Dual solutions of a boundary layer problem for MHD nanofluids through a permeable wedge with variable viscosity
 Xiaoqin Xu^{1, 2} and
 Shumei Chen^{1}Email author
Received: 16 June 2017
Accepted: 15 September 2017
Published: 11 October 2017
Abstract
Considering the effect of variable viscosity and the phenomenon of flow separation, the MHD Cu/AgWater nanofluids through a permeable wedge are investigated. The governing equations of flow and energy are reduced by similarity transformations and then solved numerically by the shooting method. It is found that dual solutions exist for negative pressure gradient. Compared with the AgWater nanofluid, the flow separation occurs later for injection, while it occurs earlier for suction in the CuWater nanofluid. The outcomes also specify that suction and small variable viscosity parameters delay the separation for the two nanofluids.
Keywords
1 Introduction
Because of the low thermal conductivity, there is a limit to the heat transfer performance of the classical heat transfer fluids such as water, ethylene glycol, and engine oil. The thermal conductivity of metals, however, is extremely higher compared with the conventional heat transfer fluids. A nanofluid, which was first proposed by Choi and Eastman [1], is a fluid that is created by the distribution of solid particles with dimensions less than 100 nm in base fluid. Choi noticed that the addition of one percent of nanoparticles by volume to the usual fluids increases the thermal conductivity of the fluid up to approximately twice. Comprehensive literature on the applications of nanofluids can be found in papers [2–5]. Magnetic nanofluid is a magnetic colloidal suspension of carrier liquid and magnetic nanoparticles. The advantage of the magnetic nanofluid is that fluid flow and heat transfer can be controlled by an external source, which makes it applicable to modern metallurgical and metalworking processes such as electronic packing, thermal engineering, and aerospace. Therefore many researchers have been contributing to the study of magnetohydrodynamic (MHD) nanofluid flow [6–8].
On the other hand, the study of the flow field in a boundary adjacent to the wedge, an essential part in the area of fluid dynamics and heat transfer, is very important in many thermal engineering applications like geothermal systems, crude oil extraction, thermal insulation heat exchangers, the storage of nuclear waste, etc. Falkner and Skan [9] were the first to analyze the steady laminar flow over a wedge, and they proposed a wellknown FalknerSkan equation to describe the flow over a wedge, which has provided many fruitful sources of information about the behavior of incompressible boundary layers. Since then, many researchers have devoted themselves to investigating the same problem and gained lots of valuable results, see [10–16]. In the past decade, Su and Zheng [17] analyzed the Hall effect on MHD flow and heat transfer of nanofluids over a stretching wedge in the presence of velocity slip and Joule heating. Srinivasacharya et al. [18] investigated the steady laminar magnetohydrodynamic (MHD) flow, heat and mass transfer characteristics in a nanofluid over a wedge in the presence of a variable magnetic field. Khan et al. [19] presented the locally similar solutions for the unsteady twodimensional FalknerSkan flow of MHD Carreau nanofluid past a static/moving wedge in the presence of convective boundary condition. They found that an increment in the pressure gradient parameter depreciates the heat and mass transfer rate both for shear thinning and shear thickening fluids.
All the abovementioned literature about MHD nanofluids flowing past a wedge only considered the accelerating or constant flow case, with positive or constant pressure gradient. In both cases, there exists no separation point in the velocity profile. However, many early researchers [20–23] pointed out that for the decelerating flow case, with negative pressure gradient, two solutions occurred in the wellknown FalknerSkan equation. Hence separation may happen in the decelerating flow. As the results pointed out in the reference [24], the occurrence of flow separation has several undesirable effects, and it leads to an increase in the drag on a body immersed in the flow. In order to reduce the drag force, injection on the boundary layer flow has been introduced and proved to be an effective way [25].
Motivated by the above research, this paper aims to explore the flow and heat transfer of MHD nanofluid past a permeable wedge with suction or injection, considering the occurrence of flow separation and variable viscosity. The physical properties of the nanofluids may change significantly with temperature [26–28]. To more accurately depict the flow behavior and heat transfer, it is necessary to take the variation of viscosity with temperature into account. By means of similarity reductions, the nonlinear equations are solved numerically by the shooting method. Besides, the effects of the governing parameters on the separation point, dimensionless velocity, temperature, skin friction coefficient, and local Nusselt number are graphically presented and discussed in detail.
2 Formulation
According to the definition of \(\theta_{r}\), we obtain \(\mu_{f} = \mu_{\infty} /[1\theta(\eta)/ \theta_{r} ]\). Since the viscosity of liquids decreases with increasing temperature, \(\theta_{r}\) is negative for nanofluids. When \(\theta_{r} \rightarrow\infty\), \(\mu_{f} \rightarrow\mu_{\infty}\), i.e., the viscosity variation in the boundary layer is negligible.
3 Results and discussion
Comparison of \(\pmb{f^{\prime\prime} ( 0 ) }\) and \(\pmb{\theta'(0)}\) for various values of m with \(\pmb{P_{r} =0.73}\) and \(\pmb{f_{w} =0 }\) when \(\pmb{\theta_{r} \rightarrow\infty}\)
m  Watanabe [ 33 ]  Deka et al. [ 24 ]  Present  

