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Stability analysis and implication of Darcy magnetic-radiative hybrid reactive nanofluid heat transfer over a shrinkable surface with Ohmic heating

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Abstract

This study focused on Al\(_2\)O\(_3\)–Cu/water-based hybrid nanofluid to address the issue related to heat-mass transfer over a Darcy shrinkable surface with the magnetic field with thermal radiation effects. Variable thermal conductivity, Joule dissipation, chemical reaction with velocity, and thermal slip condition for hybrid nanofluid are also considered here. The governing equations of this study are reduced by utilizing similarity transformations before being solved numerically by the bvp4c function in MATLAB. The results also indicate the existence of multiple solutions in the shrinking sheet region for a certain amount of mass suction parameter, where the solution of the upper branch was stable while unstable for the lower branch. The influences of the chosen parameters on the velocity, temperature, skin friction coefficient, local Nusselt number, and Sherwood number are addressed and graphically illustrated. Further, it is found that due to an increase in Al\(_2\)O\(_3\) nanoparticle volume fraction, the coefficient of skin friction with heat transfer rate boosts up, whereas the Sherwood number decays. The results also reveal that increasing values of shrinking parameter decline the local Nusselt number but upsurges skin friction coefficient. Moreover, the hybrid nanofluid velocity decreases (increases) in the first (second) solution for the enhancement of the permeability parameter. The initial solution’s positive minimum eigenvalue was revealed by stability analysis, which distinctly defined a stable and feasible flow. Boundary layer hybrid nanofluid flow in industrial applications such as extrusion processes is attributable to impulsive movement of an extensible moving surface. These results are crucial in the long term because they allow us to optimize heat transmission for cooling and heating applications, to enhance industrial growth, especially in the manufacturing and processing sectors.

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Abbreviations

\(A_1\) :

Velocity slip parameter

\(B_1\) :

Temperature slip factor

B :

Temperature slip parameter

\(B_0\) :

Magnitude of the magnetic field strength

C :

Naofluid concentrattion at the surface

\(C_\textrm{p}\) :

Specific heat in constant pressure (J kg\(^{-1}\) K\(^{-1}\))

\(C_\textrm{f}\) :

Local skin-friction coefficient

\(D_\textrm{hnf}\) :

Hybrid nanofluid mass diffusivity

\(D_\textrm{nf}\) :

Nanofluid mass diffusivity

\(D_\textrm{f}\) :

Mass diffusivity of fluid

L :

Reference length of the sheet (m)

Kc :

Chemical reaction rate

\(R_\textrm{c}\) :

Chemical reaction parameter

\(h_\textrm{f}\) :

Coefficient of heat transfer

M :

Magnetic field parameter

\(\text{Nu}_\textrm{x}\) :

Nusselt number

Nr :

Thermal radiation parameter

P :

Porous/Darcy parameter

Pr:

Prandtl number

\(Q_0\) :

Heat source/sink coefficient

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

Thermal radiative heat flux (W m\(^{-2}\))

\(\text{Re}_\textrm{x}\) :

Local Reynolds number

\(\textrm{Re}_\textrm{L}\) :

Reynolds number

S :

Suction or injection parameter

Sc:

Schmidt number

\(\textrm{Sh}_\textrm{x}\) :

Sherwood number

t :

Time (s)

\(T_\textrm{w}(x)\) :

Shrinking surface temperature (K)

T :

Nanofluid temperature (K)

\(T_{\infty }\) :

Free-stream temperature (K)

\(u_\textrm{w}(x)\) :

Shrinking sheet velocity (ms\(^{-1}\))

u,  v :

x-, y-component of fluid velocity (ms\(^{-1}\))

x, y :

Coordinates along the interface and normal to it

\({\phi _1}\) :

Al\(_{2}\)O\(_{3}\) nanoparticles’s solid volume fraction

\({\phi _2}\) :

Cu nanoparticles’s solid volume fraction

\(\eta\) :

Similarity variable

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

Mono nanofluid’s heat capacitance (J kg\(^{-1}\) K\(^{-1}\))

\((\rho C_\textrm{p})_\textrm{hnf}\) :

Hybrid nanofluid’s heat capacitance (J kg\(^{-1}\) K\(^{-1}\))

\(\kappa _\textrm{f}\) :

Thermal conductivity of fluid (Wm\(^{-1}\)K)

\(\kappa _\textrm{nf}\) :

Mono nanofluid’s thermal conductivity (Wm\(^{-1}\)K)

\(\kappa _\textrm{hnf}\) :

Hybrid nanofluid’s thermal conductivity (Wm\(^{-1}\)K)

\({\kappa ^*}_\textrm{hnf}\) :

Hybrid nanofluid’s variable thermal conductivity (Wm\(^{-1}\) K)

\(\kappa _\textrm{s}\) :

Nanoparticle’s thermal conductivity (Wm\(^{-1}\)K)

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

Fluid dynamic viscosity (kg m\(^{-1}\) s\(^{-1})\)

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

Nono nanofluid’s effective viscosity (kg m\(^{-1}\) s\(^{-1})\)

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

Hybrid nanofluid’s effective viscosity (kg m\(^{-1}\) s\(^{-1})\)

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

Kinematic fluid viscosity, \(\mu _\textrm{f}/\rho _\textrm{f}\) (\(\textrm{m}^2 \,\textrm{s}^{-1})\)

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

Fluid density (kg m\(^{-3}\))

\(\rho _\textrm{nf}\) :

Mono nanofluid’s effective density (kg m\(^{-3}\))

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

Hybrid nanofluid’s effective density of (kg m\(^{-3}\))

\(\rho _\textrm{s}\) :

Density of nanoparticles (kg m\(^{-3}\))

\(\beta\) :

Heat source/sink parameter

\(\lambda\) :

Shrinking parameter

\(\gamma\) :

Eigenvalue

\(\gamma _1\) :

Minimum eigenvalue

\(\Phi (\eta )\) :

Dimensionless concentration profile

\(\sigma ^*\) :

Stefan–Boltzmann constant

\(\sigma\) :

Electrical conductivity of the fluid (\(\Omega \,\textrm{m}^{-1}\))

\(\sigma _\textrm{nf}\) :

Mono nanofluid’s electrical conductivity (\(\Omega \,\textrm{m}^{-1}\))

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

Hybrid nanofluid’s electrical conductivity (\(\Omega \,\textrm{m}^{-1}\))

\(\theta (\eta )\) :

Dimensionless temperature

\(\psi\) :

Stream function

\(\gamma\) :

Eigenvalue parameter

\(\tau\) :

Dimensionless time variable

\(\epsilon\) :

Variable thermal conductivity parameter

f:

Fluid fraction

nf:

Mono nanofluid

hnf:

Hybrid nanofluid

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Authors and Affiliations

  1. Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal, 731 235, India

    Dulal Pal

  2. Siksha Satra, Visva-Bharati University, Sriniketan, West Bengal, 731236, India

    Gopinath Mandal

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Dulal Pal: Former Professor at Visva-Bharati University.

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Pal, D., Mandal, G. Stability analysis and implication of Darcy magnetic-radiative hybrid reactive nanofluid heat transfer over a shrinkable surface with Ohmic heating. J Therm Anal Calorim 148, 2087–2104 (2023). https://doi.org/10.1007/s10973-022-11797-4

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