that velocity and temperature in the case of a clear fluid are less than those of a CuO-water nanofluid. Increasing the unsteadiness parameter results in a decrease in the thermal boundary layer thickness. We also observe that the temperature distribution decreases with an increase in the unsteadiness parameter in the case of both a clear fluid and the CuO-water nanofluid. These results are consistent with the findings of, among others, Singh *et al.* [36].

_{2}-water and Al

_{2}O

_{3}-water nanofluids.

_{2}-water nanofluid is attained at higher values of

*ϕ*in comparison with CuO-water and Al

_{2}O

_{3}nanofluids. Further, we observe that a CuO-water nanofluid gives a higher drag force in opposition to the flow as compared to the other nanofluids. From Figure 4(b) we observe that the wall heat transfer rates for the nanofluids are increasing functions of

*ϕ*. A TiO

_{2}-water nanofluid has higher wall heat transfer rate as compared to the other nanofluids.

## 7 Conclusions

The unsteady boundary layer flow in a nanofluid due to a stretching sheet with thermal dispersion and radiation was studied. The governing equations were transformed into a set of coupled nonlinear differential equations and solved by bvp4c and a novel spectral relaxation method. To determine the convergence, accuracy and general validity of the SRM, the results were compared with the Matlab bvp4c results for selected values of the governing physical parameters. We found that velocity and temperature in the case of a clear fluid are less than those of CuO-water nanofluid. We observe that the temperature decreases throughout the boundary layer with increasing values of the radiation parameter. The convergence rate of the SRM is significantly improved by using the method in conjunction with the SOR method with $\omega <1$. Our findings show that the SRM is accurate and sufficiently robust for use in solving fluid flow problems and as an alternative to the Runge-Kutta and Keller-box schemes in finding solutions of boundary layer equations.

## Declarations

### Acknowledgements

The authors are grateful to the reviewers for their comments and constructive suggestions. We acknowledge financial support from the University of KwaZulu-Natal.

## Authors’ Affiliations

## References

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