Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
Applied Nanoscience Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Abbreviations

\(u,\,\;v\) :

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

References

  • Bellos E, Tzivanidis C (2018) Performance analysis and optimization of an absorption chiller driven by nanofluid based solar flat plate collector. J Clean Prod 174:256–272

    Article  CAS  Google Scholar 

  • Buongiorno J, Hu LW (2009) Nanofluid heat transfer enhancement for nuclear reactor applications. In: Proceedings of the ASME 2009, 2nd micro/nanoscale heat & mass transfer international conference, pp 517–522

  • Choi SUS, Eastman JA (1995) Enhancing thermal conductivity of fluids with nanoparticles: the proceedings of the 1995 ASME international mechanical engineering congress and exposition, San Francisco, USA, ASME, FED 231/MD, vol 66, pp 99–105

  • Darcy H (1856) Les Fontaines Publiques De La Ville De Dijon. Victor Dalmont, Paris

    Google Scholar 

  • Dogonchi AS, Waqas M, Ganji DD (2019) Shape effects of Copper-Oxide (CuO) nanoparticles to determine the heat transfer filled in a partially heated rhombus enclosure: CVFEM approach. Int Commun Heat Mass Transfer 107:14–23

    Article  CAS  Google Scholar 

  • Forchheimer P (1901) Wasserbewegung durch boden. Z Ver Dtsch Ing 45:1782–1788

    Google Scholar 

  • Hassan M, Faisal A, Bhatti MM (2018) Interaction of aluminum oxide nanoparticles with flow of polyvinyl alcohol solutions base nanofluids over a wedge. Appl Nanosci 8:53–60

    Article  CAS  Google Scholar 

  • Hayat T, Ali S, Alsaedi A, Alsulami HH (2016) Influence of thermal radiation and Joule heating in the Eyring–Powell fluid flow with the Soret and Dufour effects. J Appl Mech Technol Phys 57:1051–1060

    Article  CAS  Google Scholar 

  • Hayat T, Haider F, Muhammad T, Alsaedi A (2017) On Darcy–Forchheimer flow of carbon nanotubes due to a rotating disk. Int J Heat Mass Transfer 112:248–254

    Article  CAS  Google Scholar 

  • Hayat T, Khalid H, Waqas M, Alsaedi A (2018a) Numerical simulation for radiative flow of nanoliquid by rotating disk with carbon nanotubes and partial slip. Comput Methods Appl Mech Eng 341:397–408

    Article  Google Scholar 

  • Hayat T, Ullah I, Waqas M, Alsaedi A (2018b) Flow of chemically reactive magneto Cross nanoliquid with temperature-dependent conductivity. Appl Nanosci 8:1453–1460

    Article  CAS  Google Scholar 

  • Hayat T, Shah F, Alsaedi A, Waqas M (2018c) Numerical simulation for magneto nanofluid flow through a porous space with melting heat transfer. Microgravity Sci Technol 30:265–275

    Article  CAS  Google Scholar 

  • Hayat T, Aziz A, Muhammad T, Alsaedi A (2018d) An optimal analysis for Darcy-Forchheimer 3D flow of Carreau nanofluid with convectively heated surface. Results Phys 9:598–608

    Article  Google Scholar 

  • Izadi F, Ranjbarzadeh R, Kalbasi R, Afrand M (2018) A new experimental correlation for non-Newtonian behavior of COOH-DWCNTs/antifreeze nanofluid. Phys E Low-dimens Syst Nanostruct 98:83–89

    Article  CAS  Google Scholar 

  • Khan MI, Shah F, Waqas M, Hayat T, Alsaedi A (2019) The role of γAl2O3-H2O and γAl2O3-C2H6O2 nanomaterials in Darcy-Forchheimer stagnation point flow: an analysis using entropy optimization. Int J Thermal Sci 140:20–27

    Article  CAS  Google Scholar 

  • Liao SJ (2010) An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun Nonlinear Sci Numer Simulat 15:2003–2016

    Article  Google Scholar 

  • Muskat M, Meres MW (1936) The flow of heterogeneous fluids through porous media. J Appl Phys 7:346

    Google Scholar 

  • Pavlovic S, Bellos E, Loni R (2018) Exergetic investigation of a solar dish collector with smooth and corrugated spiral absorber operating with various nanofluids. J Clean Prod 174:1147–1160

