# A new numerical approach for MHD laminar boundary layer flow and heat transfer of nanofluids over a moving surface in the presence of thermal radiation

- Stanford Shateyi
^{1}Email author and - Jagdish Prakash
^{2}

**2014**:2

**DOI: **10.1186/1687-2770-2014-2

© Shateyi and Prakash; licensee Springer. 2014

**Received: **19 August 2013

**Accepted: **1 December 2013

**Published: **2 January 2014

## Abstract

Numerical analysis has been carried out on the problem of magnetohydrodynamic boundary layer flow of a nanofluid over a moving surface in the presence of thermal radiation. The governing partial differential equations were transformed into a system of ordinary differential equations using suitable similarity transformations. The resultant ordinary equations were then solved using the spectral relaxation method. Effects of the physical parameters on the velocity, temperature and concentration profiles as well as the local skin-friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form.

**MSC:** 65PXX, 76-XX.

### Keywords

MHD boundary layer flow nanofluids thermal radiation moving surface## 1 Introduction

Many engineering and industrial processes involve heat transfer by means of a flowing fluid in either laminar or turbulent regimes. A decrease in thermal resistance of heat transfer in the fluids would significantly benefit many of these applications/processes. Nanofluids have the potential to reduce thermal resistances, and industrial groups such as electronics, medical, food and manufacturing would benefit from such improved heat transfer. It is well known that conventional heat transfer fluids, such as oil, water and ethylene glycol mixture, are poor heat transfer fluids. Choi [1] introduced the technique of nanofluids by using a mixture of nanoparticles and the base fluids. The presence of the nanoparticles in the nanofluid increases the thermal conductivity and therefore substantially enhances the heat transfer characteristics of the nanofluid. Nanotechnology has been an ongoing hot topic of discussion in public health as researchers claim that nanoparticles could present possible dangers in health and environment, Mnyusiwella *et al.* [2]. There are several numerical studies on the modelling of natural convection heat transfer in nanofluids (Kaka and Pramuanjaroekij [3], Godson *et al.* [4], Olanrewaju *et al.* [5]). Gbadeyan *et al.* [6] numerically studied boundary layer flow induced in a nanofluid due to a linearly stretching sheet in the presence of thermal radiation and induced magnetic field. Khan and Aziz [7] studied natural convection flow of a nanofluid over a vertical plate with a uniform surface heat flux. Makinde and Aziz [8] investigated boundary layer flow of a nanofluid past a stretching sheet with a convective boundary conditions. Khan *et al.* [9] studied the unsteady free convection boundary layer flow of a nanofluid along a stretching sheet with thermal radiation in the presence of a magnetic field.

Thermal radiation-convection interaction problems are found in the cooling of high temperature components design where heat transfer from surfaces occurs by parallel radiation and convection, the interaction of incident solar radiation with the earth’s surface to produce complex free convection patterns. Hadey *et al.* [10] studied the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation. Khan *et al.* [11] analyzed the effects of variable viscosity and thermal conductivity on the flow and heat transfer in a laminar liquid film on a horizontal shrinking/stretching sheet. Khan *et al.* [12] investigated, by employing the homotopy perturbation transform method (HPTM) and the Padé approximation, the problem of magnetohydrodynamic (MHD) boundary layer flow over a nonlinear porous stretching sheet. Khan *et al.* [13] considered a two-dimensional, steady magnetohydrodynamic flow and heat transfer analysis of a non-Newtonian fluid in a channel with a constant wall temperature in the presence of thermal radiation.

As the governing equations modelling MHD flow and heat transfer of nanofluids are highly nonlinear, exact solutions are impossible to obtain. Over the years, numerical methods have been developed, improved, and hybred as a way of getting more accurate solutions. The current study seeks to employ a recently developed numerical technique known as spectral relaxation method [14] to solve the problem of magnetohydrodynamic boundary layer flow of nanofluids over a moving surface in the presence of thermal radiation. The current method has been successfully employed in [15–20], among others. We apply this method to the problem of MHD flow of a nanofluid past a stretching sheet in the presence of thermal radiation. The governing boundary layer equations are transformed to a system of nonlinear ordinary differential by using suitable local similarity variables.