\(\boldsymbol{f''(0)}\)  \(\boldsymbol{\theta'(0)}\)  \(\boldsymbol{f''(0)}\)  \(\boldsymbol{\theta'(0)}\)  \(\boldsymbol{f''(0)}\)  \(\boldsymbol{\theta'(0)}\)  
0  0.46960  0.42015  0.469601  0.420160  0.469590  0.420146 
0.0141  0.50461  0.42578  0.504615  0.425785  0.504607  0.425773 
0.0435  0.56898  0.43548  0.568978  0.435492  0.568970  0.435473 
0.0909  0.65498  0.44730  0.654979  0.447312  0.654968  0.447295 
0.1429  0.73200  0.45693  0.731999  0.456951  0.731987  0.456931 
0.2  0.80213  0.46503  0.802126  0.465051  0.802109  0.465026 
0.3333  0.92765  0.47814  0.927654  0.478158  0.927636  0.478131 
Thermophysical properties of nanofluids [ 8 ]
\(\boldsymbol{C_{p}}\) (J/kgK)  ρ (kg/m ^{ 3 } )  k (W/mK)  

Cu  385  8,933  400 
Ag  235  10,500  429 
Water  4,179  997.1  0.613 
Critical values \(\pmb{m_{C} }\) with representative values of \(\pmb{f_{w}}\) or \(\pmb{{\theta}_{r}}\) when \(\pmb{\phi=0.05}\) , \(\pmb{P _{r} =6.2}\) , \(\pmb{M=0}\)
\(\boldsymbol{{\theta}_{r}}\)  \(\boldsymbol{{f}_{w}}\)  \(\boldsymbol{{m}_{C}}\) (CuWater nanofluid)  \(\boldsymbol{{m}_{C}}\) (AgWater nanofluid) 

−2  −0.3  −0.0430  −0.0416 
−2  −0.2  −0.0601  −0.0591 
−2  0  −0.0955  −0.0955 
−2  0.5  −0.1875  −0.1904 
−2  1  −0.2835  −0.2889 
−1  −0.2  −0.0609  −0.0596 
−5  −0.2  −0.0590  −0.0581 
−10  −0.2  −0.0585  −0.0576 
−∞  −0.2  −0.0579  −0.0570 
−1  0.2  −0.1361  −0.1375 
−5  0.2  −0.1283  −0.1294 
−10  0.2  −0.1273  −0.1284 
−∞  0.2  −0.1263  −0.1273 
4 Conclusions

Dual solutions exist for negative pressure gradient (\(m<0\)) for the two nanofluids.

Suction and small variable viscosity parameter delay the flow separation for the two nanofluids.

Compared with the AgWater nanofluid, the flow separation occurs later for injection, while it occurs earlier for suction in the CuWater nanofluid.

Suction enhances heat transfer and skin friction, while the effect of injection is just the opposite for the two nanofluids.
Declarations
Acknowledgements
This work was supported by the Science and Technology Major Project of Fujian Province (grant number 2011HZ0061), Construction of Scientific and Technological Innovation Platform of Fujian Province (grant number 2011H2008), and Special Funds for the University Development from Central Finance of China in 2012 and 2016.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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