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RETRACTED ARTICLE: Variable viscosity on unsteady dissipative Carreau fluid over a truncated cone filled with titanium alloy nanoparticles

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This article was retracted on 20 May 2017

This article has been updated

Abstract

In this study, we proposed a theoretical investigation on the temperature-dependent viscosity effect on magnetohydrodynamic dissipative nanofluid over a truncated cone with heat source/sink. The involving set of nonlinear partial differential equations is transforming to set of nonlinear ordinary differential equations by using self-similarity solutions. The transformed governing equations are solved numerically using Runge–Kutta-based Newton’s technique. The effects of various dimensionless parameters on the skin friction coefficient and the local Nusselt number profiles are discussed and presented with the support of graphs. We also obtained the validation of the current solutions with existing solution under some special cases. The water-based titanium alloy has a lesser friction factor coefficient as compared with kerosene-based titanium alloy, whereas the rate of heat transfer is higher in water-based titanium alloy compared with kerosene-based titanium alloy. From this we can highlight that depending on the industrial needs cooling/heating chooses the water- or kerosene-based titanium alloys.

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Change history

  • 20 May 2017

    An erratum to this article has been published.

Abbreviations

u, v, w :

Velocity components in x, y and z directions, respectively (m/s)

x :

Distance along the surface (m)

y :

Distance normal to the surface (m)

\(c_\mathrm{p}\) :

Specific heat capacity at constant pressure (J/kg K)

f, g :

Dimensionless velocities

T :

Temperature of the fluid (K)

g :

Acceleration due to gravity (m/s\(^{2}\))

\(k_\mathrm{f} \) :

Thermal conductivity (W/mK)

\(\alpha \) :

Diffusion coefficient (m\(^{2}\)/s)

P :

Pressure (pa)

\(\eta \) :

Similarity variable

\(\sigma \) :

Electrical conductivity (Siemens)

\(\sigma ^{*}\) :

Stefan–Boltzmann constant (W m/K\(^{4}\))

\(k^{*}\) :

Mean absorption coefficient

\(\beta _\mathrm{T} \) :

Volumetric thermal expansion (K\(^{-1}\))

\(\theta \) :

Dimensionless temperature (K)

\(\rho _\mathrm{f} \) :

Density (kg/m\(^{3}\))

\(\nu _\mathrm{f} \) :

Kinematic viscosity (m\(^{2}\)/s)

\(\mu _\mathrm{f} \) :

Dynamic viscosity (N s/m\(^{2}\))

\(\phi \) :

Nanoparticle volume fraction

\((\rho c_\mathrm{p} )_\mathrm{f} \) :

Effective heat capacity of the fluid (kg/m\(^{3}\)K)

\((\rho c_\mathrm{p} )_\mathrm{p} \) :

Effective heat capacity of the nanoparticle medium (kg/m\(^{3}\)K)

\(Q_\mathrm{H} \) :

Heat source/sink parameter

\(Cf_x \) :

Skin friction coefficient in x direction

\(Cf_y \) :

Skin friction coefficient in y direction

\(Nu_x \) :

Local Nusselt number

\({Re}_x \) :

Local Reynolds number

\({ Pr}\) :

Prandtl number

Ec :

Eckert number

E :

Viscous variation parameter

We :

Weissenberg number

n :

Power-law index parameter

\(\gamma \) :

Half-angle parameter

\(\lambda \) :

Buoyancy parameter

\(\alpha _1 \) :

Ratio of angles

M :

Magnetic field parameter

\(Gr_x \) :

Grashof number

f:

Fluid

w:

Condition at the wall

\(\infty \) :

Condition at the free stream

nf:

Nanofluid

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Author information

Authors and Affiliations

  1. Department of Mathematics, VIT University, Vellore, 632014, India

    C. S. K. Raju

  2. Department of Mathematics, Sri Venkateswara University, Tirupati, India

    K. R. Sekhar & G. Viswanatha Reddy

  3. Department of Mathematics, GITAM University, Visakhapatnam, 530045, India

    S. M. Ibrahim

  4. Department of Engineering and Architecture, University of Parma, Parco Area Delle Scienze 181/A, 43124, Parma, Italy

    G. Lorenzini

  5. Department of Industrial Engineering, Alma Mater Studiorum-University of Bologna, Viale Risorgimento 2, 40136, Bologna, Italy

    E. Lorenzini

Authors
  1. C. S. K. Raju
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  2. K. R. Sekhar
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  3. S. M. Ibrahim
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  4. G. Lorenzini
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  5. G. Viswanatha Reddy
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  6. E. Lorenzini
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Corresponding author

Correspondence to G. Lorenzini.

Additional information

Communicated by Andreas Öchsner.

Due to the technical mistakes, some equations in the paper are wrong and the overall results are doubtful, the authors have requested to retract their article.

An erratum to this article can be found online at http://dx.doi.org/10.1007/s00161-017-0576-8

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Raju, C.S.K., Sekhar, K.R., Ibrahim, S.M. et al. RETRACTED ARTICLE: Variable viscosity on unsteady dissipative Carreau fluid over a truncated cone filled with titanium alloy nanoparticles. Continuum Mech. Thermodyn. 29, 699–713 (2017). https://doi.org/10.1007/s00161-016-0552-8

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