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Magnetohydrodynamics flow of a nanofluid driven by a stretching/shrinking sheet with suction
 U. S. Mahabaleshwar^{1}Email author,
 P. N. Vinay Kumar^{2} and
 Mikhail Sheremet^{3}
 Received: 9 February 2016
 Accepted: 20 October 2016
 Published: 2 November 2016
Abstract
The present paper investigates the effect of a mathematical model describing the aforementioned process in which the ambient nanofluid in the presence of suction/injection and magnetic field are taken into consideration. The flow is induced by an infinite elastic sheet which is stretched along its own plane. The stretching/shrinking of the sheet is assumed to be proportional to the distance from the slit. The governing equations are reduced to a nonlinear ordinary differential equation by means of similarity transformation. The consequential nonlinear equation is solved analytically. Consequences show that the flow field can be divided into a nearfield region and a farfield region. Suction on the surface plays an important role in the flow development in the nearfield whereas the farfield is responsible mainly by stretching. The electromagnetic effect plays exactly the same role as the MHD, which is to reduce the horizontal flow resulting from stretching. It is shown that the behavior of the fluid flow changes with the change of the nanoparticles type. The present study throws light on the analytical solution of a class of laminar boundary layer equations arising in the stretching/shrinking sheet problem.
Keywords
 Nonlinear differential equation
 Nanofluid
 MHD stretching/shrinking sheet
 Suction/injection
Background
Dynamics of fluid flow over a linear stretching/shrinking sheet plays very significant role in many manufacturing applications. The thin polymer sheet constitutes a continuously moving solid surface with a nonuniform surface velocity through an or else quiescent fluid. The cooling fluids in past times was selected to be the in large quantities available water, but this has the disadvantage of speedily quenching the heat leading to rapid solidification of the stretching sheet (see Andersson 1995, 2002, Fisher 1976, Siddheshwar and Mahabaleshwar 2005). From the standpoint of desirable properties of the final product water does not seem to be the ideal cooling fluid.
The word of nanofluid refers to a solid–liquid mixture with a continuous phase which is a nanometer sized nanoparticle dispersed in conventional base liquids. Nanofluids are basefluids containing suspended nanoparticles. These nanoparticles are typically mad of metals, oxides, or carbon nanotubes. There are a few wellknown correlations for predicting the thermal and physical properties of nanofluids which are often cited by researchers to calculate the convective heat transfer behaviors of the nanofluids. The word “nanofluid” coined by Choi (1995) describes a liquid suspension containing ultrafine particles (diameter less than 50 nm) (See Choi et al. 2001, Keblinski et al. 2005). The first providing for this ground of Sakiadis (1961a, b, c) he concentrated on the induced affected by the uniform motion of a continuous solid surface taking into account the laminar boundary layer approximation.
An exact analytical solution of the equation for a elastic sheet where the surface stretching velocity was proportional to the distance from the slot was given in Crane (1970). In this presentation, we will perform an analysis of a mathematical model describing the aforementioned process in which the ambient nanofluid in the presence of mass transfer is taken into consideration.
Solution of mathematical formulation
Thermophysical properties of the base fluid (water) and nanoparticles
Nanoliquid physical properties  Liquid phase (water)  Copper  Alumina  Titania 

