# Thermodiffusion effects on magneto-nanofluid flow over a stretching sheet

- Faiz G Awad
^{1}, - Precious Sibanda
^{1}Email author and - Ahmed A Khidir
^{1}

**2013**:136

https://doi.org/10.1186/1687-2770-2013-136

© Awad et al.; licensee Springer. 2013

**Received: **23 November 2012

**Accepted: **7 May 2013

**Published: **24 May 2013

## Abstract

We study the effect of thermophoresis on boundary layer magneto-nanofluid flow over a stretching sheet. The model includes the effects of Brownian motion and cross-diffusion effects. The governing partial differential equations are transformed to a system of ordinary differential equations and solved numerically using a spectral linearisation method. The effects of the magnetic influence number, the Prandtl number, Lewis number, the Brownian motion parameter, thermophoresis parameter, the modified Dufour parameter and the Dufour-solutal Lewis number on the fluid properties as well as on the heat, regular and nano mass transfer coefficients are determined and shown graphically.

## 1 Introduction

Most common fluids such as water, ethylene, glycol, toluene or oil generally have poor heat transfer characteristics owing to their low thermal conductivity. A recent technique to improve the thermal conductivity of these fluids is to suspend nano-sized metallic particles such as aluminum, titanium, gold, copper, iron or their oxides in the fluid to enhance its thermal properties, Choi [1]. The enhancement of thermal conductivity in nanofluids has been studied by, among others, Kakac and Pramuanjaroenkij [2], Choi *et al.* [3], Masuda *et al.* [4], Eapen *et al.* [5] and Fan and Wang [6]. Nield and Kuznetsov [7] analyzed the behaviour of boundary layer flow on the Chen-Minkowycz problem in a porous layer saturated with a nanofluid. Nield and Kuznetsov [8] investigated thermal instability in a porous medium saturated with nanofluid using the Brinkman model. The model incorporated the effects of Brownian motion and thermophoresis of nanoparticles. They found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount depending on whether the nanoparticle distribution is top-heavy or bottom-heavy. Aziz *et al.* [9] studied steady boundary layer flow past a horizontal flat plate embedded in a porous medium filled with a water-based nanofluid containing gyrotactic microorganisms. Cheng [10] investigated the behaviour of boundary layer flow over a horizontal cylinder of elliptic cross section in a porous medium saturated with a nanofluid. Chamkha *et al.* [11] investigated the non-similar solutions for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid.

During the last few decades, fluid flow over a stretching surface has received considerable attention because of its engineering applications such as in melt-spinning, hot rolling, wire drawing, glass-fiber production and the manufacture of polymer and rubber sheets, Altan and Gegel [12], Fisher [13], and Tidmore and Klein [14]. Nanofluid flow over a stretching surface has been investigated by many researchers. The first study on a stretching sheet in nanofluids was published by Khan and Pop [15]. Makinde and Aziz [16] performed a numerical study of boundary layer flow over a linear stretching sheet. Both Brownian motion and thermophoresis effects on the transport equations were presented. They reported that stronger Brownian motion and thermophoresis lead to an increase in the rate of heat transfer. However, the opposite was observed in the case of the rate of mass transfer. Recent studies in this area include those of Narayana and Sibanda [17] and Kameswaran *et al.* [18].

Magnetic nanofluids have numerous uses or potential applications in engineering and medicine. Using magnetic nanofluids has the potential to regulate the flow rate and heat transfer by controlling the thermo-magnetic convection current and the fluid velocity (see Shima *et al.* [19], Ganguly *et al.* [20]). The effects of a magnetic field on nanofluid flow over a stretching sheet have been investigated by, among others, Bachok *et al.* [21] and Hanad and Ferdows [22].

The aim of this study is to analyse Dufour and Soret effects in a magneto-nanofluid flow over a stretching sheet. In addition, we study Brownian motion and thermophoresis effects using a spectral linearisation method to obtain numerical solutions of the momentum, energy, concentration and mass fraction equations. The successive linearisation method (SLM) is an accurate method for solving non-linear coupled equations (see [23–25]). Recent studies such as [26–28] have suggested that the SLM is accurate and converges rapidly to the numerical results when compared to other semi-analytical methods such as the Adomian decomposition method, the variational iteration method and the homotopy perturbation method.

