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An effect of velocity slip and MHD on Hiemenz stagnation flow of ternary nanofluid with heat and mass transfer

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Abstract

The present study examines the effect of the Hiemenz stagnation point and magnetic field on the ternary nanofluid flow on the permeable stretching/shrinking surface. Ternary nanofluid composites are formed by dissolving silicon dioxide (SiO2), aluminum oxide (Al2O3), and titanium dioxide (TiO2) in base fluid water (H2O), and applying these nanoparticles to the water (base fluid) will enhance heat transfer. Furthermore, it studied the heat transmission process using variable thermal conductivity of the thermal radiation. It is noted that there is a lack of study on ternary nanofluids in the circumstances of the Hiemenz stagnation point and radiation with porous media. The novelty of the present problem is to examine the influence of magnetic field and radiation over ternary nanofluid flow. Governing equations of velocity and temperature are converted to a set of nonlinear ordinary differential equations via suitable transformations, and the obtained equations are solved using the boundary conditions, energy equation with radiation effect solved analytically by using error function and hypergeometric function. Significant physical characteristics like mass transpiration, Schmidt number, magnetic parameter, and thermal radiations, volume fraction can be discussed using the graphical analysis. The outcomes of the investigation reveal that increasing the magnetic field enhances temperature and decreases the momentum of the fluid flow. Increasing the volume fraction and thermal radiation increases the thermal boundary layer, and velocity decreases by increasing in inverse Darcy parameter, ternary nanofluids significantly increase thermal conductivity and can be used as coolants for radiators due to their improved thermal performance. Finally, adding a magnetic field causes a moving conductive fluid to conduct current, which in turn creates forces on the fluid. This is particularly useful for green and sustainable development in a variety of engineering applications as well as biomedical disciplines like medication delivery and energy efficiency.

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Abbreviations

a :

Acceleration rate (s−1)

\(A_{1} ,A_{2} ,A_{3} ,A_{4} ,A_{5}\) :

Constants (–)

\(b\) :

Slip \(\left( { = D\sqrt {\frac{U}{\nu }} } \right)\) (–)

C :

Concentration (–)

C p :

Specific heat coefficient (JK−1 kg−1)

\(C^{*}\) :

Elasticity strength parameter \(\left( { = \frac{p}{U}} \right)\) (–)

d :

Stretching/shrinking parameter (–)

\(Da^{ - 1}\) :

: Inverse Darcy number \(\left( { = \frac{{\nu_{{\text{f}}} }}{{k^{*} a}}} \right)\) (m−2)

D B :

Mass diffusivity (–)

\(f\) :

Velocity function fluid phase (ms−1)

h :

Constant (–)

j :

Constant (–)

k :

Permeability of porous medium (m2)

m :

Constant (–)

n :

Constant (–)

N r :

Radiation \(\left( { = \frac{{16\sigma * T_{\infty }^{3} }}{{3k * \kappa_{{\text{f}}} }}} \right)\) (K)

P :

Constant (–)

Pr:

Prandtl number \(\left( { = \frac{{\nu_\mathrm{f} }}{{\alpha_\mathrm{f} }}} \right)\) (Kg m−3)

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

Radiative heat flux \(\left( { = - \frac{4\sigma *}{{3k*}}\frac{{\partial T^{4} }}{\partial y}} \right)\) (Wm−2)

Sc:

Schmidt number \(\left( { = \frac{{\nu_{{\text{f}}} }}{D}} \right)\) (–)

\(T_{{\text{w}}}\) :

Surface temperature (K)

T :

Fluid temperature (K)

\(T_{\infty }\) :

Ambient temperature (K)

\(u,v\) :

x, y Axis momentum fluid phase (ms−1)

u w :

Velocity (ms−1)

v w :

Wall velocity \(v_{{\text{w}}} = - \left( {\sqrt {a\rho } } \right)\Lambda \phi_{\eta } \left( 0 \right).\) (ms−1)

x :

Coordinate along the sheet (m)

y :

Coordinate normal to the sheet (m)

\(\alpha\) :

Acceleration rate (\({\text{ms}}^{ - 1}\))

\(\beta\) :

Solution value (m)

\(\eta\) :

Similarity variable (–)

\(\gamma\) :

Heat coefficient \(\left( { = \frac{{c_{{\text{m}}} }}{{c_{{\text{p}}} }}} \right)\) (K)

\(\kappa_\mathrm{tnf}\) :

Thermal conductivity \(\left( {{\text{Wm}}^{ - 1} \;{\text{K}}^{ - 1} } \right)\)

\(\kappa^{*}\) :

Absorption coefficient \(\left( {{\text{Wm}}^{ - 1} \;{\text{K}}^{ - 1} } \right)\)

\(\mu_\mathrm{tnf}\) :

Dynamic viscosity \(\left( {{\text{Kg}}\;({\text{ms}})^{ - 1} } \right)\)

\(\nu_\mathrm{tnf}\) :

Kinematic viscosity \(\left( { = \frac{{\mu_{{{\text{tnf}}}} }}{{\rho_{{{\text{tnf}}}} }}} \right)\;\left( {{\text{m}}^{2} {\text{s}}^{ - 1} } \right)\)

\(\rho_\mathrm{tnf}\) :

Fluid density (kg m−3)

\(\left( {\rho C_{\text{p}} } \right)_\mathrm{tnf}\) :

Heat capacitance of fluid (JKg−1 K−1)

\(\psi\) :

Stream function (Kg (ms)−1)

\(\sigma^{*}\) :

Stephen Boltzmann constant \(\left( {{\text{Wm}}^{ - 2} \;{\text{K}}^{ - 4} } \right)\)

\(\xi\) :

Variable for concentration (–)

\(\phi\) :

Volume fraction (–)

\(\varphi\) :

Concentration for fluid phase (–)

θ :

Temperature for fluid phase (K)

HNF:

Hybrid nanofluid (–)

TNF:

Ternary nanofluid (–)

ODE:

Ordinary differential equation (–)

PDE:

Partial differential equation (–)

MHD:

Magnetohydrodynamic (–)

B.Cs:

Boundary conditions (–)

Sc:

Schmidt number (–)

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Acknowledgements

This work is funded by the Grant NRF2022-R1A2C2002799 of the National Research Foundation of Korea. The support provided by Davangere University, Davangere, India, and the German Jordanian University, Amman, Jordan, is highly acknowledged.

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Authors and Affiliations

  1. Department of Studies in Mathematics, Davangere University, Shivagangothri, Davangere, 577007, India

    S. M. Sachhin & U. S. Mahabaleshwar

  2. School of Electrical Engineering and Information Technology, German Jordanian University, Amman, Jordan

    Dia Zeidan

  3. School of Mechanical Engineering, Yeungnam University, Gyeongsan, 38541, Korea

    Sang W. Joo

  4. Dipartimento di Ingegneria, Universitàdegli Studidella Campania Luigi Vanvitelli, Aversa, CE, Italy

    O. Manca

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Contributions

SMS contributed to the conceptualization, data curation, investigation, methodology, writing—original draft; USM was involved in the conceptualization, investigation, methodology, supervision, writing, review, and editing; DZ assisted in writing, review, editing, supervision, methodology; SWJ performed writing, review, and editing, funding acquisition; OM contributed to writing, review, and editing.

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Sachhin, S.M., Mahabaleshwar, U.S., Zeidan, D. et al. An effect of velocity slip and MHD on Hiemenz stagnation flow of ternary nanofluid with heat and mass transfer. J Therm Anal Calorim (2024). https://doi.org/10.1007/s10973-024-12962-7

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