## 2 Mathematical formulation

*U*and that of the plate is ${U}_{w}=\lambda U$, where

*λ*is the velocity parameter. The flow is assumed to take place at $y\ge 0$, with

*y*being the coordinate measured normal to the moving surface. A uniform magnetic field is applied in the

*y*direction. At moving surface, the temperature and the nanoparticles take constant values ${T}_{w}$ and ${C}_{w}$, respectively, while the free stream values are taken to be ${T}_{\mathrm{\infty}}$ and ${C}_{\mathrm{\infty}}$, respectively. Following the nanofluid model proposed by Tiwari and Das [21] along with Boussinesq and boundary layer approximations, the governing equations for the present problem are:

where *u* and *v* are the velocity components along the *x*-axis and *y*-axis, respectively, $\alpha =k/{(\rho c)}_{f}$ is the thermal diffusivity of the fluid, *ν* is the kinematic viscosity coefficient, *k* is the thermal conductivity, ${q}_{r}$ is the heat flux, ${D}_{B}$ is the Brownian diffusion coefficient, ${D}_{T}$ is the thermophoresis diffusion coefficient, ${B}_{0}$ is the uniform magnetic field strength of the base fluid, *σ* is the electrical conductivity of the base fluid, *τ* is the ratio of the nanoparticle heat capacity and the base fluid heat capacity.

The surface moving parameter $\lambda >0$ corresponds to the downstream movement of the plate from the origin, while $\lambda <0$ corresponds to the upstream movement of the plate.

*et al.*[22]) and following Raptis [23] among other researchers, the radiative heat flux ${q}_{r}$ is given by

### 2.1 Similarity transformations

*η*is the similarity variable, $\theta (\eta )$ is the dimensionless temperature, $\varphi (\eta )$ is the dimensionless nanoparticles concentration. It is worth mentioning that the continuity equation (1) is identically satisfied from our choice of the stream function with $u=\frac{\partial \psi}{\partial y}$ and $v=-\frac{\partial \psi}{\partial x}$. Substituting the similarity variables into equations (2)-(4) gives

where $R=\frac{4\alpha \delta {T}_{\mathrm{\infty}}^{3}}{kk}$ is the radiation parameter, $\mathit{Pr}=\frac{\nu}{\alpha}$ is the Prandtl number, $\mathit{Le}=\frac{\nu}{{D}_{B}}$ is the Lewis number, $\mathit{Ha}=\frac{2x{B}_{0}^{2}}{U\rho f}$ is the local Hartman number, $\mathit{Nb}=\frac{{(\rho c)}_{p}{D}_{B}({\varphi}_{w}-{\varphi}_{\mathrm{\infty}})}{(\rho C)f\nu}$ is the Brownian motion parameter and $\mathit{Nt}=\frac{{(\rho C)}_{p}{\rho}_{T}({T}_{w}-{T}_{\mathrm{\infty}})}{(\rho c)f{T}_{\mathrm{\infty}}\nu}$ is the thermophoresis parameter.

where ${\mathit{Re}}_{x}=Ux/\nu $ is the local Reynolds number.

## 3 Method of solution

To solve the set of ordinary differential equations (11)-(13) together with the boundary conditions (14) and (15), we employ the Chebyshev pseudo-spectral method known as spectral relaxation method (SRM). This method transforms sets of nonlinear ordinary differential into sets of linear ordinary differential equations. The entire computational procedure is implemented using a program written in MATLAB computer language. The fluid velocity, temperature, the local skin-friction coefficient and the local Nusselt and Sherwood numbers are determined from these numerical computations. The SRM algorithm starts with the assumption of having a system of *m* nonlinear ordinary differential equations in *m* unknowns functions, ${z}_{i}(\eta )$, $i=1,2,\dots ,m$, where $\eta \in [a,b]$ is the independent variable. To solve the resultant iterative scheme, we then use the Chebyshev pseudo-spectral method. The details of the spectral methods can be found in Canuto *et al.* [24], Trefethen [25]. Before applying the spectral method, the domain on which the governing equation is defined is transformed to the interval $[-1,1]$ on which the spectral method can then be implemented. We use the transformation $\eta =(b-a)(\tau +1)/2$ to map the interval $[a,b]$ to $[-1,1]$.

The above equations form a system of linear decoupled equations which can be solved iteratively for $r=1,2,\dots $ , starting from initial guesses/approximations $({g}_{0}(\eta ),\theta (\eta ),\varphi (\eta ))$.

**I**is the identity matrix of size $(\overline{N}+1)\times (\overline{N}+1)$,

**f**,

**g**, ϕ and θ are the values of

*f*,

*g*,

*ϕ*and

*θ*, respectively, when evaluated at the grid points. Equations (27)-(30) constitute the SRM scheme. The initial approximation required to start the iterative process is

Accuracy of the scheme is established by increasing the number of collocation points *N* until the solutions are consistent and further increases do not change the value of the solutions.

