Skip to main content
SpringerLink
Log in

Hall current effect on unsteady rotational flow of carbon nanotubes with dust particles and nonlinear thermal radiation in Darcy–Forchheimer porous media

  • Published:
Journal of Thermal Analysis and Calorimetry Aims and scope Submit manuscript

Abstract

The present discussion is about the unsteady two-dimensional flow of mixed convection and nonlinear thermal radiation in the presence of water-based carbon nanotubes over the vertically convected stretched sheet embedded in a Darcy’ Forchheimer porous media. Saffman’s proposed model is used for the suspension of fine dust particles in the nanofluid. A strong magnetic field (MHD) is applied normal to the flow which governs the Hall current effects. Khanafer Vafai Lightstone model estimated the effect of thermal conductivity and viscosity of the carbon nanotubes. Boundary layer approximation is utilized to built the nonlinear partial differential equations (PDEs). Similarity transformation is applied to convert these PDEs into the system of ordinary differential equations. Problem is solved numerically by bvp4c, using MATLAB software. It is observed through the analysis that the thermal field of nanofluid and the temperature boundary layer are much more higher than that of the dust phase, and these are further enhanced for the higher radiation parameter.

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
Fig. 15

Similar content being viewed by others

Abbreviations

\(b,\alpha \) :

A real constant

\(B_{0}\) :

Magnetic induction \((\hbox {kg}\,\hbox {s}^{-2}\,\hbox {A}^{-1})\)

Bi :

Biot number

C :

Specific heat \((\hbox {J}\,\hbox {kg}^{-1}\,\hbox {K}^{-1})\)

E :

Intensity vector of the electric field

\(F^{\prime }\) :

Dimensionless velocity of dust

F :

Inertia coefficient of porous medium

\(F_{\mathrm{p}}\) :

Force due to dust particles

\(F_{\mathrm{b}}\) :

Body forces

\(f^{\prime }\) :

Dimensionless velocity

\(F_{\mathrm{r}}\) :

Inertia coefficient

g :

Gravitational acceleration \((\hbox {m}\,\hbox {s}^{-2})\)

\(Gr_{\mathrm{x}}\) :

Grashof number

h :

Dimensionless transverse velocity

\(h_{\mathrm{c}}\) :

Heat flux coefficient \((\hbox {W}\,\hbox {m}^{-1}\,\hbox {K}^{-1})\)

J :

Current density vector

\(K^{*}\) :

Permeability of porous medium

l :

Mass concentration of dust

\({m}_{\mathrm{p}}\) :

Mass of dust particles

Nu :

Local Nusselt number

\(n_{\mathrm{e}}\) :

Number density of electron

\(P_{\mathrm{e}}\) :

Electron pressure

P :

Pressure

Pr :

Prandtl number

\(Q_{\mathrm{p}}\) :

Thermal interaction between nanoparticles and dust phase

\(q_{\mathrm{w}}\) :

Surface heat flux

\(q_{\mathrm{r}}\) :

Radiation heat flux

Rd :

Thermal radiation parameter

\(r_{\mathrm{p}}\) :

Radius of dust particles

\(Re_{\mathrm{x}}\) :

Local Reynolds number

S :

Unsteadyness parameter

t :

Time

t :

Fluid temperature \((\hbox {K})\)

Uw :

Deformation velocity of the sheet

U(u, v, w):

Velocity vector \((\hbox {m}\,\hbox {s}^{-1})\)

V(u, v, w):

Velocity vector \((\hbox {m}\,\hbox {s}^{-1})\)

(x, y, z):

Axial and normal coordinates

\(\beta _{{\upnu }}\) :

Momentum dust parameter

\(\beta _{\mathrm{T}}\) :

Thermal expansion coefficient

\(\beta _{\mathrm{t}}\) :

Thermal dust parameter

\(\gamma \) :

Specific heat ratio

\(\delta \) :

Porosity parameter

\(\eta \) :

Dimensionless normal distance

\(\theta \) :

Dimensionless temperature

\(\theta _{\mathrm{w}}\) :

Temperature ratio parameter

\(\kappa \) :

Thermal conductivity \((\hbox {W}\,\hbox {m}^{-1}\,\hbox {K}^{-1})\)

\(\lambda \) :

Mixed convection parameter

\(\mu \) :

Dynamic viscosity

\(\sigma \) :

Electric conductivity

\(\sigma ^{*}\) :

