Skip to main content
SpringerLink
Log in

Arrhenius activation energy and Joule heating for Walter-B fluid with Cattaneo–Christov double-diffusion model

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

Abstract

The present framework examines the characteristics of Cattaneo–Christov double-diffusion model to nonlinearly convective flow of Walter-B nanofluid over a stretched sheet with Joule heating phenomena. Heat and mass transfer analysis is accomplished through activation energy, MHD, heat generation/absorption, nonlinear mixed convection, binary chemical reaction and dual stratification. The novel binary reaction model is implemented to reveal the characteristics of activation energy. The conversion of partial differential system to nonlinear ordinary differential system has been completed by utilizing appropriate transformations. The nonlinear systems have been solved through the shooting technique along with fifth-order Runge–Kutta method. The outcomes are then plotted for several values of the pertinent parameters and discussed deliberately. Furthermore, the skin friction coefficients and heat transfer rates have been computed and scrutinized. It is evident that heat transfer rate is noticeably higher for larger thermal relaxation parameter. Concentration distribution has reverse features for activation energy parameter and reaction rate constant. The thermal and solutal relaxation parameters become a source of reduction for temperature and concentration fields, respectively. From this analysis, it is observed that space- and temperature-dependent heat sinks are more suitable for cooling purposes.

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
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Abbreviations

\(A_{1}\) :

Activation energy parameter

\(B_{0}\) :

Magnetic field strength

\((B_{1} ,B_{2} )\) :

Heat generation/absorption parameters

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

Skin friction coefficient

\((D_{\text{B}} ,D_{\text{T}} )\) :

Brownian diffusion and thermophoretic coefficients

\(E_{2}^{*}\) :

Activation energy

\(k^{*}\) :

Boltzmann constant

\(\hat{k}_{2}\) :

Chemical reaction parameter

\(\hat{K}_{\text{r}}^{2}\) :

Chemical reaction rate constant

\(M\) :

Magnetic parameter

n :

Fitted rate constants,

\(\hat{N}_{\text{b}}\) :

Brownian motion parameter

\(\hat{N}_{\text{t}}\) :

Thermophoresis parameter

Pr:

Prandtl number

Sc:

Schmidt number

Shx :

Sherwood number

(S 1, S 2):

Thermal and solutal stratification parameters

U 0 :

Reference velocity

\(\alpha_{1}\) :

Viscoelastic parameter

\(\hat{\beta }\) :

Nonlinear thermal convection parameter

\(\hat{\beta }_{\text{c}}\) :

Nonlinear solutal convection parameter

\(\tau_{1}\) :

Specific heat ratio

\(\varTheta \left( {\chi_{1} } \right)\) :

Dimensionless temperature

\(\gamma_{1}\) :

Chemical reaction parameter

\(\lambda_{1}\) :

Buoyancy parameter

\(\delta\) :

Temperature relative parameter

\((\delta_{2} ,\delta_{3} )\) :

Thermal and solutal relaxation parameters

\(\xi_{1}\) :

Reaction rate constant

\(\varUpsilon \left( {\chi_{1} } \right)\) :

Dimensionless nanoparticle concentration

\((\varGamma_{1} ,\varGamma_{3} )\) :

Linear thermal expansion coefficient

\((\varGamma_{2} ,\varGamma_{4} )\) :

Nonlinear solutal expansion coefficient

\(\varPsi_{1}\) :

Stream function

References

  1. Fourier JBJ. Théorie Analytique De La Chaleur, Paris, 1822.

  2. Fick R. Medizinische Physik, 1856.

  3. Cattaneo C. Some aspects of diffusion theory, atti semin. Mat Fis Univ. Modena Reggio Emilia. 1948;3:83–101.

    Google Scholar 

  4. Christov CI. On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction. Mech Res Commun. 2009;39:481–6.

    Article  Google Scholar 

  5. Ijaz M, Ayub M. Nonlinear convective stratified flow of Maxwell nanofluid with activation energy. Heliyon. 2019. https://doi.org/10.1016/j.heliyon.2019.e01121.

    Article  PubMed  PubMed Central  Google Scholar 

  6. Hayat T, Muhammad T, Alsaedi A, Ahmad B. Three-dimensional flow of nanofluid with Cattaneo–Christov double diffusion. Results Phys. 2016;6:897–903.

    Article  Google Scholar 

  7. Sui J, Zheng L, Zhang X. Boundary layer heat and mass transfer with Cattaneo–Christov double-diffusion in upper-convected Maxwell nanofluid past a stretching sheet with slip velocity. Int J Therm Sci. 2016;104:461–8.

