International Journal of Dynamics and Control

, Volume 7, Issue 4, pp 1293–1305 | Cite as

Heat and mass transfer analysis of MHD couple stress fluid flow through expanding or contracting porous pipe with cross diffusion effects

  • Adigoppula Raju
  • Odelu OjjelaEmail author
  • N. Naresh Kumar
  • Shankar Rao Munjam


This article investigates the effects of an incompressible laminar magnetohydrodynamic flow of couple stress fluid through an expanding or contracting porous pipe with cross diffusion has been considered. It is assumed that the pipe wall expands or contracts uniformly at time dependent rate. The system of governing partial differential equations and with associate boundary conditions are transformed into non-linear coupled system of equations by using suitable similarity transformations and obtained a solution by homotopy analysis method based Mathematica package BVPh2.0. The behaviour of various non-dimensional important parameters on velocity profiles, temperature and concentration distributions and also the rate of heat and mass transfers are studied and exhibited in the form of graphs. We noticed that the temperature of the fluid is enhanced by cross diffusion effects, whereas the concentration of the fluid decreased. In addition, the concentration of the fluid is decreased as Hartmann and inverse Darcy’s parameters increase.


Magneto hydrodynamic Couple stress fluid Porous pipe Homotopy analysis method Soret and Dufour 

List of symbols



\( a(t) \)

Radius of the pipe

\( \dot{a}(t) \)

Time dependent rate

\( B_{0} \)

Applied magnetic field

\( k_{T} \)

Thermal diffusion ratio

\( c_{s} \)

Susceptibility of the concentration


Specific heat at constant pressure

\( T_{m} \)

Mean temperature


Injection/suction coefficient


Dimensional pressure


Thermal conductivity






Coefficient of mass diffusivity


Velocity component along z direction


Velocity component along r direction


Injection/suction permeation Reynolds number, \( \frac{{av_{w} }}{\upsilon } = A\alpha \)


Hartmann number, \( B_{0} a\sqrt {\frac{\sigma }{\mu }} \)

\( D^{ - 1} \)

Inverse Darcy parameter,\( \frac{{a^{2} }}{{k_{1} }} \)


Prandtl number, \( \frac{\mu c}{k} \)


Eckert number, \( \frac{{\upsilon^{2} }}{{c\left( {T_{w} - T_{1} } \right)a^{2} }} \)


Dufour number, \( \frac{{\rho Dk_{T} \left( {C_{w} - C_{1} } \right)}}{{\mu cc_{s} \left( {T_{w} - T_{1} } \right)}} \)


Schmidt number, \( \frac{\upsilon }{D} \)


Soret number, \( \frac{{Dk_{T} \left( {T_{w} - T_{1} } \right)}}{{T_{m} \upsilon \left( {C_{w} - C_{1} } \right)}} \)

Greek letters

\( \xi \)

Dimensional less axial variable, \( \frac{z}{a} \)

\( \lambda \)

Dimensionless radial variable, \( \frac{r}{a} \)

\( \eta,\eta_{1} \)

Couple stress fluid parameters

\( \upsilon \)

Kinematic viscosity

\( \mu \)

Dynamic viscosity

\( \alpha \)

Wall expansion ratio, \( \frac{{a\dot{a}}}{\upsilon } \)

\( \beta \)

Couple stress parameter, \( \sqrt {\frac{\eta }{{\mu a^{2} }}} \)

\( \rho \)

Density of the fluid

\( \sigma \)

Electrical conductivity


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Adigoppula Raju
    • 1
  • Odelu Ojjela
    • 1
    Email author
  • N. Naresh Kumar
    • 2
  • Shankar Rao Munjam
    • 3
  1. 1.Department of Applied MathematicsDefence Institute of Advanced Technology (Deemed University)PuneIndia
  2. 2.Department of MathematicsSASTRA Deemed UniversityThanjavurIndia
  3. 3.Department of MathematicsVisvesvaraya National Institute of TechnologyNagpurIndia

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