Stefan–Boltzmann constant \((\hbox {m}^{2}\,\hbox {s}^{-1})\)

\(\tau _{\mathrm{e}}\) :

Electron collision time

\(\tau _{\mathrm{T}}\) :

Thermal relaxation time of dust phase

\(\tau _{\upnu }\) :

Momentum relaxation time of dust phase

\(\tau _{\mathrm{wx}}\) :

Wall shear stress in x direction

\(\omega \) :

Frequency

\(\omega _{\mathrm{e}}\) :

Electron frequency

\(\varkappa \) :

Nanoparticle volume fraction

\(\psi \) :

Stream function \((\hbox {m}^{2}\,\hbox {s}^{-1}) \)

\(\varOmega \) :

Rotational parameter

\(\rightthreetimes \) :

Absorption coefficient

nf:

Nanofluid

f:

Base fluid

s:

Nanoparticles (CNTs)

p:

Dust phase/particles

w:

Wall

e:

Charge on electron

References

  1. Doronin GG, Larkin NA. Mathematical problems for a dusty gas flow. Boletim Da Sociedade Paranaense de Matemtica Terceira Srie. 2004;22(1):21–9.

    Google Scholar 

  2. Farbar L, Morley MJ. Heat transfer to flowing gas-solid mixtures in a circular tube. Ind Eng Chem. 1957;49:1143–50.

    CAS  Google Scholar 

  3. Saffman PG. On the stability of laminar flow of a dusty gas. J Fluid Mech. 1962;13:120–8.

    Google Scholar 

  4. Attia HA, Abbas W, Abdeen MAM. Ion slip effect on unsteady Couette flow of a dusty fluid in the presence of uniform suction and injection with heat transfer. J Braz Soc Mech Sci Eng. 2016;38(8):2381–91.

    Google Scholar 

  5. Koneri LK, Gireesha BJ, Mahanthesh B, Gorla RSR. Influence of nonlinear thermal radiation and magnetic field on upper-convected Maxwell fluid flow due to a convectively heated stretching sheet in the presence of dust particles. Commun Numer anal. 2016;1:57–73.

    Google Scholar 

  6. Sandeep N, Sulochana C, Kumar BR. Unsteady MHD radiative flow and heat transfer of a dusty nanofluid over an exponentially stretching surface. Eng Sci Technol Int J. 2016;19:227–40.

    Google Scholar 

  7. Pop I, Hamid RA, Nazar R. Boundary layer flow of a dusty fluid over a permeable shrinking surface. Int J Numer Methods Heat Fluid Flow. 2017;27(4):1–19.

    Google Scholar 

  8. Ghadikolaei SS, Hosseinzadeh K, Yassari M, Sadeghi H, Ganji DD. Boundary layer analysis of micropolar dusty fluid with \(TiO_{2}\) nanoparticles in a porous medium under the effect of magnetic field and thermal radiation over a stretching sheet. J Mol Liq. 2017;244:374–89.

    CAS  Google Scholar 

  9. Siddiqa S, Begum N, Ouazzi A, Hossain MA, Gorla RSR. Heat transfer analysis of Casson dusty fluid flow along a vertical wavy cone with radiating surface. Int J Heat Mass Transf. 2018;127:589–96.

    Google Scholar 

  10. Gireesha BJ, Mahanthesh B, Thammanna GT, Sampathkumar PB. Hall effects on dusty nanofluid two-phase transient flow past a stretching sheet using KVL model. J Mol Liq. 2018;256:139–47.

    CAS  Google Scholar 

  11. Bear J. Dynamics of fluids in porous media. New York: Dover; 1972.

    Google Scholar 

  12. Hellstrom JGI, Lundstrom TS. Flow through porous media at moderate Reynolds number. In: 4th Int. Sci. Colloquium: Model. Mat. Proc. Uni. Latvia, Riga, Latvia; 2006. p. 8–9.

  13. Sobieski W, Trykozko A. Darcy’s and Forchheimer’s laws in practice. Part 1. The experiment. Tech. Sci. 2014;17(4):321–35.

    Google Scholar 

  14. Chapman RE. Geology and water: an introduction to fluid mechanics for geologists. The Hague: Martinus Nijho & Dr. W. Junk Publishers; 1981.

    Google Scholar 

  15. Forchheimer P. Wasserbewegung durch boden. Z. Ver D. Ing. 1901;45:1782–8.

    Google Scholar 

  16. Ewing R, Lazarov R, Lyonss L, Papavassiliou DV, Pasciak J, Qin GX. Numerical well model for non-Darcy flow. Comput. Geosci. 1999;3(3):185–204.