## 2 Mathematical formulation

*a*is a real positive number. The coordinate system is assumed to define the

*x*-axis along the surface of the sheet and

*y*is the coordinate normal to the surface of the sheet. The surface temperature ${T}_{w}$ and nanoparticle concentration ${\stackrel{\u02c6}{\varphi}}_{w}$ are higher than the ambient values ${T}_{\mathrm{\infty}}$ and ${\stackrel{\u02c6}{\varphi}}_{\mathrm{\infty}}$, respectively. The governing equations for the problem can be written in the form

*u*and

*v*are the velocity components along the

*x*and

*y*direction respectively,

*σ*is the electrical conductivity, ${B}_{0}$ is magnetic field flux density,

*ν*kinematic viscosity of the base fluid,

*α*is the thermal diffusivity of the porous medium, ${D}_{B}$ is the Brownian diffusion coefficient, ${D}_{T}$ is thermophoresis diffusion coefficient, ${D}_{CT}$ and ${D}_{TC}$ are the Soret and Dufour diffusivities, ${D}_{S}$ is the solutal diffusivity,

*T*is the fluid temperature,

*C*is the solutal concentration, $\stackrel{\u02c6}{\varphi}$ is the nanoparticle volume fraction, ${(\rho c)}_{f}$ and ${(\rho c)}_{p}$ are the heat capacity of the fluid and the effective heat capacity of the nanoparticle material respectively,

*τ*is a parameter defined by ${(\rho c)}_{f}/{(\rho c)}_{P}$. Using the similarity variables

*η*:

*M*, the Prandtl number

*Pr*, the Lewis number

*Le*, the Brownian motion parameter

*Nb*, the thermophoresis parameter

*Nt*, the nanofluid Lewis number

*Ln*, the modified Dufour parameter

*Nd*and the Dufour-solutal Lewis number

*Ld*. These parameters are defined as

*Nur*, the Sherwood number $\stackrel{\u02c6}{\mathit{Sh}}$ and the reduced Sherwood

*Shr*defined as

## 3 Method of solution

*L*is a scaling parameter used to invoke the boundary condition at infinity. This is achieved by using the mapping

*N*is the number of collocation points used. The functions ${F}_{i}$, ${\mathrm{\Theta}}_{i}$, ${\tilde{S}}_{i}$ and ${\mathrm{\Phi}}_{i}$ for $i\ge 1$ are approximated at the collocation points as follows:

*k*th Chebyshev polynomial given by

*r*is the order of differentiation and $\mathbf{D}=\frac{2}{L}\mathcal{D}$ with $\mathcal{D}$ being the Chebyshev spectral differentiation matrix (see, for example, [31–33]), whose entries are defined as

*T*stands for transpose, ${\mathbf{a}}_{k,i-1}$ ($k=1,\dots ,3$), ${\mathbf{b}}_{k,i-1}$ ($k=1,\dots ,4$), ${\mathbf{c}}_{k,i-1}$ ($k=1,\dots ,3$), ${\mathbf{d}}_{k,i-1}$ ($k=1,\dots ,3$) and ${\mathbf{r}}_{k,i-1}$ ($k=1,\dots ,4$) are diagonal matrices of order $(N+1)\times (N+1)$,

**I**is an identity matrix of order $(N+1)\times (N+1)$ and $[\mathbf{0}]$ is a zero matrix of order $(N+1)\times (N+1)$. The solution is obtained as

## 4 Results and discussion

*M*. As is now well known, the velocity decreases with increases in the magnetic field parameter due to an increase in the Lorentz drag force that opposes the fluid motion.

**Comparison of results for the reduced Nusselt number**
$\mathbf{-}\mathit{\theta}\mathbf{(}\mathbf{0}\mathbf{)}$
**with**
$\mathit{M}\mathbf{=}\mathbf{0}$
**,**
$\mathit{Pr}\mathbf{=}\mathbf{10}$
**,**
$\mathit{Le}\mathbf{=}\mathbf{10}$

Nb | Nt | −θ(0) | ||||
---|---|---|---|---|---|---|

Khan and Pop [15] | Present results | |||||

Ord 2 | Ord 4 | Ord 5 | Ord 6 | |||

0.1 | 0.1 | 0.9524 | 0.954110803008 | 0.952376830835 | 0.952376830835 | 0.952376830835 |

0.2 | 0.6932 | 0.696282777163 | 0.693174335745 | 0.693174335745 | 0.693174335745 | |

0.3 | 0.5201 | 0.523772737719 | 0.520079246363 | 0.520079246361 | 0.520079246361 | |

0.4 | 0.4026 | 0.406474865249 | 0.402579651548 | 0.402579651503 | 0.402579651503 | |

