Continuum Mechanics and Thermodynamics

, Volume 29, Issue 3, pp 699–713 | Cite as

RETRACTED ARTICLE: Variable viscosity on unsteady dissipative Carreau fluid over a truncated cone filled with titanium alloy nanoparticles

  • C. S. K. Raju
  • K. R. Sekhar
  • S. M. Ibrahim
  • G. LorenziniEmail author
  • G. Viswanatha Reddy
  • E. Lorenzini
Original Article


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.


Viscous dissipation Nanofluid Temperature-dependent viscosity Truncated cone Heat source/sink Magnetic field parameter 



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


Distance along the surface (m)


Distance normal to the surface (m)


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


Dimensionless velocities


Temperature of the fluid (K)


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

\(k_\mathrm{f} \)

Thermal conductivity (W/mK)

\(\alpha \)

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


Pressure (pa)

Greek symbols

\(\eta \)

Similarity variable

\(\sigma \)

Electrical conductivity (Siemens)

\(\sigma ^{*}\)

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


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)

Nondimensional parameters

\(Q_\mathrm{H} \)

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


Eckert number


Viscous variation parameter


Weissenberg number


Power-law index parameter

\(\gamma \)

Half-angle parameter

\(\lambda \)

Buoyancy parameter

\(\alpha _1 \)

Ratio of angles


Magnetic field parameter

\(Gr_x \)

Grashof number





Condition at the wall

\(\infty \)

Condition at the free stream




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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • C. S. K. Raju
    • 1
  • K. R. Sekhar
    • 2
  • S. M. Ibrahim
    • 3
  • G. Lorenzini
    • 4
    Email author
  • G. Viswanatha Reddy
    • 2
  • E. Lorenzini
    • 5
  1. 1.Department of MathematicsVIT UniversityVelloreIndia
  2. 2.Department of MathematicsSri Venkateswara UniversityTirupatiIndia
  3. 3.Department of MathematicsGITAM UniversityVisakhapatnamIndia
  4. 4.Department of Engineering and ArchitectureUniversity of ParmaParmaItaly
  5. 5.Department of Industrial EngineeringAlma Mater Studiorum-University of BolognaBolognaItaly

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