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RETRACTED ARTICLE: Magnetohydrodynamic 3D slip flow in a suspension of carbon nanotubes over a slendering sheet with heat source/sink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

Abstract

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.

Keywords

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

Nomenclature

uvw

Velocity components in xy and z directions

\(C_\mathrm{p}\)

Specific heat capacity at constant pressure

fg

Dimensionless velocities

A

Coefficient related to stretching sheet

m

Velocity power index parameter

B(x)

Magnetic field parameter

T

Temperature of the fluid

k

Thermal conductivity

\(D_\mathrm{m}\)

Molecular diffusivity of the species concentration

\(C_\mathrm{s}\)

Concentration susceptibility

C

Concentration of the fluid

\(T_\mathrm{m}\)

Mean fluid temperature

\(T_\infty \)

Temperature of the fluid in the free stream

\(C_\infty \)

Concentration of the fluid in the free stream

\(j_1^*\)

Dimensional velocity slip parameter

\(j_2^*\)

Dimensional temperature jump parameter

\(j_3^*\)

Dimensional concentration jump parameter

\(f_1\)

Maxwell’s reflection coefficient

a

Thermal accommodation coefficient

b

Physical parameter related to stretching sheet

d

Concentration accommodation coefficient

m

Velocity power index parameter

Pr

Prandtl number

\(Q_\mathrm{H}\)

Heat source/sink parameter

B(x)

Dimensional magnetic field parameter

M

Magnetic interaction parameter

K

Porosity parameter

\(N_\mathrm{t}\)

Thermophoresis parameter

Le

Lewis number

\(N_\mathrm{b}\)

Brownian motion parameter

\(j_1\)

Dimensionless velocity slip parameter

\(j_2\)

Dimensionless temperature jump parameter

\(j_3\)

Dimensionless concentration jump parameter

\(C_\mathrm{f}\)

Wall skin friction coefficient

\(Nu_x\)

Local Nusselt number

\(Sh_x\)

Local Sherwood number

\(Re_x\)

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

\(k_{\mathrm{nf}}\)

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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