Skip to main content
SpringerLink
Log in

Second-grade fluid with carbon nanotubes flowing over an elongated curve surface possessing thermal radiation and internal heat generation effects

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

Abstract

The goal of the current research is to analyze heat transmission of radiative nanofluid with reference to boundary layer nature. Carbon nanotube's (CNT's) reliant liquid is being tested, and it flows on top of a curved extending surface. To scrutinize thermal transmission through the flow additional impacts of thermal radiation as well as internal heat generation have been incorporated. Dual nature of CNTs, that is, single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs) have been employed in conjunction with slurry mixture (base fluid) for the formulation of nanofluid. Second-grade fluid model is engaged in order to capture the rheological properties of slurry mixtures (base fluid). To acquire the numerical solution of designed mathematical model, NDSolve approach is engaged using software Mathematica. Various parameters occurring in governing equations makes an impact on focused physical quantities. Graphs have been employed to capture these impacts for both SWCNTs and MWCNTs. In like manner, the impact of numerous factors on skin friction coefficient as well as Nusselt number have been examined using numerical charts. An increment in dimensionless curvature parameter causes a decline in fluid’s velocity profile as well as temperature profile. However, both fluid’s velocity and temperature get enhanced as an upsurge in solid volume fraction of carbon nanotubes, radiation parameter and heat generation 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

Similar content being viewed by others

Abbreviations

\(\left( \grave{u},\grave{v}\right)\) :

Velocity components (ms−1)

\(\grave{R}_{\text{c}}^{*}\) :

Radius of curvature \(\text {(m)}\)

a :

Stretching constant \(\text {(s}^{-1})\)

\(\grave{\alpha }_{1}\) :

Second-grade fluid parameter

\((\rho C_{\text{p}})_{\text{nf}}\) :

Nanofluid heat capacity

\(\mu _{\text{f}}\) :

Base fluid dynamic viscosity (kg ms−1)

\(k_{\text{f}}\) :

Base fluid thermal conductivity W m−1 K−1

\(C_{\text{p}}\) :

Specific heat J kg−1 K−1

\(k_{\text{CNT}}\) :

Thermal conductivity of CNTs W m−1 K−1

\(\alpha _{\text{nf}}\) :

Thermal diffusivity of nanofluid (m2 s−1)

\(\grave{a}_{\text{R}}\) :

Rooseland mean approximation coefficient

\(\rho _{\text{f}}\) :

Fluid density (kg m−3)

\(\rho _{_{\text{CNT}}}\) :

Carbon nanotubes density (kg m−3)

\(\nu _{\text{f}}\) :

Base fluid kinematic viscosity (m2 s−1)

\(\zeta\) :

Similarity variable

\(\kappa\) :

Dimensionless radius of curvature

l :

Characteristic length \(\text {(m)}\)

A :

Reference temperature \(\text {(K)}\)

\(\tau _{\grave{\text{r}}\grave{\text{s}}}\) :

Wall shear stress \(\text {(Pa)}\)

\(\hbox {Re}_{\acute{\text{s}}}\) :

Local Reynolds number

\(\grave{r},\grave{s}\) :

Coordinate axes \(\text {(m)}\)

P :

Pressure \(\text {(Pa)}\)

\(\grave{\gamma }\) :

Volumetric heat generation \(\text {(J)}\)

\(\grave{q}_{\text{r}}\) :

Radiative heat flux \(\text {(J)}\)

\(\phi\) :

Nanoparticles concentration

\(\mu _{\text{nf}}\) :

Nanofluid dynamic viscosity (kg ms−1)

\(k_{\text{nf}}\) :

Nanofluid thermal conductivity W m−1 K−1

\(\grave{T}\) :

Temperature \(\text {(K)}\)

\(\grave{T}_{\text{w}}\) :

Surface temperature \(\text {(K)}\)

\(\grave{T}_{\infty }\) :

Free stream temperature \(\text {(K)}\)

\(\grave{\sigma }^{*}\) :

Stefan–Boltzmann constant (W m−2 K−4)

\(R_{\text{D}}\) :

Radiation parameter

\(\Pr\) :

