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Nanofluid flow by a permeable stretching/shrinking cylinder

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Abstract

The purpose of this paper is to theoretically investigate the steady nanofluid flow due to a permeable stretching/shrinking cylinder using the mathematical nanofluid model proposed by Buongiorno. The effects of the stretching/shrinking parameter as well as the suction and curvature parameters are studied. A similarity transformation is used to reduce the governing partial differential equations to a set of nonlinear ordinary (similarity) differential equations which are then solved numerically using the function bvp4c from Matlab for different values of the governing parameters. It is found that the solution is unique for stretching case; however, multiple (dual) solutions exist for the shrinking case. A closed form analytical solution is also obtained when the curvature parameter α = 0.  For α small (α ≪ 1) dual solutions exist for both stretching and shrinking cylinder, while for α large (α ≫ 1) dual solutions are found only for the shrinking cylinder(λ < 0).

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Abbreviations

a :

radius of the cylinder

C :

nanoparticle volume fraction

C ∞ :

ambient nanoparticle volume fraction

c p :

specific heat due to constant pressure

D B :

Brownian diffusion coefficient

D T :

thermophoretic diffusion coefficient

k :

thermal conductivity of the nanofluid

l :

characteristic length

Nb :

Brownian motion parameter

Nt :

thermophoresis parameter

Nu x :

local Nusselt number

Pr :

Prandtl number

q w :

heat flux from the surface of the cylinder

Re x :

local Reynolds number

S :

suction/injection parameter

Sc :

Schmidt number

T :

temperature of the nanofluid

T w :

temperature at the surface of the cylinder

T ∞ :

temperature of the ambient fluid

u :

velocity along the x axis

v :

velocity along the r axis

uw(x):

stretching/shrinking velocity

u 0 :

characteristic velocity

(x, r):

axial coordinate x and radial coordinate r normal to the x

α :

curvature parameter

λ :

constant shrinking/stretching parameter

ρ :

density of the nanofluid

(ρcp)f :

heat capacity of the base fluid

(ρcp)p :

effective heat capacity of the nanoparticles

υ :

kinematic coefficient of viscosity of nanofluid

η :

similarity variable

θ :

dimensionless temperature

φ :

dimensionless rescaled nanoparticle volume fraction

τ w :

skin frictions or shear stresses in the x - direction of the stretching/shrinking cylinder

γ :

nanoparticle heat capacity ratio

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Acknowledgements

The work of N.C. Roşca, A.V. Roşca and I. Pop has been supported from the grant PN-III-P4-ID-PCE-2016-0036, UEFISCDI, Romania. The authors wish also to express their thanks to the very competent Reviewers for the valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics and Computer Science, Babeș-Bolyai University, 400084, Cluj-Napoca, Romania

    N. C. Roşca & I. Pop

  2. Department of Statistics-Forecasts-Mathematics, Faculty of Economics and Business Administration, Babeș-Bolyai University, 400084, Cluj-Napoca, Romania

    A. V. Roşca

  3. Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    J. H. Merkin

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  1. N. C. Roşca
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Correspondence to I. Pop.

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Roşca, N.C., Roşca, A.V., Pop, I. et al. Nanofluid flow by a permeable stretching/shrinking cylinder. Heat Mass Transfer 56, 547–557 (2020). https://doi.org/10.1007/s00231-019-02730-x

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