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Numerical scrutinization of Darcy–Forchheimer flow for trihybrid nanofluid comprising of \({\text{GO}} + {\text{ZrO}}_{2} + {\text{SiO}}_{2}\)/kerosene oil over the curved surface

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Abstract

Significant applications of chemical reaction in flow using a curve sheet include metabolic, plastic extrusion and biochemical engineering. A few instances include the management of moisture and temperature supplies during an agricultural fair and the transmission of energy in a drizzly cooling tower. The current study aims to examine the heat and mass transmission for Darcy–Forchheimer flow induced by the curved stretch surface looped in a circle of radius \(R\). The flow is influenced by heat generation and thermal radiation with the effects of chemical reactions. Kerosene oil (KO) is used as the base fluid while three different types of nanoparticles, \({\text{GO}}, \,{\text{ZrO}}_{2}\) and \({\text{SiO}}_{2}\) are mixed with it. Convective heat conditions are incorporated in this model. A curvilinear coordinates system is studied to build the mathematical model of the flow condition under certain assumptions and then solved using the bvp4c approach in MATLAB. The effects of physical and geometrical characteristics relevant to this analysis on dimensionless physical quantities of interest are discussed using the necessary graphs and tables. According to the results, the Darcy–Forchheimer and porosity parameters caused a fall in the velocity profile, whereas the curvature parameter was found to cause an enhancement in the velocity profile. Our finding exhibits that \({\text{GO}}\) has a greater influence on \({\text{ZrO}}_{2} \;{\text{and}}\;{\text{SiO}}_{2}\). Thermal distribution augments with growth in radiative flow and Biot number while has declined with the upsurge in curvature factor. Growth in curvature factor has supported the behavior of concentration distribution and has been opposed by an upsurge in chemically reactive factor and Schmidt number. A comparison of the results obtained here with those that have already been published in the relevant literature has been made with outstanding achievement.

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Data availability

All data used in this manuscript have been presented within the article.

Abbreviations

\({\text{Pr}}\) :

Prandtl number, (Dimensionless)

\({\text{Rd}}\) :

Radiation parameter, (Dimensionless)

\({K}^{*}\) :

Mean absorption, (Dimensionless)

\({\text{Sc}}\) :

Schmidt number, (Dimensionless)

\(\boldsymbol{\alpha }\) :

Curvature parameter, (Dimensionless)

\({\text{Fr}}\) :

Forchheimer number, (Dimensionless)

\({\varvec{\lambda}}\) :

Porosity parameter, (Dimensionless)

\({\text{Rd}}\) :

Thermal radiation, (Dimensionless)

\({{\text{Bi}}}_{1}\) :

Biot number, (Dimensionless)

\({\epsilon }_{1}\) :

Heat source/sink parameter, (Dimensionless)

\({{\text{Nu}}}_{{\text{s}}}\) :

Nusselt number, (Dimensionless)

\({{\text{Sh}}}_{{\text{s}}}\) :

Sherwood number, (Dimensionless)

\({\text{Kr}}\) :

Chemical reaction parameter, (Dimensionless)

\(F^{\prime}\) :

Velocity, (Dimensionless)

\(\vartheta \) :

Dimensionless temperature field, (Dimensionless)

\(\phi \) :

Dimensionless concentration field, (Dimensionless)

\({{\text{Cf}}}_{{\text{s}}}\) :

Skin friction coefficient, (Dimensionless)

\({\varphi }_{1}\) :

Nanoparticle volume fraction silicon dioxide (\({{\text{SiO}}}_{2}\)), (-)

\({\varphi }_{2}\) :

Nanoparticle volume fraction graphene oxide (Go), (-)

\({\varphi }_{3}\) :

Nanoparticle volume fraction of zirconium dioxide \({{\text{ZrO}}}_{2}\), (-)

R :

Radius, (m)

\({\tau }_{{\text{w}}}\) :

Wall shear stress, (\(\mathrm{kg }\,{{\text{m}}}^{-1} \,{{\text{s}}}^{-2}\))

\(u,v\) :

Component of velocity, (\(\mathrm{m }\,{{\text{s}}}^{-1}\))

\({c}_{{\text{p}}}\) :

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

\(g\) :

Acceleration due to gravity, (\(\mathrm{m }\,{{\text{s}}}^{-2}\))

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

Dynamic viscosity, (\({\text{kg}}\,{({\text{ms}})}^{-1}\))