Prandtl number

\(\nu _{\text{nf}}\) :

Nanofluid kinematic viscosity (m2 s−1)

\(\lambda _{1}\) :

Heat generation parameter

\(\theta\) :

Dimensionless temperature

\(f^{\prime }\) :

Dimensionless velocities

\(\grave{q}_{\text{w}}\) :

Wall heat flux (W m−2)

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

Skin friction coefficient

\(Nu_{\grave{\text{s}}}\) :

Local Nusselt number

References

  1. Choi SUS. Enhancing thermal conductivity of fluids with nanoparticles. Dev Appl Non Newton Flows. 1995;232:99–105.

    Google Scholar 

  2. Kim J, Kang YT, Choi CK. Analysis of convective instability and heat transfer characteristics of nanofluids. Phys Fluids. 2004;16:2395–401.

    ADS  CAS  Google Scholar 

  3. Buongiorno J. Convective transport in nanofluids. J Heat Transf. 2006;128:240–50.

    Google Scholar 

  4. Tiwari RK, Das MK. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int J Heat Mass Transf. 2007;50:2002–18.

    CAS  Google Scholar 

  5. Khan WA, Pop I. Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transf. 2010;53:2477–83.

    CAS  Google Scholar 

  6. Kuznetsov AV, Nield DA. Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci. 2010;49:243–7.

    CAS  Google Scholar 

  7. Raza J, Rohni AM, Omar Z. Numerical investigation of copper-water (Cu-Water) nanofluid with different shapes of nanoparticles in a channel with stretching wall: slip effects. Math Comput Appl. 2016;21:43.

    MathSciNet  Google Scholar 

  8. Hayat T, Rashid M, Alsaedi A. MHD convective flow of magnetite-Fe3O4 nanoparticles by curved stretching sheet. Results Phys. 2017;7:3107–15.

    ADS  Google Scholar 

  9. Saif RS, Muhammad T, Sadia H, Ellahi R. Hydromagnetic flow of Jeffrey nanofluid due to a curved stretching surface. Phys A: Stat Mech. 2020;551:0378–4371.

    MathSciNet  Google Scholar 

  10. Saif RS, Hashim M, Zaman M. Ayaz. Thermally stratified flow of hybrid nanofluids with radiative heat transport and slip mechanism: multiple solutions. Commun Theor Phys. 2022;74: 015801.

    ADS  MathSciNet  CAS  Google Scholar 

  11. Maxwell JC. A Treatise on Electricity and Magnetism. 3rd ed. Oxford, UK: Oxford University Press; 1904.

    Google Scholar 

  12. Xue QZ. Model for thermal conductivity of carbon nanotube-based composites. Phys B Condens Matter. 2005;368:302–7.

    ADS  CAS  Google Scholar 

  13. Khan U, Ahmed N, Mohyud-Din ST. Stoke’s first problem for carbon nanotubes suspended nanofluid flow under the effect of slip boundary condition. J Nanofluids. 2016;5(2):239–44.

    Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  15. Wakif A, Boulahia Z, Sehaqui R. A semi-analytical analysis of electro-thermo-hydrodynamic stability in dielectric nanofluids using Buongiorno’s mathematical model together with more realistic boundary conditions. Results Phys. 2018;9:1438–54.

    ADS  Google Scholar 

  16. Khan MI, Shah F, Hayat T, Alsaedi A. Transportation of CNTs based nanomaterial flow confined between two coaxially rotating disks with entropy generation. Phys A: Stat Mech. 2019;527: 121154.

    MathSciNet  CAS  Google Scholar 

  17. Acharya N, Bag R, Kundu PK. On the mixed convective carbon nanotube flow over a convectively heated curved surface. Heat Transf Res. 2020;49:1713–35.

    Google Scholar 

  18. Al-Hanaya AM, Sajid F, Abbas N, Nadeem S. Effect of SWCNT and MWCNT on the flow of micropolar hybrid nanofluid over a curved stretching surface with induced magnetic field. Sci Rep. 2020;10:8488.