C _{ p } (J/kg K)  4179  385  765  686.2 
\(\rho\) (kg/m^{3})  997.1  8933  3970  42.50 
k (W/m K)  0.613  400  40  8.9538 
The bases for present analysis laminar boundary layer equations for an incompressible nanofluid.
Laminar boundary layer flows induced by a continuous surface stretching with velocity \(u_{w} (x)\), \(v_{c}\) is the mass flux velocity with \(v_{c} < 0\) for suction, \(v_{c} > 0\) for injection and \(v_{c} = 0\) is the case when the surface is impermeable.
So in the case of a shrinking sheet if the inequality (13a) is not satisfied it is impossible to find \(\beta\) and the analytical solution of the required form does not exist.
Suppose now that the discriminant is ≥0 and distinguish some cases.
Case (i): If V _{ c } = 0 and \(\Gamma_{2} \lambda + Q > 0,\) then \(\beta = \Gamma_{1} \sqrt {\Gamma_{2} \lambda + Q}\), if V _{ c } = 0 and \(\Gamma_{2} \lambda + Q \le 0\) it is not possible to find \(\beta\).
Therefore in this case the problem (9) and (10) admits two analytical solutions. If \(V_{c} < 0\), the two roots are negative and so the problem does not admit a solution in closed form. If \(\Gamma_{2} \lambda + Q = 0,\) and \(V_{c} > 0\), then \(\beta = \Gamma_{1} \Gamma_{2} V_{c}\); if \(\Gamma_{2} \lambda + Q = 0\) and \(V_{c} < 0\), then it is impossible to find \(\beta\).
Finally if \(\lambda < 0,\) and the discriminant is equal to 0, then in the case \(V_{c} > 0\) we have \(\beta = \frac{{\Gamma_{1} \Gamma_{2} V_{c} }}{2}\), while in the case \(V_{c} < 0\) it is impossible to find \(\beta\). When it is impossible to find \(\beta\) one can try to solve the problem numerically. More over the possibility of two values of \(\beta\) is not surprising because in the studies on the flows of the classical fluids with a stretching/shrinking sheet dual solutions have been found in the literature.
Skin friction
Results and discussion
The present article is the generalization of the classical work of Crane (1970) flow and nanofluid driven by stretching/shrinking sheet with external magnetic field and suction. The classical Crane solution of the linear stretching sheet is extensive to include nanofluid, shrinking and suction/injection of weakly electrically conducting Newtonian fluids and also three types nanofluids, namely Copper (Cu), alumina (Al_{2}O_{3}) and Titania (TiO_{2}) in water as the base fluid. The basic boundary layer equation of momentum field is mapped into highly nonlinear ordinary differential equations via similarity transformations. Similarity solution is obtained for the velocity distribution. The velocities are decreasing function of \(\eta\) as it is an exponential function with negative argument. It is apparent from Eq. (11), that is \(\beta\), which is function of the suction/injection parameter \(V_{c}\), with \(V_{c} < 0\) for suction, \(V_{c} > 0\) for injection and \(V_{c} = 0\) is the case when the surface is impermeable, stretching/shrinking parameter \(\lambda\), \(\lambda > 0\) for stretching sheet, \(\lambda < 0\) for a shrinking and \(\lambda = 0\) for fixed surface and Chandrasekhar number Q, shows to the slope of above exponentially decreasing velocity profiles.
Concluding remarks
 1.
The axial velocity and transverse velocity, is a decreasing function of \(\eta\) as it is an exponential function with negative argument.
 2.
Increasing values of the Q results in pulling down of velocity profiles.
 3.
Velocity profiles decrease with an increase in Q (Ferraro and Plumpton 1961) and (Borrelli et al. 2015).
 4.
The velocity components transverse velocity \(f\) and axial \(f_{\eta }\) are reveals for different values of the Q, the velocity decreases with increases in the Q due to an increase in the Lorentz drag force that opposes the fluid motion.
 5.
The increase of Q leads to the increase of skin friction parameter in all the cases of suction/injection.
 6.
The classical Crane (1970) flow is recovered from Eq. (13) for \(V_{c} = Q = \phi = 0\) and \(\lambda = \Gamma_{1} = \Gamma_{2} = 1\).
 7.
The classical Pavlov (1974) flow is recovered from Eq. (13) for \(V_{c} = \phi = 0\) and \(\lambda = \Gamma_{1} = \Gamma_{2} = 1\).
 8.
The Gupta and Gupta (1977) flow is recovered from Eq. (13) for \(\phi = 0\) and \(\lambda = \Gamma_{1} = \Gamma_{2} = 1\).
 9.
The skin friction is lower for stretching and higher for shrinking sheets.
 10.
The effect of increasing the \(V_{c}\) and the Q is to increase the velocity and decrease the laminar boundary layer thickness in shrinking case (Borrelli et al. 2012, 2013a, b).
 11.
The heat transfer at the surface of the sheet increases with the increasing suction/injection \(V_{c}\) and the nanoparticle solid volume fraction \(\phi\).
List of symbols
 C _{ f } :

skin friction coefficient
 B _{0} :

magnetic field (w m^{−2)}
 f :

dimensionless stream function
 J :

current density
 q _{ i } and q _{ j } :

velocity components
 \(\text{Re}_{x}\) :

local Reynolds number \(\left( {\text{Re}_{x} = \frac{{xu_{w} }}{{\nu_{f} }}} \right)\)
 u :

axial velocity part along xaxis (m s^{−1})
 v :

transverse velocity part along yaxis (m s^{−1})
 \(V_{c}\) :

constant suction/injection parameter \(V_{C} = \frac{{v_{c} }}{{\sqrt {\alpha \,v_{f} } }}\)
 x :

horizontal coordinate (m)
 y :

vertical coordinate (m)
Greek symbols
 α :

constant in the sheet coefficient (s^{−1}), \((\alpha > 0)\)
 \(\lambda\) :

constant, represents stretching/shrinking parameter
 \(\eta \,\) :

similarity variable = \(\left( {\sqrt {\frac{\alpha }{{\nu_{f} }}} } \right)y\)
 \(\mu_{nf}\) :

viscosity of the nanofluid (kg m^{−1} s^{−1})
 \(\nu_{f}\) :

kinematic viscosity of the fluid (m^{2} s^{−1})
 \(\rho_{nf}\) :

density of the nanofluid (kg m^{−3})
 \(\rho_{f}\) :

density of the fluid (kg m^{−3})
 \(\rho_{s}\) :

density of the nanosolid particles
 \(\sigma\) :

electrical conductivity of fluid (mho m^{−1})
 \(\tau_{w}\) :

wall shearing stress (m^{2} s^{−1})
 \(\phi\) :

nanoparticle volume fraction
 \(\psi\) :

physical stream function (m^{2} s^{−1})
Subscripts/superscripts
 0:

origin
 f :

fluid
 s:

solid
 w :

wall condition
 \(\infty\) :

for from the sheet
 \(f_{\eta }\) :

first derivative w.r. t. \(\eta\)
 \(f_{\eta \eta }\) :

second derivative w.r. t. \(\eta\)
 \(f_{\eta \eta \eta }\) :

third derivative w.r. t. \(\eta\)
Declarations
Authors’ contributions
The authors have contributed equally in conception and design of the study as well as analyzing and drafting of the manuscript. All authors read and approved the final manuscript.
Acknowledgements
Author Dr. U. S. Mahabaleshwar (USM) thanks and wishes to Bharat Ratna Prof. Dr. C. N. R. Rao F.R.S, Honorable President VGST, Department of IT, BT Science and Technology, Bangalore, INDIA, for supporting this work under Seed Money to Young Scientists for Research (# VGST/SMYSR/GRD304/201314) and USM would like to acknowledge the receipt of Asia Bridge Fellowship, to thank to Professor Dr. Akira Nakayama, Shizuoka University, Japan for his hospitality.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis 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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