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

Kinematic viscosity of KO, (\({{{\text{m}}}^{2}\,\mathrm{ s}}^{-1}\))

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

Thermal diffusivity, (\({{{\text{m}}}^{2}\,\mathrm{ s}}^{-1}\))

\(k\) :

Thermal conductivity, (\(\mathrm{kgm }\,\,{{\text{K}}}^{-1} \,{{\text{s}}}^{-3}\))

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

Density of nanofluid, (\(\mathrm{kg }\,{{\text{m}}}^{-3}\))

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

Stefan–Boltzmann constant, (\({{\mathrm{W\,m}}^{-1}\,\mathrm{ K}}^{-1}\))

\(T\) :

The temperature of the fluid, (\(K\))

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

Wall temperature, (\({\text{K}}\))

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

Ambient temperature, (\({\text{K}}\))

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

Surface heat, (\({\text{K}}\))

\(h\) :

Coefficient of heat transfer, (\({\mathrm{W \,k}}^{-1}\,{\mathrm{ m}}^{-2}\))

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

Radiative heat flux, (\( {\text{W}}\,({\text{m}})^{{ - 2}} \))

\(P\) :

Pressure, (\({\mathrm{N\, m}}^{-2}\))

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

Dynamic viscosity of KO, (\( {\text{kg}}\,({\text{ms}})^{{ - 1}} \))

\({v}_{{\text{thnf}}}\) :

Kinematic viscosity of KO, (\({{{\text{m}}}^{2}\,\mathrm{ s}}^{-1}\))

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

Thermal diffusivity of KO, (\({{{\text{m}}}^{2}\,\mathrm{ s}}^{-1}\))

\({k}_{{\text{thnf}}}\) :

Thermal conductivity of KO, (\(\mathrm{kg\,m }\,\,{{\text{K}}}^{-1} {{\text{s}}}^{-3}\))

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

Density of Nanofluid of KO, (\(\mathrm{kg }\,{{\text{m}}}^{-3}\))

\({{(C}_{{\text{p}}})}_{{\text{thnf}}}\) :

Specific heat capacity of KO, (\({\text{kg}}\,{\mathrm{ m}}^{-3}\))

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

Dynamic viscosity of KO, (\( {\text{kg}}\,({\text{ms}})^{{ - 1}} \))

\({v}_{{\text{hnf}}}\) :

Kinematic viscosity of KO, (\({{{\text{m}}}^{2}\,\mathrm{ s}}^{-1}\))

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

Thermal diffusivity of KO, (\({{{\text{m}}}^{2}\,\mathrm{ s}}^{-1}\))

\({k}_{{\text{hnf}}}\) :

Thermal conductivity of KO, (\(\mathrm{kg\,m }\,{{\text{K}}}^{-1}\,{\mathrm{ s}}^{-3}\))

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

Density of nanofluid of KO, (\({\text{kg}}\,{\mathrm{ m}}^{-3}\))

\({{(C}_{{\text{p}}})}_{{\text{hnf}}}\) :

Specific heat capacity of KO, (\({\text{kg}}\,{\mathrm{ m}}^{-3}\))

\(f\) :

Fluid base

\({\text{thnf}}\) :

Trihybrid nanofluid

\({\text{hnf}}\) :

Hybrid nanofluid

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

Nanofluid

\(s\) :

Solid phase

PDE:

Partial differential equation

ODE:

Ordinary differential equation

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Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP.2/505/44).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Federal Urdu University of Arts, Sciences and Technology, Gulshan-e-Iqbal Karachi, 75300, Pakistan

    Farhan Ali

  2. Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia

    Ali M. Mahnashi & Waleed Hamali

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

    Zehba Raizah

  4. Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Khyber Pakhtunkhwa, Pakistan

    Anwar Saeed

  5. College of Aeronautical Engineering, National University of Sciences and Technology (NUST), Sector H-12, Islamabad, 44000, Pakistan

    Arshad Khan

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Ali, F., Mahnashi, A.M., Hamali, W. et al. Numerical scrutinization of Darcy–Forchheimer flow for trihybrid nanofluid comprising of \({\text{GO}} + {\text{ZrO}}_{2} + {\text{SiO}}_{2}\)/kerosene oil over the curved surface. J Therm Anal Calorim (2024). https://doi.org/10.1007/s10973-024-13103-w

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