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A comparative investigation of magnetohydrodynamics slip \(F{e}_{3}{O}_{4}-{H}_{2}O\) and \(A{l}_{2}{O}_{3}-{H}_{2}O\) nanofluids flow and heat transfer over an unsteady vertically curved stretching surface

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Abstract

This study investigates the impact of various shapes of \(F{e}_{3}{O}_{4}\) and \(A{l}_{2}{O}_{3}\) nanometers immersed in water \(({H}_{2}O)\) base fluid over an unsteady vertically curved stretching surface. The article also covers the effects of velocity slip and variable magnetohydrodynamics (MHD). The mathematical model is based on fundamental principles, specifically the conservation of mass, momentum, and energy. We have transformed the modeled governing partial differential equations into highly nonlinear ordinary differential equations using suitable similarity conversions. The BVP4C program in MATLAB is used to generate numerical solutions for significant physical parameters, which can be visualized as graphs. The bar graphs are also drawn to analyze the impact of physical emerging parameters on skin friction coefficient and Nusselt Numbers. The result is also validated by comparing the present results with the available literature and found to be in good accord. It has been found that the addition of \(F{e}_{3}{O}_{4}\) and \(A{l}_{2}{o}_{3}\) nanoparticles improves the thermal conductivity of the base fluid \({H}_{2}O\). This rise is more pronounced for platelet-shaped \(F{e}_{3}{O}_{4}\) nanoparticles. The shape and the size of the nanoparticles play a crucial role in the heat transfer mechanism. Furthermore, the brick-shaped \(A{l}_{2}{O}_{3}-{H}_{2}O\) nanofluid exhibits the lowest velocity, the greatest temperature, and the lowest skin friction coefficient.

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Abbreviations

\(r,s,z\) :

Curvilinear coordinates

\(U\) :

Stretching velocity \([{{\text{ms}}}^{-1}]\)

\(k\) :

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

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

Specific heat \([{{\text{Jkg}}}^{-1} {{\text{K}}}^{-1} ]\)

\(\kappa\) :

Slip parameter

\(S\) :

Unsteady parameter

\(M\) :

Magnetic parameter

\({B}_{0}\) :

Strength of magnetic field

\(T\) :

Fluid temperature \([{\text{K}}]\)

\(F{e}_{3}{O}_{4}\) :

Ferrous ferric oxide or Iron oxide

\(A{l}_{2}{O}_{3}\) :

Aluminum oxide

\({H}_{2}O\) :

Hydrogen oxide (water)

\(a, \alpha , L\) :

Positive dimensional constants \([{{\text{s}}}^{-1}]\)

\(\theta\) :

Non-dimensional temperature

\(\sigma\) :

Electrical conductivity \([{{\text{Sm}}}^{-1}]\)

\(\rho\) :

Density \([{{\text{kgm}}}^{-3}]\)

\(\alpha\) :

Thermal diffusivity

\(m\) :

Shape factor

R:

Radius of curvature

\({C}_{{f}_{s}}\) :

Local skin friction coefficient

\(N{u}_{s}\) :

Local Nusselt number

\(Gr\) :

Grashof number

\(K\) :

Curvature

\(Pr\) :

Prandtl number

\(\mu\) :

Dynamic viscosity \([{{\text{kgm}}}^{-1}{{\text{s}}}^{-1}]\)

\(u,v\) :

Components of velocity [\({{\text{ms}}}^{-1}\)]

\(P\) :

Pressure \([{{\text{kgm}}}^{-1}{{\text{s}}}^{-2}]\)

\(\phi\) :

Volumetric fraction

\(\eta\) :

Non-dimensional space variables

\(\tau\) :

Sheer stress \([{{\text{Nm}}}^{-2}]\)

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

Heat flux \([{{\text{Wm}}}^{-2}]\)

\(\nu\) :

Kinematic viscosity \([{{\text{m}}}^{2}{{\text{s}}}^{-1}]\)

BL:

Boundary layer

\(MHD\) :

Magnetohydrodynamic

\(nf, f\),ref:

Nanofluid, fluid, reference

\(s\) :

Solid nanoparticles

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Authors and Affiliations

  1. Department of Mathematics, The Sahara University, Narowal, 51600, Pakistan

    Shakil Shaiq

  2. Department of Basic Sciences, University of Engineering and Technology, Taxila, 47050, Pakistan

    Azeem Shahzad & Umer Hayat

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  1. Shakil Shaiq
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  2. Azeem Shahzad
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  3. Umer Hayat
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Contributions

Shakil Shaiq Modled the Problem and also wrote the discussion section. Azeem Shahzad wrote the Introduction Section and Numerical Procedure. Umer Hayat Prepared all the Figures. All authors reviewed the manuscript.

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Correspondence to Shakil Shaiq.

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Shaiq, S., Shahzad, A. & Hayat, U. A comparative investigation of magnetohydrodynamics slip \(F{e}_{3}{O}_{4}-{H}_{2}O\) and \(A{l}_{2}{O}_{3}-{H}_{2}O\) nanofluids flow and heat transfer over an unsteady vertically curved stretching surface. Multiscale and Multidiscip. Model. Exp. and Des. (2024). https://doi.org/10.1007/s41939-023-00352-9

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