Continuum Mechanics and Thermodynamics

, Volume 29, Issue 3, pp 835–851 | Cite as

RETRACTED ARTICLE: Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes over a slendering sheet with heat source/sink

  • R. V. M. S. S. Kiran Kumar
  • S. Vijaya Kumar Varma
  • C. S. K. Raju
  • S. M. Ibrahim
  • G. LorenziniEmail author
  • E. Lorenzini
Original Article


Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. These cylindrical carbon molecules have unusual properties, which are valuable for nanotechnology, electronics, optics and other fields of materials science and technology. With this intention, we investigate the three-dimensional magnetohydrodynamic convective heat and mass transfer of nanofluid over a slendering stretching sheet filled with porous medium and heat source/sink. For balancing the flow, temperature and concentration slip mechanisms are also taken into account. In this investigation simulation performed by mixing the two types of carbon nanotubes, namely single- and multi-walled carbon nanotubes, into water as base fluid. The governing system of partial differential equations is transformed into nonlinear ordinary differential equations which answered by using R–K–Fehlberg-integration scheme. The impact of various pertinent parameters on velocity, temperature and concentration as well as the friction factor coefficient, local Nusselt and local Sherwood number is derived and discussed through graphs and tables for both single- and multi-walled carbon nanotubes cases. It is found that the momentum boundary layer thickness of SWCNTs is thicker than MWCNTs. These results can help us to conclude that SWCNTs are helpful for minimizing the friction between the particles, whereas MWCNTs are helpful for boosting the heat and mass transfer rate.


Carbon nanotubes Magnetohydrodynamic Porous medium Multiple slips Variable thickness sheet Heat source 



Velocity components in xy and z directions


Specific heat capacity at constant pressure


Dimensionless velocities


Coefficient related to stretching sheet


Velocity power index parameter


Magnetic field parameter


Temperature of the fluid


Thermal conductivity


Molecular diffusivity of the species concentration


Concentration susceptibility


Concentration of the fluid


Mean fluid temperature

\(T_\infty \)

Temperature of the fluid in the free stream

\(C_\infty \)

Concentration of the fluid in the free stream


Dimensional velocity slip parameter


Dimensional temperature jump parameter


Dimensional concentration jump parameter


Maxwell’s reflection coefficient


Thermal accommodation coefficient


Physical parameter related to stretching sheet


Concentration accommodation coefficient


Velocity power index parameter


Prandtl number


Heat source/sink parameter


Dimensional magnetic field parameter


Magnetic interaction parameter


Porosity parameter


Thermophoresis parameter


Lewis number


Brownian motion parameter


Dimensionless velocity slip parameter


Dimensionless temperature jump parameter


Dimensionless concentration jump parameter


Wall skin friction coefficient


Local Nusselt number


Local Sherwood number


Local Reynolds number

Greek symbols

\(\phi \)

Dimensionless concentration

\(\eta \)

Similarity variable

\(\sigma \)

Electrical conductivity of the fluid

\(\gamma \)

Ratio of specific heats

\(\theta \)

Dimensionless temperature

\(\rho _{\mathrm{nf}}\)

Density of the nanofluid


Thermal conductivity of the nanofluid

\(\mu _{\mathrm{nf}}\)

Dynamic viscosity of nanofluid

\(\upsilon _\mathrm{f}\)

Kinematic viscosity

\(\delta \)

Wall thickness parameter

\(\xi _1 ,\xi _2\)

Mean free path (constant)

\(\xi _3 ,\xi _4\)

Mean free path (constant)

\(\varGamma \)

Positive characteristic time


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • R. V. M. S. S. Kiran Kumar
    • 1
  • S. Vijaya Kumar Varma
    • 1
  • C. S. K. Raju
    • 2
  • S. M. Ibrahim
    • 3
  • G. Lorenzini
    • 4
    Email author
  • E. Lorenzini
    • 5
  1. 1.Department of MathematicsSri Venkateswara UniversityTirupatiIndia
  2. 2.Department Of MathematicsVIT UniversityVelloreIndia
  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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