0.5 | 0.3211 | 0.325192006813 | 0.321057339674 | 0.321057339175 | 0.321057339175 | |

0.2 | 0.1 | 0.5056 | 0.507610155261 | 0.505578818179 | 0.505578818179 | 0.505578818179 |

0.2 | 0.3654 | 0.367853633248 | 0.365368345283 | 0.365368345283 | 0.365368345283 | |

0.3 | 0.2731 | 0.275387707176 | 0.273079280934 | 0.273079280931 | 0.273079280931 | |

0.4 | 0.2110 | 0.213054455507 | 0.210961536564 | 0.210961536512 | 0.210961536512 | |

0.5 | 0.1681 | 0.170080726148 | 0.168004798568 | 0.168004798105 | 0.168004798105 | |

0.3 | 0.1 | 0.2522 | 0.253142109948 | 0.252145911886 | 0.252145911886 | 0.252145911886 |

0.2 | 0.1816 | 0.182448890243 | 0.181611610633 | 0.181611610633 | 0.181611610633 | |

0.3 | 0.1355 | 0.136247585715 | 0.135548634738 | 0.135548634736 | 0.135548634736 | |

0.4 | 0.1046 | 0.105143130395 | 0.104494777320 | 0.104494777289 | 0.104494777289 | |

0.5 | 0.0833 | 0.083748729977 | 0.083300228592 | 0.083300228332 | 0.083300228332 | |

0.4 | 0.1 | 0.1194 | 0.119563178994 | 0.119374160613 | 0.119374160613 | 0.119374160613 |

0.2 | 0.0859 | 0.086351287057 | 0.085925168149 | 0.085925168149 | 0.085925168149 | |

0.3 | 0.0641 | 0.064969735925 | 0.064079763378 | 0.064079763377 | 0.064079763377 | |

0.4 | 0.0495 | 0.050308680850 | 0.049312783009 | 0.049312782995 | 0.049312782995 | |

0.5 | 0.0394 | 0.040068826105 | 0.039480432439 | 0.039480432335 | 0.039480432335 | |

0.5 | 0.1 | 0.0543 | 0.055006822209 | 0.054252883744 | 0.054252883744 | 0.054252883744 |

0.2 | 0.0390 | 0.041220738923 | 0.039039843265 | 0.039039843265 | 0.039039843265 | |

0.3 | 0.0291 | 0.032448207734 | 0.029136702982 | 0.029136702982 | 0.029136702982 | |

0.4 | 0.0225 | 0.025991804960 | 0.022499022345 | 0.022499022340 | 0.022499022340 | |

0.5 | 0.0179 | 0.020636326976 | 0.017899977204 | 0.017899977138 | 0.017899977138 |

**Comparison of results for the reduced Sherwood number**
$\mathbf{-}\mathit{\varphi}\mathbf{(}\mathbf{0}\mathbf{)}$
**with**
$\mathit{M}\mathbf{=}\mathbf{0}$
**,**
$\mathit{Pr}\mathbf{=}\mathbf{10}$
**,**
$\mathit{Le}\mathbf{=}\mathbf{10}$

Nb | Nt | −ϕ(0) | ||||
---|---|---|---|---|---|---|

Khan and Pop [15] | Present results | |||||

Ord 2 | Ord 4 | Ord 5 | Ord 6 | |||

0.1 | 0.1 | 2.1294 | 2.127980595220 | 2.129393826738 | 2.129393826738 | 2.129393826738 |

0.2 | 2.2740 | 2.269600795082 | 2.274021155237 | 2.274021155237 | 2.274021155237 | |

0.3 | 2.5286 | 2.522442790300 | 2.528634341968 | 2.528634341973 | 2.528634341973 | |

0.4 | 2.7952 | 2.789547614977 | 2.795197381386 | 2.795197381518 | 2.795197381518 | |

0.5 | 3.0351 | 3.031692110921 | 3.035086541257 | 3.035086542806 | 3.035086542806 | |

0.2 | 0.1 | 2.3819 | 2.381135534775 | 2.381870765082 | 2.381870765082 | 2.381870765082 |

0.2 | 2.5152 | 2.513872542870 | 2.515221791508 | 2.515221791508 | 2.515221791508 | |

0.3 | 2.6555 | 2.654621334344 | 2.655461783297 | 2.655461783300 | 2.655461783300 | |

0.4 | 2.7818 | 2.782448136707 | 2.781787213285 | 2.781787213347 | 2.781787213347 | |