    Article  Google Scholar 

  8. Ijaz M, Ayub M. Activation energy and dual stratification effects for Walter-B fluid flow in view of Cattaneo–Christov double diffusion. Heliyon. 2019;5:e01815. https://doi.org/10.1016/j.heliyon.2019.e01815.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Khan AA, Bukhari SR, Marin M, Ellahi R. Effects of chemical reaction on third grade magnetohydrodynamics fluid flow under the influence of heat and mass transfer with variable reactive index. Heat Transf. Res. 2019;50(11):1061–80.

    Article  Google Scholar 

  10. Ijaz M, Ayub M, Zubair M, Malik MY. Numerical simulation of Joule heating and Arrhenius activation energy for nonlinear radiative flow of Casson nanofluid with Cattaneo–Christov heat flux model. Scr: Phys; 2019.

    Google Scholar 

  11. Khan AA, Masood F, Ellahi R, Bhatti MM. Mass transport on chemicalized fourth-grade fluid propagating peristaltically through a curved channel with magnetic effects. J Mol Liq. 2018;258:186–95.

    Article  CAS  Google Scholar 

  12. Nayak MK, Dash GC, Singh LP. Heat and mass transfer effects on MHD viscoelastic fluid over a stretching sheet through porous medium in presence of chemical reaction. Propul. Power Res. 2016;5(1):70–80.

    Article  Google Scholar 

  13. Ravindran R, Samyuktha N. Unsteady mixed convection flow over stretching sheet in presence of chemical reaction and heat generation or absorption with non-uniform slot suction or injection. Appl. Math. Mech. 2010;36:1253–72.

    Article  Google Scholar 

  14. Zubair M, Ijaz M, Abbas T, Riaz A. Analysis of modified Fourier law in flow of ferromagnetic Powell-Eyring fluid considering two equal magnetic dipoles. Can J Phys. 2018. https://doi.org/10.1139/cjp-2018-0586.

    Article  Google Scholar 

  15. Abbas N, Malik MY, Nadeem S. Transportation of magnetized micropolar hybrid nanomaterial fluid flow over a riga curface surface. Comput Methods Prog Biomed. 2019;185:105136. https://doi.org/10.1016/j.cmpb.2019.105136.

    Article  Google Scholar 

  16. Khan MN, Nadeem S. Theoretical treatment of bio-convective Maxwell nanofluid over an exponentially stretching sheet. Can J Phys. 2019. https://doi.org/10.1139/cjp-2019-0380.

    Article  Google Scholar 

  17. Khan LA, Raza M, Mir NA, Ellahi R. Effects of different shapes of nanoparticles in peristaltic flow of MHD nanofluids filled in an asymmetric channel: a novel mode for heat transfer enhancement. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-019-08348-9.

    Article  Google Scholar 

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

    Article  Google Scholar 

  19. 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.

    Article  CAS  Google Scholar 

  20. Alblawi A, Malik MY, Nadeem S, Abbas N. Buongiorno’s nanofluid model over a curved exponentially stretching surface. Process. 2019;7(10):665. https://doi.org/10.3390/pr7100665.

    Article  CAS  Google Scholar 

  21. Ijaz M, Ayub M. Simulation of magnetic dipole and dual stratification in radiative flow of ferromagnetic Maxwell fluid. Heliyon. 2009;5:e01465. https://doi.org/10.1016/j.heliyon.2019.e01465.

    Article  Google Scholar 

  22. Subhani M, Nadeem S. Numerical investigation of unsteady MHD flow of micropolar hybrid nanofluid in porous medium. Phys. Scr. 2019. https://doi.org/10.1088/1402-4896/ab154a.

    Article  Google Scholar 

  23. Nadeem S, Abbas N, Elmasry Y, Malik M. Y (2020) Numerical analysis of water based CNTs flow of micropolar fluid through rotating frame. Comput Method Prog Biomed 186: 105194. https://doi.org/10.1016/j.cmpb.2019.105194.

  24. Ellahi R, Sait SM, Shehzad N, Mobin N. Numerical simulation and mathematical modeling of electroosmotic Couette-Poiseuille flow of MHD power-law nanofluid with entropy generation. Symmetry. 2019;11(8):1038.

    Article  CAS  Google Scholar 

  25. Ahmed Z, Nadeem S, Saleem S, Ellahi R. Numerical study of unsteady flow and heat transfer CNT based MHD nanofluid with variable viscosity over a permeable shrinking surface. Int J Numer Methods Heat Fluid Flow. 2019;29(12):4607–23.

    Article  Google Scholar 

  26. Beard WD, Walters K. Elastico-viscous boundary layer flows. Part I. Two-dimensional flow near a stagnation point. Proc Camb Philos Soc 1964;60:667–74.