    Article  CAS  Google Scholar 

  • Rafati M, Hamidi A, Niaser MS (2012) Application of nanofluids in computer cooling systems (heat transfer performance of nanofluids). Appl Therm Eng 45:9–14

    Article  Google Scholar 

  • Sadiq MA, Waqas M, Hayat T (2017) Importance of Darcy-Forchheimer relation in chemically reactive radiating flow towards convectively heated surface. J Mol Liq 248:1071–1077

    Article  CAS  Google Scholar 

  • Sadiq MA, Waqas M, Hayat T, Alsaedi A (2019) Modeling and analysis of Maxwell nanofluid considering mixed convection and Darcy–Forchheimer relation. Appl Nanosci. https://doi.org/10.1007/s13204-019-00968-9

    Article  Google Scholar 

  • Seyyedi SM, Dogonchi AS, Tilehnoee MH, Asgharand Z, Ganji DD (2019) A computational framework for natural convective hydromagnetic flow via inclined cavity: an analysis subjected to entropy generation. J Mol Liq 287:110863

    Article  CAS  Google Scholar 

  • Sulochana C, Sandeep N (2016) Stagnation point flow and heat transfer behavior of Cu–water nanofluid towards horizontal and exponentially stretching/shrinking cylinders. Appl Nanosci 6:451–459

    Article  CAS  Google Scholar 

  • Waqas M, Farooq M, Khan MI, Alsaedi A, Hayat T, Yasmeen T (2016) Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition. Int J Heat Mass Transfer 102:766–772

    Article  Google Scholar 

  • Waqas M, Khan MI, Hayat T, Alsaedi A (2017) Numerical simulation for magneto Carreau nanofluid model with thermal radiation: a revised model. Comput Methods Appl Mech Eng 324:640–653

    Article  Google Scholar 

  • Waqas M, Hayat T, Alsaedi A (2018) A theoretical analysis of SWCNT–MWCNT and H2O nanofluids considering Darcy–Forchheimer relation. Appl Nanosci. https://doi.org/10.1007/s13204-018-0833-6

    Article  Google Scholar 

  • Waqas M, Jabeen S, Hayat T, Khan MI, Alsaedi A (2019a) Modeling and analysis for magnetic dipole impact in nonlinear thermally radiating Carreau nanofluid flow subject to heat generation. J Magn Magn Mater 485:197–204

    Article  CAS  Google Scholar 

  • Waqas M, Shehzad SA, Hayat T, Khan MI (2019b) Simulation of magnetohydrodynamics and radiative heat transport in convectively heated stratified flow of Jeffrey nanofluid. J Phys Chem Solids 133:45–51

    Article  CAS  Google Scholar 

  • Waqas M, Khan MI, Hayat T, Farooq S, Alsaedi A (2019c) Interaction of thermal radiation in hydromagnetic viscoelastic nanomaterial subject to gyrotactic microorganisms. Appl Nanosci. https://doi.org/10.1007/s13204-018-00938-7

    Article  Google Scholar 

  • Yu W, France DM, Routbort JL, Choi SUS (2008) Review and comparison of nanofluid thermal conductivity and heat transfer enhancements. Heat Transfer Eng 29:432–460

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. NUTECH School of Applied Sciences and Humanities, National University of Technology, Islamabad, 44000, Pakistan

    M. Waqas & M. Mudassar Gulzar

  2. Department of Mechanical Engineering, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran

    A. S. Dogonchi

  3. Department of Mathematics and Statistics, The University of Lahore Gujrat Campus, Gujrat, 50700, Pakistan

    M. Asif Javed

  4. Department of Mathematics, Mohi-ud-Din Islamic University, Narian Shareef, Azad Jammu and Kashmir, Pakistan

    W. A. Khan

  5. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    W. A. Khan

Authors
  1. M. Waqas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Mudassar Gulzar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. S. Dogonchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Asif Javed
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. W. A. Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Waqas.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Waqas, M., Gulzar, M.M., Dogonchi, A.S. et al. Darcy–Forchheimer stratified flow of viscoelastic nanofluid subjected to convective conditions. Appl Nanosci 9, 2031–2037 (2019). https://doi.org/10.1007/s13204-019-01144-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13204-019-01144-9

Keywords

Search

Navigation