0.5 | 2.8883 | 2.891077315907 | 2.888289878800 | 2.888289879328 | 2.888289879328 | |

0.3 | 0.1 | 2.4100 | 2.409868561539 | 2.410018897249 | 2.410018897249 | 2.410018897249 |

0.2 | 2.5150 | 2.515064990923 | 2.514994504216 | 2.514994504216 | 2.514994504216 | |

0.3 | 2.6088 | 2.609550527921 | 2.608824244439 | 2.608824244440 | 2.608824244440 | |

0.4 | 2.6876 | 2.689475214512 | 2.687604301826 | 2.687604301841 | 2.687604301841 | |

0.5 | 2.7519 | 2.755453212842 | 2.751842541500 | 2.751842541544 | 2.751842541544 | |

0.4 | 0.1 | 2.3997 | 2.399691610597 | 2.399650250624 | 2.399650250624 | 2.399650250624 |

0.2 | 2.4807 | 2.480840530130 | 2.480738445269 | 2.480738445269 | 2.480738445269 | |

0.3 | 2.5486 | 2.548758207066 | 2.548611975329 | 2.548611975329 | 2.548611975329 | |

0.4 | 2.6038 | 2.604477947716 | 2.603832566300 | 2.603832566297 | 2.603832566297 | |

0.5 | 2.6483 | 2.650218941812 | 2.648243871234 | 2.648243871122 | 2.648243871122 | |

0.5 | 0.1 | 2.3836 | 2.383468564586 | 2.383571426509 | 2.383571426509 | 2.383571426509 |

0.2 | 2.4468 | 2.446168708773 | 2.446806984545 | 2.446806984545 | 2.446806984545 | |

0.3 | 2.4984 | 2.497045285759 | 2.498378497565 | 2.498378497565 | 2.498378497565 | |

0.4 | 2.5399 | 2.538409035362 | 2.539849811783 | 2.539849811777 | 2.539849811777 | |

0.5 | 2.5731 | 2.572599764241 | 2.573109330795 | 2.573109330658 | 2.573109330658 |

*Nt*increases, the temperature within the boundary layer increases. The fast flow from the stretching sheet carries with it nanoparticles leading to an increase in the mass volume fraction boundary layer thickness.

*Le*, and the Dufour-solutal Lewis number

*Ld*on the species concentration in the boundary layer. The concentration profiles significantly contract as the Lewis number increases. The effect of the random motion of the nanoparticles suspended in the fluid on the temperature and nanoparticle volume fraction is shown in Figures 4(a) and 4(b). As expected, the increased Brownian motion of the nanoparticles carries with it heat and the thickness of the thermal boundary layer increases. The Brownian motion of the nanoparticles increases thermal transport which is an important mechanism for the enhancement of thermal conductivity of nanofluids. However, we note that increasing the Brownian motion parameter leads to a clustering of the nanoparticles near the stretching sheet. An increase in the Brownian motion of the nanoparticles leads to a decrease in the mass volume fraction profiles.

*Pr*and mass volume fraction profile for several values of the modified Dufour number

*Nd*. The temperature profiles decrease as the Prandtl number increases since, for high Prandtl numbers, the flow is governed by momentum and viscous diffusion rather than thermal diffusion. On the other hand, the thickness of the mass volume fraction boundary layer increases with an increase in

*Nd*.

*Nt*, the Lewis number

*Le*, the magnetic field parameter

*M*, the Prandtl number

*Pr*and the modified Dufour number

*Nd*on the wall heat and mass fraction transfer rates. It can be seen that the thermal boundary layer thickness increases when the thermophoresis parameter

*Nt*increases, thus decreasing the reduced Nusselt number. However, increasing the Lewis number

*Le*leads to a decrease in the reduced Nusselt number. On the other hand, the results show that the reduced Nusselt number increases with increasing Prandtl numbers. Increasing both the magnetic field parameter

*M*and the modified Dufour parameter

*Nd*leads to an increase in the thermal boundary layer thickness, thus reducing the Nusselt number.

*Ld*and the nanofluid Lewis number

*Ln*on the reduced Nusselt number

*Nur*as the Brownian motion parameter

*Nb*increases. We note a decrease in the reduced Nusselt number when

*Ln*increases, and an increase in the reduced Nusselt number when

*Ld*increases.

