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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

Abstract

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.

Keywords

Viscoelastic nanoliquid Hydromagnetic flow Stratifications Non-Darcian characteristics HAM simulations 

List of symbols

\(u,\,\;v\)

Velocity components

\(x,\,\;y\)

Space coordinates

\(\rho_{\text{f}}\)

Density of base liquid

\(\nu\)

Kinematic viscosity

\(\alpha\)

Thermal diffusivity

\(\mu\)

Dynamic viscosity

\(k\)

Thermal conductivity

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

Liquid heat capacity

\(\tau\)

Heat capacity ratio

\(K\)

Porous medium permeability

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

Drag coefficient

\(\alpha_{1}\)

Normal stress moduli

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

Drag coefficient per unit length

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

Stretching velocity

\(c\)

Stretching rate

\(D_{\text{B}}\)

Brownian movement coefficient

\(D_{\text{T}}\)

Thermophoresis diffusion coefficient

\(\sigma\)

Electrical conductivity

\(B_{0}\)

Magnetic field strength

T

Temperature

C

Concentration

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

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

\(T_{\text{f}}\)

Hot fluid temperature

\(C_{\text{f}}\)

Hot fluid concentration

\(T_{\infty }\)

Ambient fluid temperature

\(C_{\infty }\)

Ambient fluid concentration

\(T_{0}\)

Reference temperature

\(C_{0}\)

Reference concentration

\(F_{\text{r}}\)

Local inertia coefficient

\(\beta\)

Material parameter of second grade fluid

\(\lambda\)

Porosity parameter

\(\gamma_{1}\)

Thermal Biot number

\(N_{{\text{t}}}\)

Thermophoretic variable

\(\gamma_{2}\)

Solutal Biot number

\(N_{\text{b}}\)

Brownian motion variable

\(Ha\)

Hartman number

\(Sc\)

Schmidt number

\(S_{1}\)

Thermal stratified variable

\(S_{2}\)

Solutal stratified variable

\(Pr\)

Prandtl number

\(Re_{x}\)

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

\(\eta\)

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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