## 4 Results and discussion

The system of ordinary differential equations (12)-(13) subject to the boundary conditions (14)-(15) are numerically solved by using spectral relaxation method (SRM). This is a recently developed method, and details of the method are found in [26]. The SRM results presented in this work were obtained using $N=50$ collocation points, and also the convergence was achieved after as few as five iterations. We also take 15 to be the infinity value ${\eta}_{\mathrm{\infty}}$. We use these default values for the parameters $\mathit{Pr}=0.71$, $\mathit{Nt}=\mathit{Nb}=0.3$, $\mathit{Le}=2$, $R=2$, $\lambda =0.1$.

*Le*on the heat and mass transfer coefficients. The Lewis number is defined as the ratio of thermal diffusivity to mass diffusivity or the ratio of the Schmidt number to the Prandtl number. As can be clearly seen in Table 2, increasing the Lewis number means that the fluid becomes more viscous therefore causing increases in the rate of mass transfer at the surface. As expected, increasing the Lewis number reduces the rate of heat transfer on the surface as this corresponds to the reduced Prandtl number. In Table 3, we display the effect of increasing the values of the moving surface parameter

*λ*on the rates of heat and mass transfer. These rates both increase with increasing values of the surface moving parameter. Table 4 depicts the effects of Brownian motion parameter

*Nb*and the thermophoresis parameter on the heat and mass transfer coefficients. The rate of heat transfer on the surface as expected is reduced as the Brownian motion parameter increases, while the rate of mass transfer slightly increases as this parameter increases. Also in Table 4, we depict the effect of the thermophoresis parameter

*Nt*on the Nusselt and Sherwood numbers. By definition thermophoresis is the migration of a colloidal particle in a solution in response to a macroscopic temperature gradient. Both the Nusselt number and the Sherwood number are reduced as the values of this parameter increases.

**Comparison of SRM solutions for** $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ **and** $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ **against those of the** $\mathtt{b}\mathtt{v}\mathtt{p}\mathtt{4}\mathtt{c}$ **as well as those obtained by Olanrewaju** et al. [5] **for different values of** Pr

Pr | [5] | bvp4c $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | Present (SRM) | [5] | bvp4c $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | Present (SRM) |
---|---|---|---|---|---|---|

1 | 0.35300909 | 0.35300909 | 0.35300909 | 0.60037641 | 0.60037641 | 0.60037641 |

2 | 0.44343465 | 0.44343465 | 0.44343465 | 0.56245704 | 0.56245704 | 0.56245704 |

6 | 0.59307165 | 0.59307165 | 0.59307165 | 0.481913727 | 0.481913727 | 0.481913727 |

**Comparison of SRM solutions for** $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ **and** $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ **against those of the** $\mathtt{b}\mathtt{v}\mathtt{p}\mathtt{4}\mathtt{c}$ **as well as those obtained by Olanrewaju** et al. [5] **for different values of** Le

Le | [5] | bvp4c $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | Present (SRM) | [5] | bvp4c $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | Present (SRM) |
---|---|---|---|---|---|---|

10 | 0.31104609 | 0.31104594 | 0.31104594 | 1.22044717 | 1.22044716 | 1.22044716 |

20 | 0.31066665 | 0.31066665 | 0.31066665 | 1.6237386 | 1.62037385 | 1.62037385 |

50 | 0.310302944 | 0.31030279 | 0.31030279 | 2.36541737 | 2.36541736 | 2.36541736 |

**Comparison of SRM solutions for** $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ **and** $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ **against those of the** $\mathtt{b}\mathtt{v}\mathtt{p}\mathtt{4}\mathtt{c}$ **as well as those obtained by Olanrewaju** et al. [5] **for different values of** λ

λ | [5] | bvp4c $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | Present (SRM) | [5] | bvp4c $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | Present (SRM) |
---|---|---|---|---|---|---|

0.5 | 0.36940306 | 0.36940302 | 0.36940302 | 0.81838542 | 0.81838541 | 0.81838541 |

1 | 0.42412772 | 0.42412771 | 0.42412771 | 1.0624088 | 1.0624088 | 1.0624088 |

2 | 0.50948651 | 0.50948651 | 0.50948651 | 1.32551731 | 1.32551731 | 1.32551731 |

**Computations of**
$\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**and**
$\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**at different values of**
Nb
**and**
Nt

