 Research
 Open Access
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
 flow separation
 MHD
 nanofluid
 permeable wedge
 variable viscosity
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
References
 Choi, SUS, Eastman, JA: Enhancing thermal conductivity of fluids with nanoparticles. Mater. Sci. 231, 99105 (1995) Google Scholar
 Hussein, AM, Bakar, RA, Kadirgama, K: Study of forced convection nanofluid heat transfer in the automotive cooling system. Case Stud. Therm. Eng. 2, 5061 (2014) View ArticleGoogle Scholar
 Frank, M, Drikakis, D, Asproulis, N: Thermal conductivity of nanofluid in nanochannels. Microfluid. Nanofluid. 19(5), 10111017 (2015) View ArticleGoogle Scholar
 Serna, J: Heat and mass transfer mechanisms in nanofluids boundary layers. Int. J. Heat Mass Transf. 92, 173183 (2016) View ArticleGoogle Scholar
 Hayat, T, Khan, MI, Waqas, M, Alsaedi, A: Newtonian heating effect in nanofluid flow by a permeable cylinder. Results Phys. 7, 256262 (2017) View ArticleGoogle Scholar
 Hatami, M, Sheikholeslami, M, Hosseini, M, Ganji, DD: Analytical investigation of MHD nanofluid flow in nonparallel walls. J. Mol. Liq. 194(2), 251259 (2014) View ArticleGoogle Scholar
 Hayat, T, Imtiaz, M, Alsaedi, A: MHD 3D flow of nanofluid in presence of convective conditions. J. Mol. Liq. 212, 203208 (2015) View ArticleGoogle Scholar
 Mabood, F, Khan, WA: Analytical study for unsteady nanofluid MHD flow impinging on heated stretching sheet. J. Mol. Liq. 219, 216223 (2016) View ArticleGoogle Scholar
 Falkner, VM, Skan, SW: Some approximate solutions of the boundary layer equations. Philos. Mag. 12, 865896 (1931) View ArticleMATHGoogle Scholar
 Yih, KA: MHD forced convection flow adjacent to a nonisothermal wedge. Int. Commun. Heat Mass Transf. 26(6), 819827 (1999) View ArticleGoogle Scholar
 Martin, MJ, Boyd, ID: Falknerskan flow over a wedge with slip boundary conditions. J. Thermophys. Heat Transf. 24(2), 263270 (2010) View ArticleGoogle Scholar
 Sattar, MA: A local similarity transformation for the unsteady twodimensional hydrodynamic boundary layer equations of a flow past a wedge. Int. J. Appl. Math. Mech. 7(1), 1528 (2011) MATHGoogle Scholar
 Kandasamy, R, Muhaimin, I, Khamis, AB, Roslan, RB: Unsteady Hiemenz flow of Cunanofluid over a porous wedge in the presence of thermal stratification due to solar energy radiation: Lie group transformation. Int. J. Therm. Sci. 65, 196205 (2013) View ArticleGoogle Scholar
 Turkyilmazoglu, M: Slip flow and heat transfer over a specific wedge: an exactly solvable FalknerSkan equation. J. Eng. Math. 92(1), 7381 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Raju, CSK, Sandeep, NA: Comparative study on heat and mass transfer of the Blasius and FalknerSkan flow of a bioconvective Casson fluid past a wedge. Eur. Phys. J. Plus 131(11), 405 (2016) View ArticleGoogle Scholar
 Raju, CSK, Sandeep, N: Nonlinear radiative magnetohydrodynamic FalknerSkan flow of Casson fluid over a wedge. Alex. Eng. J. 55(3), 20452054 (2016) View ArticleGoogle Scholar
 Su, X, Zheng, L: Hall effect on MHD flow and heat transfer of nanofluids over a stretching wedge in the presence of velocity slip and Joule heating. Cent. Eur. J. Phys. 11(12), 16941703 (2013) Google Scholar
 Srinivasacharya, D, Mendu, U, Venumadhav, K: MHD boundary layer flow of a nanofluid past a wedge. Proc. Eng. 127, 10641070 (2015) View ArticleGoogle Scholar
 Khan, M, Azam, M, Munir, A: On unsteady FalknerSkan flow of MHD Carreau nanofluid past a static/moving wedge with convective surface condition. J. Mol. Liq. 230, 4858 (2017) View ArticleGoogle Scholar
 Hartree, DR: On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer. Proc. Camb. Philos. Soc. 33(2), 223239 (1937) View ArticleMATHGoogle Scholar
 Stewartson, K: Further solutions of the FalknerSkan equation. Math. Proc. Camb. Philos. Soc. 50(3), 454465 (1954) MathSciNetView ArticleMATHGoogle Scholar
 Steinheuer, J: Similar solutions for the laminar wall jet in a decelerating outer flow. AIAA J. 6(11), 21982200 (1968) View ArticleGoogle Scholar
 Cebeci, T, Keller, HB: Shooting and parallel shooting methods for solving the FalknerSkan boundarylayer equation. J. Comput. Phys. 7(2), 289300 (1971) View ArticleMATHGoogle Scholar
 Deka, RK, Basumatary, M: Effect of variable viscosity on flow past a porous wedge with suction or injection: new results. Afr. Math. 26(7), 12631279 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Schlichting, H, Gersten, K: BoundaryLayer Theory. McGrawHill, New York (1979) MATHGoogle Scholar
 Khamis, S, Makinde, DO, NkansahGyekye, Y: Unsteady flow of variable viscosity CuWater and Al_{2}O_{3}Water nanofluids in a porous pipe with buoyancy force. Int. J. Numer. Methods Heat Fluid Flow 25(7), 16381657 (2015) View ArticleMATHGoogle Scholar
 Makinde, OD, Iskander, T, Mabood, F, Khan, WA, Tshehla, MS: MHD CouettePoiseuille flow of variable viscosity nanofluids in a rotating permeable channel with Hall effects. J. Mol. Liq. 221, 778787 (2016) View ArticleGoogle Scholar
 Huda, AB, Akbar, NS, Beg, OA, Khan, MY: Dynamics of variableviscosity nanofluid flow with heat transfer in a flexible vertical tube under propagating waves. Results Phys. 7, 413425 (2017) View ArticleGoogle Scholar
 Brinkman, HC: The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 20(4), 571 (1952) View ArticleGoogle Scholar
 Sourtiji, E, GorjiBandpy, M, Ganji, DD, Hosseinizadeh, SF: Numerical analysis of mixed convection heat transfer of Al_{2}O_{3}Water nanofluid in a ventilated cavity considering different positions of the outlet port. Mindfulness 5(4), 381391 (2014) Google Scholar
 Garnett, JCM: Colours in metal glasses and in metallic films. Philos. Trans. R. Soc. Lond. 203, 385420 (1904) View ArticleMATHGoogle Scholar
 Soundalgekar, VM, Takhar, HS, Das, UN, Deka, RK, Sarmah, A: Effect of variable viscosity on boundary layer flow along a continuously moving plate with variable surface temperature. Heat Mass Transf. 40(5), 421424 (2004) View ArticleGoogle Scholar
 Watanabe, T: Thermal boundary layer over a wedge with uniform suction or injection in forced flow. Acta Mech. 83(3), 119126 (1990) View ArticleGoogle Scholar