Applied Nanoscience

, Volume 9, Issue 8, pp 2031–2037 | Cite as

Darcy–Forchheimer stratified flow of viscoelastic nanofluid subjected to convective conditions

  • M. WaqasEmail author
  • M. Mudassar Gulzar
  • A. S. Dogonchi
  • M. Asif Javed
  • W. A. Khan
Original Article


Homotopy analysis method (HAM) simulations for hydromagnetic stratified viscoelastic nanoliquid in frames of non-Darcian characteristics are addressed in this attempt. The nanoliquid model comprising Brownian and thermophoretic movements is taken into account for formulation and analysis. The novel aspect featuring convective conditions and stratifications is introduced. Boundary-layer idea presented by Prandtl is implemented to simplify the complex systems which are then converted to ordinary ones employing transformation procedure. Besides, a detailed discussion is elaborated for various non-dimensional variables against significant profiles.


Viscoelastic nanoliquid Hydromagnetic flow Stratifications Non-Darcian characteristics HAM simulations 

List of symbols


Velocity components


Space coordinates


Density of base liquid


Kinematic viscosity


Thermal diffusivity


Dynamic viscosity


Thermal conductivity

\(\left( {\rho c} \right)_{\text{f}}\)

Liquid heat capacity


Heat capacity ratio


Porous medium permeability


Drag coefficient


Normal stress moduli

\(C_{\text{b}}^{ * }\)

Drag coefficient per unit length

\(u_{\text{w}} (x)\)

Stretching velocity


Stretching rate


Brownian movement coefficient


Thermophoresis diffusion coefficient


Electrical conductivity


Magnetic field strength





\(a_{1} ,\;a_{2} ,\;d_{1} ,\;d_{2}\)

Dimensional constants

\(C_{{{\text{f}}_{x} }} Re_{x}^{1/2}\)

Dimensionless drag force

\({\text{Nu}}\;Re_{x}^{ - 1/2}\)

Dimensionless heat transfer rate

\({\text{Sh}}\;Re_{x}^{{ - \tfrac{1}{2}}}\)

Dimensionless mass transfer rate


Hot fluid temperature


Hot fluid concentration

\(T_{\infty }\)

Ambient fluid temperature

\(C_{\infty }\)

Ambient fluid concentration


Reference temperature


Reference concentration


Local inertia coefficient


Material parameter of second grade fluid


Porosity parameter


Thermal Biot number


Thermophoretic variable


Solutal Biot number


Brownian motion variable


Hartman number


Schmidt number


Thermal stratified variable


Solutal stratified variable


Prandtl number


Reynolds numbers

\(f\left( \eta \right)\)

Dimensionless velocity

\(\theta \left( \eta \right)\)

Dimensionless temperature

\(\phi \left( \eta \right)\)

Dimensionless concentration

\(\hbar_{\text{f}} ,\;\hbar_{\theta } ,\;\hbar_{\phi }\)

Auxiliary variables


Dimensionless variable

\(C_{\infty }\)

Ambient fluid concentration


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Copyright information

© King Abdulaziz City for Science and Technology 2019

Authors and Affiliations

  • M. Waqas
    • 1
    Email author
  • M. Mudassar Gulzar
    • 1
  • A. S. Dogonchi
    • 2
  • M. Asif Javed
    • 3
  • W. A. Khan
    • 4
    • 5
  1. 1.NUTECH School of Applied Sciences and HumanitiesNational University of TechnologyIslamabadPakistan
  2. 2.Department of Mechanical EngineeringAliabad Katoul Branch, Islamic Azad UniversityAliabad KatoulIran
  3. 3.Department of Mathematics and StatisticsThe University of Lahore Gujrat CampusGujratPakistan
  4. 4.Department of MathematicsMohi-ud-Din Islamic University, Narian ShareefAzad Jammu and KashmirPakistan
  5. 5.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina

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