Nb | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|

0.1 | 0.27455519 | 0.48306117 |

0.5 | 0.24471306 | 0.53071472 |

1 | 0.21121513 | 0.53632970 |

Nt | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|

0.1 | 0.27455519 | 0.48306117 |

0.5 | 0.25536198 | 0.32985814 |

1 | 0.23358969 | 0.25949965 |

**Comparison of SRM solutions for**
$\mathbf{-}{\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**against those of the**
$\mathtt{b}\mathtt{v}\mathtt{p}\mathtt{4}\mathtt{c}$
**for different values of**
λ

λ | bvp4c ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | Present (SRM) |
---|---|---|

0.1 | 0.46251223 | 0.46251223 |

0.2 | 0.44315478 | 0.44315478 |

0.4 | 0.32874112 | 0.32874112 |

*λ*corresponds to accelerated movement of the plate. This in turn results in reduction of skin friction. The effects of physical parameters of importance in this study on the velocity, temperature and nanoparticles distributions are depicted in Figures 1-9. Figure 1 displays the effect of the Hartman number

*Ha*on the dimensionless velocity. The presence of a magnetic field normal to the flow in an electrically conducting fluid gives rise to a Lorentz force, which acts against the flow. Thus the velocity distributions of the nanofluid are greatly reduced with increasing values of the Hartman number.

*λ*on the velocity profiles. Increasing the down stream movement of the plate from the origin, as expected, enhances the fluid velocity. The influence of the magnetic field parameter on the temperature distribution is shown in Figure 3. From this figure, we observe that the temperature profiles increase with increasing values of the magnetic field parameter. This implies that the applied magnetic field tends to heat the fluid, thus reducing the heat transfer from the wall.

*R*on the temperature distribution. We observe in this figure that the thickness of the thermal boundary layer increases as the values of the radiation parameter are increased. Thus thermal radiation enhances thermal diffusion.

*Nb*results in thickening of the thermal boundary layer. Thus enhancing the temperature of the nanofluid as can be easily seen in Figure 6.

*Ha*on the nanoparticle volume fraction profiles is shown in Figure 7. Application of a normal magnetic field results in slowing down of the nanofluid flow thereby resulting in volumetric increase of the nanoparticles within the flow as less particles are now transported per given time.

## 5 Conclusion

Numerical analysis has been carried out on MHD boundary layer flow of nanofluids over a moving surface in the presence of thermal radiation. The governing partial differential equations were transformed into a system of ordinary differential equations using suitable similarity transformations. The resultant equations were then solved using the spectral relaxation method. The accuracy of the SRM is validated against the MATLAB in-built $\mathtt{b}\mathtt{v}\mathtt{p}\mathtt{4}\mathtt{c}$ routine for solving boundary value problems as well as previously obtained results. An excellent agreement was observed between our results and those obtained using other methods giving confidence to our present results. We observed that the local temperature rises as the Brownian motion, thermophoresis and radiation effects intensify. Lastly, the Nusselt number decreases while the Sherwood number increases as the Brownian motion and thermophoresis effects are increased.

## Greek letters

## Nomenclature

## Declarations

### Acknowledgements

The authors wish to acknowledge financial support from the University of Venda, NRF and University of Botswana.