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  19. Ahmad S, Nadeem S, Muhammad N, Khan MN. Cattaneo-Christov heat flux model for stagnation point flow of micropolar nanofluid toward a nonlinear stretching surface with slip effects. J Therm Anal Calorim. 2021;143:1187–99.

    CAS  Google Scholar 

  20. Bhatti MM, Ellahi R, Doranehgard MH. Numerical study on the hybrid nanofluid (Co3O4-Go/H2O) flow over a circular elastic surface with non-Darcy medium: Application in solar energy. J Mol Liq. 2022;361: 119655.

    CAS  Google Scholar 

  21. Bhatti MM, Öztop HF, Ellahi R. Study of the magnetized hybrid nanofluid flow through a flat elastic surface with applications in solar energy. Materials. 2022;15:7507.

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  22. Bhatti MM, Ellahi R. Numerical investigation of non-Darcian nanofluid flow across a stretchy elastic medium with velocity and thermal slips. Numer Heat Transfer. 2023;83(5):323–43.

    ADS  Google Scholar 

  23. Hayat T, Sajid M. Analytic solution for axisymmetric flow and heat transfer of a second-grade fluid past a stretching sheet. Int J Heat Mass Transf. 2007;50(1):75–84.

    Google Scholar 

  24. Saif RS, Hayat T, Ellahi R, Muhammad T, Alsaedi A. Stagnation-point flow of second grade nanofluid towards a nonlinear stretching surface with variable thickness. Results Phys. 2017;7:2821–30.

    ADS  Google Scholar 

  25. Wakif A. A novel numerical procedure for simulating steady MHD convective flows of radiative Casson fluids over a horizontal stretching sheet with irregular geometry under the combined influence of temperature-dependent viscosity and thermal conductivity. Math Problem Eng. 2020. https://doi.org/10.1155/2020/1675350.

    Article  MathSciNet  Google Scholar 

  26. Hayat T, Ahmad S, Khan MI, Alsaedi A. Non-Darcy Forchheimer flow of ferromagnetic second-grade fluid. Results Phys. 2017;7:3419–24.

    ADS  Google Scholar 

  27. Jamshed W, Nisar KS, Gowda RJP, Kumar RN, Prasannakumara BC. Radiative heat transfer of second grade nanofluid flow past a porous flat surface: a single-phase mathematical model. Phys Scr. 2021;96(6): 064006.

    ADS  Google Scholar 

  28. Gowda RJP, Baskonus HM, Kumar RN, Prasannakumara BC, Prakasha DG. Computational investigation of Stefan blowing effect on flow of second-grade fluid over a curved stretching sheet. Int J Appl Comput Math. 2021;7(3):109.

    MathSciNet  Google Scholar 

  29. Khan SU, Tlili I, Waqas H, Imran M. Effects of nonlinear thermal radiation and activation energy on modified second-grade nanofluid with Cattaneo-Christov expressions. J Therm Anal Calorim. 2021;143:1175–86.

    Google Scholar 

  30. Khan AA, Khan MN, Ahsan N, Khan MI, Muhammad T, Rehman A. Heat and mass transfer features of transient second-grade fluid flow through an exponentially stretching surface. Pramana-J Phys. 2022;96:58.

    ADS  Google Scholar 

  31. Abbas SZ, Waqas M, Thaljaoui A, Zubair M, Riahi A, Chu YM, Khan WA. Modeling and analysis of unsteady second-grade nanofluid flow subject to mixed convection and thermal radiation. Soft Comput. 2022;26:1033–42.

    Google Scholar 

  32. Noureddine E, Cherlacola SR, Ahmed A, Sayed E, Taseer M, Wakif A. A passive control approach for simulating thermally enhanced Jeffery nanofluid flows nearby a sucked impermeable surface subjected to buoyancy and Lorentz forces. Case Stud Thermal Eng. 2023;47: 103106.