    Google Scholar 

  17. Shehzad SA, Abbasi FM, Hayat T, Alsaedi A. Cattaneo-Christov heat flux model for Darcy-Forchheimer flow of an Oldroyd-B fluid with variable conductivity and non-linear convection. J Mol Liq. 2016;224:274–8.

    CAS  Google Scholar 

  18. Rami YJ, Fawzi A, Fahmi AAR. Darcy-Forchheimer mixed convection heat and mass transfer in fluid saturated porous media. Int J Numer Methods Heat Fluid Flow. 2001;11(6):600–18.

    Google Scholar 

  19. Sobieski W, Trykozko A. Sensitivity aspects of Forchheimers approximation. Trans Porous Media. 2011;89(2):155–64.

    CAS  Google Scholar 

  20. Hayat T, Haider F, Muhammad T, Alsaedi A. Darcy-Forchheimer flow with Cattaneo-Christov heat flux and homogeneous heterogeneous reactions. PLoS ONE. 2017;12(4):e0174938.

    PubMed  Google Scholar 

  21. Muhammad T, Alsaedi A, Shehzad SA, Hayat T. A revised model for Darcy Forchheimer flow of Maxwell nanofluid subject to convective boundary condition. Chin J Phys. 2017;55(3):963–76.

    CAS  Google Scholar 

  22. Sajid T, Sagheer M, Hussain S, Bilal M. Darcy-Forchheimer flow of Maxwell nanofluid flow with nonlinear thermal radiation and activation energy. AIP Adv. 2018;8:035102.

    Google Scholar 

  23. Alzahrani AK. Importance of Darcy Forchheimer porous medium in 3D convective flow of carbon nanotubes. Phys Lett A. 2018;382(42):2938–43.

    CAS  Google Scholar 

  24. Majeed A, Zeeshan A, Alamri SZ, Ellah R. Heat transfer analysis in ferromagnetic viscoelastic fluid flow over a stretching sheet with suction. Neural Comput Appl. 2018;30(6):1947–55.

    Google Scholar 

  25. Alamri SZ, Ellahi R, Shehzad N, Zeeshan A. Convective radiative plane Poiseuille flow of nanofluid through porous medium with slip: an application of Stefan blowing. J Mol Liq. 2018;273:292–304.

    Google Scholar 

  26. Shehzad N, Zeeshan A, Ellahi R, Rashidi S. Modelling study on internal energy lossd due to entropy generation for non-Darcy poiseuille flow of silver-water nanofluid: an application of purification. Entropy. 2018;20:851.

    CAS  Google Scholar 

  27. Hassan M, Marinb M, Alsharif A, Ellahi R. Convective heat transfer flow of nanofluid in a porous medium over wavy surface. Phys Lett A. 2018;382(38):2749–53.

    CAS  Google Scholar 

  28. Hassan M, Marinb M, Alsharif A, Ellahi R. Numerical investigation and optimization of mixed convection in ventilated square cavity filled with nanofluid of different Inlet and outlet port. Int J Numer Methods Heat Fluid Flow. 2017;27(9):2053–69.

    Google Scholar 

  29. Rashidi S, Akar S, Bovand M, Ellahi R. Volume of fluid model to simulate the nanofluid flow and entropy generation in a single slope solar still. Renew Energy. 2018;115:400–10.

    CAS  Google Scholar 

  30. Khanafer K, Vafai K. Applications of nanofluids in porous medium. J Therm Anal Calorim. 2019;135:1479–92.

    CAS  Google Scholar 

  31. Freidoonimehr N, Rahimi AB. Brownian motion effect on heat transfer of a three-dimensional nanofluid flow over a stretched sheet with velocity slip. J Therm Anal Calorim. 2019;135:207–22.

    CAS  Google Scholar 

  32. Ramzan M, Bilal M, Chung JD. Radiative Williamson nanofluid flow over a convectively heated Riga plate with chemical reaction—a numerical approach. Chin J Phys. 2017;55:1663–73.

    CAS  Google Scholar 

  33. Ramzan M, Bilal M, Chung JD. Effects of thermal and solutal stratification on Jeffrey magneto-nanofluid along an inclined stretching cylinder with thermal radiation and heat generation/absorption. Int J Mech Sci. 2017;131:317–24.