## Authors’ Affiliations

## References

- Choi SUS: Enhancing thermal conductivity of fluids with nanoparticles. FED 231.
*Developments and Applications of Non-Newtonian Flows*1995, 99-105. MD, vol. 66Google Scholar - Mnyusiwalla A, Daar AS, Singer PA: Mind the gap: science and ethics in nanotechnology.
*Nanotechnology*2003, 14: R9-R13. 10.1088/0957-4484/14/3/201View ArticleGoogle Scholar - Kakaç S, Pramuanjaroenkij A: Review of convective heat transfer enhancement with nanofluids.
*Int. J. Heat Mass Transf.*2009, 52: 3187-3196. 10.1016/j.ijheatmasstransfer.2009.02.006View ArticleGoogle Scholar - Godson L, Raja B, Lal DM, Wongwises S: Enhancement of heat transfer using nanofluid - An overview.
*Renew. Sustain. Energy Rev.*2010, 14: 629-641. 10.1016/j.rser.2009.10.004View ArticleGoogle Scholar - Olanrewaju PO, Olanrewaju MA, Adesanya AO: Boundary layer flow of nanofluids over a moving surface in a flowing fluid in the presence of radiation.
*Int. J. Appl. Sci. Technol.*2012, 2(1):274-285.Google Scholar - Gbadeyan JA, Olanrewaju MA, Olanrewaju PO: Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition in the presence of magnetic field and thermal radiation.
*Aust. J. Basic Appl. Sci.*2011, 5(9):1323-1334.Google Scholar - Khan WA, Aziz A: Natural convection flow of a nanofluid over a vertical plate with uniform surface heat flux.
*Int. J. Therm. Sci.*2011, 50: 1207-1214. 10.1016/j.ijthermalsci.2011.02.015View ArticleGoogle Scholar - Makinde OD, Aziz A: Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition.
*Int. J. Therm. Sci.*2011, 50: 1326-1332. 10.1016/j.ijthermalsci.2011.02.019View ArticleGoogle Scholar - Khan MS, Karim I, Ali LE, Islam A: Unsteady MHD free convection boundary-layer flow of a nanofluid along a stretching sheet with thermal radiation and viscous dissipation effects.
*Int. Nano Lett.*2012., 2: Article ID 24 10.1186/2228-5326-2-24Google Scholar - Hady FM, Ibrahim FS, Abdel-Gaied SM, Eid MR: Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet.
*Nanoscale Res. Lett.*2012, 7: 229-242. 10.1186/1556-276X-7-229View ArticleGoogle Scholar - Khan Y, Wua Q, Faraz N, Yildirim A: The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet.
*Comput. Math. Appl.*2011, 61(11):3391-3399. 10.1016/j.camwa.2011.04.053MathSciNetView ArticleGoogle Scholar - Khan Y, Abdou MA, Faraz N, Yildirim A, Wu Q: Numerical solution of MHD flow over a nonlinear porous stretching sheet.
*Iran. J. Chem. Chem. Eng.*2012, 31(3):125-132.Google Scholar - Khan Y, Wu Q, Faraz N, Yildirim A, Mohyud-Din ST: Heat transfer analysis on the magnetohydrodynamic flow of a non-Newtonian fluid in the presence of thermal radiation: an analytic solution.
*Z. Naturforsch. A*2012, 67a: 147-152.Google Scholar - Motsa SS, Dlamini P, Khumalo M: A new multistage spectral relaxation method for solving chaotic initial value systems.
*Nonlinear Dyn.*2013, 72: 265-283. 10.1007/s11071-012-0712-8MathSciNetView ArticleGoogle Scholar - Motsa SS, Dlamini PG, Khumalo M: Solving hyperchaotic systems using the spectral relaxation method.
*Abstr. Appl. Anal.*2012. 10.1155/2012/203461Google Scholar - Motsa SS, Dlamini P, Khumalo M: A new multistage spectral relaxation method for solving chaotic initial value systems.
*Nonlinear Dyn.*2013. 10.1007/s11071-012-0712-8Google Scholar - Motsa SS, Makukula ZG: On spectral relaxation method approach for steady von Kármán flow of a Reiner-Rivlin fluid with Joule heating, viscous dissipation and suction/injection.
*Cent. Eur. J. Phys.*2013. 10.2478/s11534-013-0182-8Google Scholar - Shateyi S, Makinde OD: Hydromagnetic stagnation-point flow towards a radially stretching convectively heated disk.
*Math. Probl. Eng.*2013. 10.1155/2013/616947Google Scholar - Shateyi S, Marewo GT: A new numerical approach of MHD flow with heat and mass transfer for the UCM fluid over a stretching surface in the presence of thermal radiation.
*Math. Probl. Eng.*2013. 10.1155/2013/670205Google Scholar - Shateyi S: A new numerical approach for MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction.
*Bound. Value Probl.*2013. 10.1186/1687-2770-2013-196Google Scholar - Tiwari RK, Das MK: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids.
*Int. J. Heat Mass Transf.*2007, 50: 2002-2018. 10.1016/j.ijheatmasstransfer.2006.09.034View ArticleGoogle Scholar - Hossain MA, Takhar HS: Radiation effect on mixed convection along a vertical plate with uniform surface temperature.
*Heat Mass Transf.*1996, 31: 243-248. 10.1007/BF02328616View ArticleGoogle Scholar - Raptis A: Flow of a micropolar fluid past a continuously moving plate by the presence of radiation.
*Int. J. Heat Mass Transf.*1998, 41: 2865-2866. 10.1016/S0017-9310(98)00006-4View ArticleGoogle Scholar - Canuto C, Hussaini MY, Quarteroni A, Zang TA:
*Spectral Methods in Fluid Dynamics*. Springer, Berlin; 1988.View ArticleGoogle Scholar - Trefethen LN:
*Spectral Methods in MATLAB*. SIAM, Philadelphia; 2000.View ArticleGoogle Scholar - Motsa SS: A new spectral relaxation method for similarity variable nonlinear boundary layer flow systems.
*Chem. Eng. Commun.*2013. 10.1080/00986445.2013.766882Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.