Slip flow of hydromagnetic micropolar nanofluid between two disks with characterization of porous medium

  • Z. Abbas
  • T. Mushtaq
  • S. A. Shehzad
  • A. RaufEmail author
  • R. Kumar
Technical Paper


An analysis is performed for axisymmetric, incompressible, hydromagnetic micropolar nanofluid flow through infinite porous parallel disks subjected to porous medium. The flow is imposed via uniform injection through surface of disks. The utilization of similarity transformations converts the system of partial differential equations with combined boundary conditions into highly nonlinear ordinary ones. Boundary conditions of velocity, temperature and concentration slips are executed. Both slip and no-slip cases are discussed. Runge–Kutta–Fehlberg (RKF45) numerical method is effectuated to obtain the results in pictorial representations and tabular forms.


Axisymmetric flow Microrotation Nanofluid MHD Porous medium 

List of symbols

\( \left( {u,\,w} \right)\,\left[ {\frac{L}{T}} \right] \)

Velocity components

\( \mu \,\,\left[ {\frac{M}{LT}} \right] \)

Dynamic viscosity

\( k\,\,\,\left[ {\frac{ML}{{T^{3} \,{\text{Kelvin}}}}} \right] \)

Thermal conductivity

(C1, C2) [1]

Concentration at lower and upper disk

\( D_{\text{T}} \,\,\left[ {\frac{{L^{2} }}{T}} \right] \)

Thermophoresis diffusion coefficient


Effective nanoparticles heat capacity


Vortex viscosity


Microinertia density


Radiative heat flux


Velocity coefficient


Concentration coefficient

\( R = \frac{k}{\mu } \)

Vortex viscosity parameter

\( A = \frac{j}{{a^{2} }} \)

Microinertia density parameter

\( M = \sqrt {\frac{{\sigma_{e} a\,\beta_{0}^{2} }}{{\rho V_{0} }}} \)

Magnetic parameter

\( Pr = \frac{{\mu \,c_{p} }}{{k_{0} }} \)

Prandtl number

\( Nt = \frac{{\tau \,D_{\text{B}} }}{\upsilon }\left( {C_{1} - C_{2} } \right) \)

Thermophoretic parameter

\( Sc = \frac{\upsilon }{{D_{\text{B}} }} \)

Schmidt number

\( \gamma_{3} = \frac{{\gamma_{1} }}{a} \)

Thermal slip parameter


Shear stress

\( \rho \,\left[ {\frac{M}{{L^{3} }}} \right] \)


\( \sigma_{e} \,\left[ {\frac{{T^{3} {\text{Ampere}}^{2} }}{{L^{3} M}}} \right] \)

Electrical conductivity

(T1, T2) [Kelvin]

Temperature at lower and upper disk

\( B_{0}^{2} \,\left[ {\frac{M}{{T^{2} {\text{Ampere}}}}} \right] \)

Magnetic field strength

\( D_{\text{B}} \,\,\left[ {\frac{{L^{2} }}{T}} \right] \)

Brownian constant


Liquid heat capacity




Spin gradient viscosity


Thermal diffusivity


Thermal coefficient


Constant injection velocity

\( D = \frac{\gamma }{{\mu \,a^{2} }} \)

Spin gradient viscosity parameter

\( Re = \frac{{\rho V_{0} a}}{\mu } \)

Reynolds number

\( P = \frac{{\mu \,a\,\phi_{1} }}{{\rho \,V_{0} k_{1} }} \)

Porosity parameter

\( Rd = \frac{{4\,\sigma \,T_{2}^{3} }}{{k_{0} k^{*} }} \)

Radiation parameter

\( Nb = \frac{{\tau \,D_{\text{T}} }}{{\upsilon T_{m} }}\left( {T_{1} - T_{2} } \right) \)

Brownian motion parameter

\( \beta_{2} = \frac{{\beta_{1} }}{a} \)

Velocity slip parameter

\( \gamma_{4} = \frac{{\gamma_{2} }}{a} \)

Concentration slip parameter


Couple stress



There are no funders to report for this submission.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Z. Abbas
    • 1
  • T. Mushtaq
    • 2
  • S. A. Shehzad
    • 3
  • A. Rauf
    • 1
    • 3
    Email author
  • R. Kumar
    • 4
  1. 1.Department of MathematicsThe Islamia University of BahawalpurBahawalpurPakistan
  2. 2.Department of MathematicsCOMSATS University IslamabadVehariPakistan
  3. 3.Department of MathematicsCOMSATS University IslamabadSahiwalPakistan
  4. 4.Department of MathematicsCentral University of Himachal PradeshDharamshalaIndia

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