    Article  Google Scholar 

  27. Hayat T, Ijaz M, Qayyum S, Ayub M, Alsaedi A. Mixed convective stagnation point flow of nanofluid with Darcy-Fochheimer relation and partial slip. Results Phys. 2018;9:771–8.

    Article  Google Scholar 

  28. Ijaz M, Ayub M. Thermally stratified flow of Jeffrey fluid with homogeneous-heterogeneous reactions and non-Fourier heat flux model. Heliyon. 2019;5:e02303.

    Article  CAS  Google Scholar 

  29. Hayat T, Nadeem S. Flow of 3D Eyring-Powell fluid by utilizing Cattaneo–Christov heat flux model and chemical processes over an exponentially stretching surface. Results Phys. 2017;8:397–403.

    Article  Google Scholar 

  30. Ijaz M, Ayub M. Cattaneo–Christov double diffusion model for viscoelastic nanofluid with activation energy and nonlinear thermal radiation. Multidiscip Model Mater Struct. 2019. https://doi.org/10.1108/MMMS-03-2019-0046.

    Article  Google Scholar 

  31. Ellahi R, Zeeshan A, Hussain F, Asadollahi A. Peristaltic blood flow of couple stress fluid suspended with nanoparticles under the influence of chemical reaction and activation energy. Symmetry. 2019;11(2):11020276.

    Article  Google Scholar 

  32. Muhammad N, Nadeem S, Issakhov A. Finite volume method for mixed convection flow of Ag-ethylene glycol nanofluid flow in a cavity having thin central heater. Physica A: Stat Mech Appl 2020; 537 122738. https://doi.org/10.1016/j.physa.2019.122738.

  33. Yousaf MA, Ismael HF, Abbas T, Ellahi R. Numerical study of momentum and heat transfer of MHD Carreau nanofluid over exponentially stretched plate with internal heat source/sink and radiation. Heat Transf Res. 2019;50(7):649–58.

    Article  Google Scholar 

  34. Ijaz M, Ayub M, Zubair M, Riaz A. On stratified flow of ferromagnetic nanofluid with heat generation/absorption. Phys Scr. 2019. https://doi.org/10.1088/1402-4896/aaf6df.

    Article  Google Scholar 

  35. Shahzadi I, Suleman S, Saleem S, Nadeem S. Utilization of Cu-nanoparticles as medication agent to reduce atherosclerotic lesions of a bifurcated artery having compliant walls. Comput Methods Prog Biomed. 2019;184:105123. https://doi.org/10.1016/j.cmpb.2019.105123.

    Article  Google Scholar 

  36. Asadollahi A, Esfahani JA, Ellahi R. Evacuation liquid coatings from a diffusive oblique fin in micro/mini-channels: an application of condensation cooling process. J Therm Anal Calorim. 2019;138:255–63.

    Article  CAS  Google Scholar 

  37. Nadeem S, Alblawi A, Muhammad N, Alarifi IM, Issakhov A, Mustafa MT. A computational model for suspensions of motile micro-organisms in the flow of ferrofluid. J Mol Liq. 2019. https://doi.org/10.1016/j.molliq.2019.112033.

    Article  Google Scholar 

  38. Rashidi S, Eskandarian M, Mahian O, Poncet S. Combination of nanofluid and inserts for heat transfer enhancement. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7070-9.

    Article  Google Scholar 

  39. Sarafraz MM, Pourmehran O, Yang B, Arjomandi M, Ellahi R. Pool boiling heat transfer characteristics of iron oxide nano-suspension under constant magnetic field. Int J Therm Sci. 2020; 147: 106131. https://doi.org/10.1016/j.ijthermalsci.2019.106131.

  40. Makinde OD, Aziz A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int J Therm Sci. 2011;50:1326–32.

    Article  CAS  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through general research program under Grant Number (G.R.P. 70-41).

Author information

Authors and Affiliations

  1. Department of Mathematics, Quaid-I-Azam University 45320, Islamabad, 44000, Pakistan

    M. Ijaz

  2. Avicenna Medical College, Lahore, Pakistan

    M. Yousaf

  3. Chemistry Department, Faculty of Science and Arts, King Khalid University, Sarat Ebidah, Saudi Arabia

    A. M. El Shafey

Authors
  1. M. Ijaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Yousaf
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. M. El Shafey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Ijaz.

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

Ijaz, M., Yousaf, M. & El Shafey, A.M. Arrhenius activation energy and Joule heating for Walter-B fluid with Cattaneo–Christov double-diffusion model. J Therm Anal Calorim 143, 3687–3698 (2021). https://doi.org/10.1007/s10973-020-09270-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10973-020-09270-1

Keywords

Search

Navigation