*Ld*for different values of the parameters

*Nt*,

*Nb*and

*Le*. We observe that $-{\theta}^{\prime}(0)$ increases in the absence of the Brownian motion and the thermophoresis parameter while $-{\theta}^{\prime}(0)$ decreases in the presence of Brownian motion and thermophoresis parameters. An increase in $-{S}^{\prime}(0)$ is observed in the presence of both the Brownian motion and the thermophoresis parameter. Figures 9(a) and 9(b) show the effect of increasing

*Nt*and

*Nb*respectively on the reduced Sherwood number $-\varphi (0)$.

## 5 Conclusions

A numerical study of the magneto-nanofluid boundary layer flow over a stretching sheet was carried out. We determined the effects of various parameters on the fluid properties as well as on the heat, and the regular and nano mass transfer rates. We have shown that increasing the magnetic field parameter *M* tends to retard the fluid flow within the boundary layer. The effects of the Prandtl number, the Lewis number, the Brownian motion parameter, the thermophoresis parameter, the nanofluid Lewis number, the modified Dufour parameter and the Dufour-solutal Lewis number on the heat, regular and nano mass transfer coefficients and fluid flow characteristics have been studied. We have shown *inter alia* that:

– the thermal boundary layer thickness increases with the thermophoresis parameter;

– increasing the Lewis number reduces the heat transfer coefficient;

– the heat transfer coefficient increases in the absence of the Brownian motion and the thermophoresis parameter and decreases in the presence of Brownian motion and thermophoresis parameters.

## Declarations

### Acknowledgements

The authors wish to thank the University of KwaZulu-Natal for financial support.