    Google Scholar 

  33. Sakiadis BC. Boundary-layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface. AIChE J. 1961;7:221–5.

    ADS  CAS  Google Scholar 

  34. Crane LJ. Flow past a stretching plate. J Appl Math Phys (ZAMP). 1970;21:645–7.

    Google Scholar 

  35. Sajid M, Ali N, Javed T, Abbas Z. Stretching a curved surface in a viscous fluid. Chin Phys Lett. 2010;27: 024703.

    ADS  Google Scholar 

  36. Hayat T, Saif RS, Ellahi R, Muhammad T, Ahmad B. Numerical study for Darcy-Forchheimer flow due to a curved stretching surface with Cattaneo-Christov heat flux and homogeneous-heterogeneous reactions. Results Phys. 2017;7:2886–92.

    ADS  Google Scholar 

  37. Hayat T, Saif RS, Ellahi R, Muhammad T, Ahmad B. Numerical study of boundary-layer flow due to a nonlinear curved stretching sheet with convective heat and mass conditions. Results Phys. 2017;7:2601–6.

    ADS  Google Scholar 

  38. Reddy JVR, Sugunamma V, Sandeep N. Effect of frictional heating on radiative ferrofluid flow over a slendering stretching sheet with aligned magnetic field. Eur Phys J Plus. 2017;132:7.

    Google Scholar 

  39. Hayat T, Saif RS, Ellahi R, Alsaedi A, Muhammad T. Homogeneous-heterogeneous reactions in MHD flow of micropolar fluid by a curved stretching surface. J Mol Liq. 2017;240:209–20.

    CAS  Google Scholar 

  40. Saba F, Ahmed N, Hussain S, Khan U, Mohyud-Din ST, Darus M. Thermal analysis of nanofluid flow over a curved stretching surface suspended by carbon nanotubes with internal heat generation. Appl Sci. 2018;8:395.

    Google Scholar 

  41. Saif RS, Muhammad T, Sadia H, Ellahi R. Hydromagnetic flow of Jeffrey nanofluid due to a curved stretching surface. Phys A: Stat Mech Appl. 2020;551:0378–4371.

    MathSciNet  Google Scholar 

  42. Raza R, Mabood F, Naz R, Abdelsalam SI. Thermal transport of radiative Williamson fluid over stretchable curved surface. Therm Sci Eng Prog. 2021;23:2451–9049.

    Google Scholar 

  43. Rosseland S. Astrophysik Und Atom-Theoretische Grundlagen. Berlin, Germany: Springer; 1931.

    Google Scholar 

  44. Magyari E, Pantokratoras A. Note on the effect of thermal radiation in the linearized Rosseland approximation on the heat transfer characteristics of various boundary layer flows. Int Commun Heat Mass Transf. 2011;38:554–6.

    Google Scholar 

  45. Nagaraja B, Gireesha BJ. Exponential space-dependent heat generation impact on MHD convective flow of Casson uid over a curved stretching sheet with chemical reaction. J Therm Anal Calorim. 2021;143:4071–9.

    CAS  Google Scholar 

  46. Abbas Z, Naveed M, Sajid M. Heat transfer analysis for stretching flow over a curved surface with magnetic field. J Eng Thermophys. 2013;22:337–45.

    CAS  Google Scholar 

Download references

Acknowledgment

The authors extend their appreciation to the research unit at King Khalid University for funding this work through Project number 495 and the authors acknowledge the Research Center for Advanced Materials Science (RCAMS) at King Khalid University, Saudi Arabia for their valuable technical support.

Author information

Authors and Affiliations

  1. School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad, Pakistan

    Zain Ul Abideen

  2. Department of Humanities and Sciences, School of Electrical Engineering and Computer Science (SEECS), National University of Sciences and Technology (NUST), Islamabad, Pakistan

    Rai Sajjad Saif

  3. Department of Mathematics, College of Sciences, King Khalid University, Abha, 61413, Saudi Arabia

    Taseer Muhammad

Authors
  1. Zain Ul Abideen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rai Sajjad Saif
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Taseer Muhammad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rai Sajjad Saif.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abideen, Z.U., Saif, R.S. & Muhammad, T. Second-grade fluid with carbon nanotubes flowing over an elongated curve surface possessing thermal radiation and internal heat generation effects. J Therm Anal Calorim 149, 1239–1250 (2024). https://doi.org/10.1007/s10973-023-12779-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-023-12779-w

Keywords

Search

Navigation