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Unsteady MHD flow of a Williamson nanofluid on a permeable stretching surface with radiation and chemical reaction effects

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Abstract

In the present work, we emphasize the impacts of an inclined magnetic field, viscous dissipation and radiation on the unsteady flow of a Williamson nanofluid over a vertical stretching porous surface with the presence of non-uniform heat source/sink and chemical reaction. In this study, we considered different kinds of nanoparticles such as silver, copper, aluminium oxide (\(\hbox {Al}_{2}\hbox {O}_{3})\), titanium oxide (\(\hbox {TiO}_{2})\), and magnesium oxide (MgO). The basic equations of this investigation are transmuted into a system of nonlinear and coupled ODEs using suitable similarity variables and elucidated numerically by R.-K. Fehlberg-based shooting technique. Influences of the pertinent parameters on the velocity, the temperature and the concentration distributions are deliberated with the assistance of graphs and tables. This study depicts that \(\hbox {Al}_{2}\hbox {O}_{3}\) nanofluid has greater velocity since it has less dense nanoparticles compared to other nanoparticles. However, Cu-nanofluid has greater heat transfer due to greater thermal conductivity. Further, we identified that the thermal boundary layer thickness can be increased with the help of the viscous dissipation parameter. The inclination angle of the magnetic field strengthens the magnetic field on the fluid flow

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Abbreviations

u, v :

Velocities in x- and y-direction (m/s)

g :

Acceleration due to gravity (\(\hbox {ms}^{{-2}})\)

\(\phi \) :

Nanoparticle solid volume fraction (nm)

\(\rho _\mathrm{f},\rho _\mathrm{s} \) :

Densities of the base fluid and solid nanoparticles \((\hbox {kg}/\hbox {m}^{3})\)

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

Kinematic viscosity of the fluid (\(\hbox {m}^{2}\)/s)

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

Dynamic viscosity of the base fluid (kg/ms)

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

Density of the nanofluid (kg/ \(\hbox {m}^{3}\)K)

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

Effective dynamic viscosity of nanofluid (kg/ms)

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

Electrical conductivity of the nanofluid (S/m)

\(\alpha _\mathrm{nf}\) :

Thermal diffusivity of the nanofluid

\(\beta _\mathrm{nf}\) :

Thermal expansion of the nanofluid

\(k_\mathrm{nf}\) :

Thermal conductivity of the nanofluid (W/m K)

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

Specific heat capacity (J/kg K)

\(\sigma _\mathrm{f},\sigma _\mathrm{s}\) :

Electrical conductivities of the base fluid and solid fraction

\((\rho \beta )_\mathrm{f} ( \rho \beta )_\mathrm{s}\) :

Thermal expansion coefficient of the base fluid and solid fraction

\({{(\rho C}_\mathrm{p})}_\mathrm{f}, {{(\rho C}_\mathrm{p})}_\mathrm{s}\) :

Heat capacities of the base fluid and solid fraction

\({k}_\mathrm{{f}}\), \({k}_{\mathrm{s}}\) :

Thermal conductivities of the base fluid and Solid fraction

k :

Thermal conductivity

\(\eta \) :

Similarity variable

f :

Dimensionless velocity

\(\theta \) :

Dimensionless temperature

\({\Phi }\) :

Dimensionless concentration

\(A=\frac{\alpha }{a}\) :

Unsteadiness parameter

\(\mathrm{We}=\sqrt{2} \Gamma \sqrt{\frac{a^{3}}{\nu _\mathrm{f}{(1-\alpha t)}^{3}}}\) :

Fluid parameter

H :

Magnetic parameter

\(\lambda \) :

Buoyancy or convection parameter

\(\gamma \) :

Angle of inclination

K :

Permeability parameter

\(A*\) :

Space-dependent heat source/sink

\(B*\) :

Temperature-dependent heat source/sink

R :

Radiation parameter

\(\mathrm{Ec}=\frac{U_\mathrm{w}^{2}}{(T_\mathrm{w}-T_{\infty })}\) :

Eckert number (viscous dissipation parameter)

\(\mathrm{Pr}=\frac{\left( \rho C_\mathrm{p} \right) \nu _\mathrm{f}}{k_\mathrm{f}}\) :

Prandtl number

\(\mathrm{Sc}=\frac{\nu _\mathrm{f}}{D_\mathrm{B}}\) :

Schmidt number

\(\mathrm{Re}_{x}=\frac{u_\mathrm{w}x}{\nu _\mathrm{f}}\) :

Local Reynolds number

\(K_\mathrm{r}\) :

Chemical reaction parameter

\(C_\mathrm{fx}\) :

Skin friction coefficient

\(\mathrm{Nu}_{x}\) :

Nusselt number

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

Sherwood number

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Authors and Affiliations

  1. Department of Mathematics, SAS, VIT, Vellore, 632014, India

    R. Meenakumari & P. Lakshminarayana

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816-1364, USA

    K. Vajravelu

Authors
  1. R. Meenakumari
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  3. K. Vajravelu
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Contributions

The authors R.M. and P.L. provided the idea, formulation of the problem, methodology and worked on the details of the problem. The author K.V. worked on the details of the problem, several drafts of the manuscript and presented the final version of the manuscript. All authors approve the final version for publication.

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Correspondence to P. Lakshminarayana.

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Meenakumari, R., Lakshminarayana, P. & Vajravelu, K. Unsteady MHD flow of a Williamson nanofluid on a permeable stretching surface with radiation and chemical reaction effects. Eur. Phys. J. Spec. Top. 230, 1355–1370 (2021). https://doi.org/10.1140/epjs/s11734-021-00039-7

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