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An unsteady MHD Williamson fluid flow in a vertical porous channel with porous media and thermal radiation

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Abstract

The current article covers the unsteady Williamson fluid flow towards vertical parallel porous plates through porous media. The Cartesian coordinate system is utilised to model the flow equations. Along with thermal radiation, the effect of the applied magnetic field is also taken into account. Furthermore, the vertical walls of the channel are employed to achieve velocity slip and temperature jump conditions. Convection is applied to assume that the stable plates swap heat with an outer fluid. The mathematical model is numerically solved via the bvp4c shooting technique in the computer software MATLAB. For various values \(-1\), 0, 1 of the ratio of the fluid temperature to the wall temperature, the impacts of numerous physical characteristics affecting temperature and velocity profile including Weissenberg number, magnetic permeability, Prandtl number, suction parameter, Knudsen number, Radiation parameter, frequency of oscillation, magnetic parameter and the fluid–wall conversation parameter are discussed graphically. The concluding results suggest that the fluid velocity at the solid–fluid interface is influenced by the magnetic field, alterations in suction or injection parameters and specific values of velocity slip coupled with temperature jump conditions.

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Abbreviations

\(b\) :

Channel width

\(B _{0}\) :

Magnetic field

\(C _{p}\) :

Specific heat at constant pressure

\(\textrm{Ln}\) :

Fluid–wall interaction parameter

\(M\) :

Hartmann number

\(\textrm{Nr}\) :

Radiation parameter

\(\textrm{Pr}\) :

Prandtl number

\(\textrm{Re}\) :

Reynolds number

\(U\) :

Dimensionless velocity

\(\textrm{We}\) :

Weissenberg number

S :

Suction parameter

T :

Temperature

\(T_{1}\) :

Heated wall temperature

\(T_{2}\) :

Cold wall temperature

u :

X direction velocity

\({g _{t}}\) :

Thermal accommodation coefficients

\({g _{v}}\) :

Tangential momentum coefficients

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

Surface heat flux

\(\beta\) :

Thermal expansion coefficient

\(\chi ^2\) :

Magnetic permeability

\(\Gamma\) :

Constant for the Williamson fluid

\(\gamma _{s}\) :

Ratio of specific heat

\(\kappa\) :

Thermal conductivity

\(\lambda\) :

Molecular mean free path

\(\mu\) :

Dynamic viscosity

\(\nu\) :

Kinematic viscosity

\(\omega\) :

Frequency of the oscillation

\(\rho\) :

Density

\(\sigma\) :

Electrical conductivity

\(\sigma *\) :

The Stefan–Boltzmann constant

\(\theta\) :

Dimensionless temperature

\(\xi\) :

Ratio of the fluid temperature to the wall temperature

\(\kappa *\) :

Mean absorption coefficient

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Acknowledgements

This work has been carried out with the financial support of UGC, India, in the form of a Junior Research Fellowship under a research scheme grant (NTA Ref. No.: 201610200062) awarded to one of the authors Kalu Ram Sharma.

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  1. Department of Mathematics, University of Rajasthan, Jaipur, 302004, India

    Kalu Ram Sharma & Shalini Jain

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Sharma, K.R., Jain, S. An unsteady MHD Williamson fluid flow in a vertical porous channel with porous media and thermal radiation. Int J Adv Eng Sci Appl Math (2024). https://doi.org/10.1007/s12572-024-00371-w

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