## Authors’ Affiliations

## References

- Choi SUS:
**Enhancing thermal conductivity of fluid with nanoparticles.23.**In*Developments and Applications of Non-Newtonian Flows*. FED, New York; 1995:99–105.Google Scholar - Kakac S, Pramuanjaroenkij A:
**Review of convective heat transfer enhancement with nanofluid.***Int. J. Heat Mass Transf.*2009, 52: 3187–3196. 10.1016/j.ijheatmasstransfer.2009.02.006View ArticleGoogle Scholar - Choi SUS, Zhang ZG, Yu W, Lockwood FE, Grulke EA:
**Anomalously thermal conductivity enhancement in nanotube suspensions.***Appl. Phys. Lett.*2001, 79: 2252–2254. 10.1063/1.1408272View ArticleGoogle Scholar - Masuda H, Ebata A, Teramae K, Hishinuma N:
**Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles.***Netsu Bussei*1993, 7: 227–233. 10.2963/jjtp.7.227View ArticleGoogle Scholar - Eapen J, Rusconi R, Piazza R, Yip S:
**The classical nature of thermal conduction in nanofluids.***J. Heat Transf.*2010., 132: Article ID 102402Google Scholar - Fan J, Wang L:
**Effective thermal conductivity of nanofluids: the effects of microstructure.***J. Phys. D, Appl. Phys.*2010., 43: Article ID 165501Google Scholar - Nield DA, Kuznetsov AV:
**The Cheng-Minkowycz problem for natural convective boundary-layer flow over a porous medium saturated by a nanofluid.***Int. J. Heat Mass Transf.*2010, 52: 5792–5795.View ArticleGoogle Scholar - Nield DA, Kuznetsov AV:
**Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model.***Transp. Porous Media*2010, 81: 409–422. 10.1007/s11242-009-9413-2MathSciNetView ArticleGoogle Scholar - Aziz A, Khan WA, Pop I:
**Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms.***Int. J. Therm. Sci.*2012, 56: 48–57.View ArticleGoogle Scholar - Cheng C-Y:
**Free convection boundary layer flow over a horizontal cylinder of elliptic cross section in porus media saturated by a nanofluid.***Int. Commun. Heat Mass Transf.*2012, 39: 931–936. 10.1016/j.icheatmasstransfer.2012.05.014View ArticleGoogle Scholar - Chamkha A, Gorla RSR, Ghodeswar K:
**Non-similar solution for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid.***Transp. Porous Media*2011, 86: 13–22. 10.1007/s11242-010-9601-0MathSciNetView ArticleGoogle Scholar - Altan T, Oh S, Gegel H:
*Metal Forming Fundamentals and Applications*. Am. Soc. Metals, Metals Park; 1979.Google Scholar - Fisher EG:
*Extrusion of Plastics*. Wiley, New York; 1976.Google Scholar - Tidmore Z, Klein I
**Polymer Science and Engineering Series.**In*Engineering Principles of Plasticating Extrusion*. Van Norstrand, New York; 1970.Google Scholar - Khan WA, Pop I:
**Boundary layer flow of a nanofluid past a stretching sheet.***Int. J. Heat Mass Transf.*2010, 53: 2477–2483. 10.1016/j.ijheatmasstransfer.2010.01.032View 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 - Narayana M, Sibanda P:
**Laminar flow of a nanoliquid film over an unsteady stretching sheet.***Int. J. Heat Mass Transf.*2012, 55: 7552–7560. 10.1016/j.ijheatmasstransfer.2012.07.054View ArticleGoogle Scholar - Kameswaran PK, Narayana N, Sibanda P, Murthy PVSN:
**Hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects.***Int. J. Heat Mass Transf.*2012, 55: 7587–7595. 10.1016/j.ijheatmasstransfer.2012.07.065View ArticleGoogle Scholar - Shima PD, Philip J, Raj B:
**Magnetically controllable nanofluid with tunable thermal conductivity and viscosity.***Appl. Phys. Lett.*2009., 95: Article ID 133112Google Scholar - Ganguly R, Sen S, Puri IK:
**Heat transfer augmentation using a magnetic fluid under the influence of a line dipole.***J. Magn. Magn. Mater.*2004, 271: 63–73. 10.1016/j.jmmm.2003.09.015View ArticleGoogle Scholar - Bachok N, Ishak A, Pop I:
**Unsteady boundary-layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet.***Int. J. Heat Mass Transf.*2012, 55: 2102–2109. 10.1016/j.ijheatmasstransfer.2011.12.013View ArticleGoogle Scholar - Hamad MAA, Ferdows M:
**Similarity solutions to viscous flow and heat transfer of nanofluid over nonlinearly stretching sheet.***Appl. Math. Mech.*2012, 33: 923–930.MathSciNetView ArticleGoogle Scholar - Makukula ZG, Motsa SS, Sibanda P:
**On a new solution for the viscoelastic squeezing flow between two parallel plates.***J. Adv. Res. Appl. Math.*2010, 2: 31–38.MathSciNetView ArticleGoogle Scholar - Awad FG, Sibanda P, Motsa SS, Makinde OD:
**Convection from an inverted cone in a porous medium with cross-diffusion effects.***Comput. Math. Appl.*2011, 61: 1431–1441. 10.1016/j.camwa.2011.01.015MathSciNetView ArticleGoogle Scholar - Makukula ZG, Sibanda P, Motsa SS:
**A novel numerical technique for two-dimensional laminar flow between two moving porous walls.***Math. Probl. Eng.*2010., 2010: Article ID 528956. doi:10.1155/2010/528956Google Scholar - Makukula ZG, Motsa SS, Sibanda P:
**A novel numerical technique for two-dimensional laminar flow between two moving porous walls.***Math. Probl. Eng.*2010., 2010: Article ID 528956. doi:10.1155/2010/528956Google Scholar - Awad FG, Sibanda P, Narayana M, Motsa SS:
**Convection from a semi-finite plate in a fluid saturated porous medium with cross-diffusion and radiative heat transfer.***Int. J. Phys. Sci.*2011, 6: 4910–4923.Google Scholar - Motsa SS, Sibanda P, Shateyi S:
**On a new quasi-linearization method for systems of nonlinear boundary value problems.***Math. Methods Appl. Sci.*2011, 34: 1406–1413. 10.1002/mma.1449MathSciNetView ArticleGoogle Scholar - Khan WA, Aziz A:
**Double-diffusive natural convective boundary layer flow in a porous medium saturated with a nanofluid over a vertical plate: prescribed surface heat, solute and nanoparticle fluxes.***Int. J. Therm. Sci.*2011, 50: 2154–2160. 10.1016/j.ijthermalsci.2011.05.022View ArticleGoogle Scholar - Shateyi S, Motsa SS:
**Variable viscosity on magnetohydrodynamic fluid flow and heat transfer over an unsteady stretching surface with hall effect.***Bound. Value Probl.*2010., 2010: Article ID 257568. doi:10.1155/2010/257568Google Scholar - Canuto C, Hussaini MY, Quarteroni A, Zang TA:
*Spectral Methods in Fluid Dynamics*. Springer, Berlin; 1988.View ArticleGoogle Scholar - Don WS, Solomonoff A:
**Accuracy and speed in computing the Chebyshev collocation derivative.***SIAM J. Sci. Comput.*1995, 16: 1253–1268. 10.1137/0916073MathSciNetView ArticleGoogle Scholar - Trefethen LN:
*Spectral Methods in MATLAB*. SIAM, Philadelphia; 2000.View ArticleGoogle Scholar

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