    Google Scholar 

  34. Turkyilmazoglu M. Buongiorno model in a nanofluid filled a symmetric channel fulfilling zero net particle flux at the walls. Int J Heat Mass Transf. 2018;126:974–9.

    CAS  Google Scholar 

  35. Turkyilmazoglu M. Fluid flow and heat transfer over a rotating and vertically moving disk. Phys Fluids. 2018;30:063605.

    Google Scholar 

  36. Zeeshan A, Ijaz N, Abbas T, Ellahi R. The sustainable characteristic of Bio-Bi-phase flow of peristaltic transport of MHD Jeffrey fluid in the human body. Sustainability. 2018;10:2671.

    CAS  Google Scholar 

  37. Turkyilmazoglu M. Equivalences and correspondences between the deforming body induced flow and heat in two-three dimensions. Phys Fluids. 2016;28:043102.

    Google Scholar 

  38. Turkyilmazoglu M. Analytical solutions to mixed convection MHD fluid flow induced by a nonlinearly deforming permeable surface. Commun Nonlinear Sci Numer Simul. 2018;63:373–9.

    Google Scholar 

  39. Timofeeva VE, Routbort JL, Signh D. Particle shape effects on thermophysical properties of alumina nanofluids. J Appl Phys. 2009;106(1):014304.

    Google Scholar 

  40. Murshed SMS, Nietode Castro CA, Lourenco MJV, Lopes MLM, Santos FJV. A review of boiling and convective heat transfer with nanofluids. Renew Sustain Energy Rev. 2011;15(5):2342–54.

    CAS  Google Scholar 

  41. Hussien AA, Yusop NM, Abdullah MZ, Al-Nimr MA, Khavarian M. Study on convective heat transfer and pressure drop of MWCNTs water nanofluid in mini-tube. J Therm Anal Calorim. 2019;135:123–32.

    CAS  Google Scholar 

  42. Padilha GS, Campos VAB, Costa MC, Franco TT. Multi-walled carbon nanotubes used as support for lipase from Burkholderia cepacia. J Therm Anal Calorim. 2018;134(2):1021–9.

    CAS  Google Scholar 

  43. Choi SUS, Zang ZG, Yu Z, Lookwood FE, Grulke EA. Anomalous thermal conductivity enhancement in nanotube suspensions. Appl Nanosci Appl Phys Lett. 2001;79:2252–4.

    CAS  Google Scholar 

  44. Kandasamy R, Mohammad R, Muhaimin I. Carbon nanotubes on unsteady MHD non-Darcy flow over porous wedge in presence of thermal radiation energy. Appl Math Mech Engl. 2016;37(8):1031–40.

    Google Scholar 

  45. Bilal M, Hussain S, Sagheer M. Boundary layer flow of magneto-micropolar nanofluid flow with Hall and ion-slip effects using variable thermal diffusivity. Bull Pol Acad Tech Sci. 2017;65(3):383–90.

    CAS  Google Scholar 

  46. Ramzan M, Bilal M, Chung JD. Soret and Dufour effects on three dimensional upper-convected Maxwell fluid with chemical reaction and non-linear radiative heat flux. Int J Chem Reactor Eng. 2017;15(3):20160136.

    CAS  Google Scholar 

  47. Lu DC, Ramzan M, Bilal M, Chung JD, Farooq U. A numerical investigation of 3-D MHD rotating flow with binary chemical reaction, activation energy and non-Fourier heat flux. Commun Theor Phys. 2018;70:89–96.

    CAS  Google Scholar 

  48. Hayat T, Muhammad T, Shehzad SA, Alsaedi A. On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip effect: a numerical study. Comput Methods Appl Mech Eng. 2017;315:467–77.

    Google Scholar 

  49. Turkyilmazoglu M. Magnetohydrodynamics two-phase dusty fluid flow and heat model over deforming isothermal surfaces. Phys Fluids. 2017;29:013302.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Lahore, Chenab Campus, Gujrat, Pakistan

    M. Bilal

  2. Bahria University, Islamabad, Pakistan

    M. Ramzan

Authors
  1. M. Bilal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Ramzan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Bilal.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bilal, M., Ramzan, M. Hall current effect on unsteady rotational flow of carbon nanotubes with dust particles and nonlinear thermal radiation in Darcy–Forchheimer porous media. J Therm Anal Calorim 138, 3127–3137 (2019). https://doi.org/10.1007/s10973-019-08324-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-019-08324-3

Keywords

Search

Navigation