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Dual solutions of a mixed convection flow near the stagnation point region over an exponentially stretching/shrinking sheet in nanofluids

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Abstract

The aim of this paper is to study the development of mixed convection flow near the stagnation point region over an exponentially stretching/shrinking sheet in nanofluids. The external flow, stretching velocity and wall temperature are assumed to vary as prescribed exponential functions. Using the local similarity method, it has been shown that dual solutions of velocity and temperature exist for certain values of suction/injection, mixed convection, nanoparticle volume fraction and stretching/shrinking parameters. The transformed non-linear ordinary differential equations along with the boundary conditions form a two point boundary value problem and are solved using Shooting method, by converting into an initial value problem. In this method, the system of equations is converted into a set of first order system which is solved by fourth-order Runge–Kutta method. Three different types of nanoparticles, namely copper (Cu), aluminum oxide (Al2O3) and titanium oxide (TiO2) are considered by using water-based fluid with Prandtl number Pr = 6.2. It is also found that the skin friction coefficient and the heat transfer rate at the surface are highest for Copper–water nanofluids as compared to Al2O3. The effect of the solid volume fraction parameter φ of the nanofluids on the heat transfer characteristics is also investigated. The results indicate that dual solutions exist only for shrinking sheet. The effects of various parameters on the velocity and temperature profiles are also presented here.

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Abbreviations

a,b :

Constants

ɛ :

Skin friction coefficient

C p :

Specific heat at constant pressure

f :

Dimensionless streamfunction

F :

Dimensionless velocity

g :

Acceleration due to gravity

Gr :

Grash of number

k :

Thermal conductivity

L :

Reference length

Nu :

Nusselt number

Pr:

Prandtl number, Pr = ν f /α f

q w :

Surface heat flux

Re:

Reynolds number, Re = u e L/v f

T:

Fluid temperature in the boundary layer

u, v :

Velocity components in the x- and y- directions, respectively

u e :

Velocity of the external flow

u w :

Velocity of the stretching surface

x, y :

Cartesian coordinates measured along the surface and normal to it, respectively

α :

Thermal diffusivity

β :

Volumetric coefficient of thermal expansion

ɛ :

Stretching/shrinking parameter, ɛ = b/a

η :

Similarity variable

θ :

Dimensionless temperature

λ :

Mixed convection parameter

μ :

Dynamic viscosity

ν :

Kinematic viscosity

ρ :

Fluid density

τ w :

Surface heat flux

φ :

Nanoparticle volume fraction

ψ :

Dimensional streamfunction

e :

Condition at the edge of the boundary layer

w :

Condition at the wall

∞:

Freestream condition

nf :

Nanofluid

f :

Fluid

s :

Solid

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Authors and Affiliations

  1. Department of Mathematics, Anna University, Chennai, 600 025, India

    S. V. Subhashini & R. Sumathi

  2. School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag – 3, Wits, Johannesburg, 2050, South Africa

    E. Momoniat

Authors
  1. S. V. Subhashini
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  2. R. Sumathi
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  3. E. Momoniat
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Correspondence to S. V. Subhashini.

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Subhashini, S.V., Sumathi, R. & Momoniat, E. Dual solutions of a mixed convection flow near the stagnation point region over an exponentially stretching/shrinking sheet in nanofluids. Meccanica 49, 2467–2478 (2014). https://doi.org/10.1007/s11012-014-0016-9

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