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Impact of Hall Current on a 3D Casson Nanofluid Flow Past a Rotating Deformable Disk with Variable Characteristics

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Abstract

The significance of the present investigation is to explore the impacts of the Hall current and variable traits of thermal conductivity and the mass diffusion instead of the constant on a three-dimensional (3D) Darcy–Forchheimer–Casson nanofluid flow past a deformable rotating disk extendable in the radial direction. However, the motion of the flow is induced owing to the rotation and deformation of the disk. The process of heat transfer is inspected in the presence of the Cattaneo–Christov heat and mass fluxes, thermal radiation, and the activation energy amalgamated with chemical reaction added to witness the effect on the mass transfer. The novelty of the model is enhanced with the additional effects of momentum slip and convective boundary conditions. On utilizing the Von Karmann similarity variable, the formulated problem is transformed into ordinary differential equations. A numerical solution for the system of the differential equations is attained by employing the bvp4c function in MATLAB. The upshot of sundry parameters versus involved profiles is deliberated graphically. It is perceived that on escalating the Brownian motion and temperature-dependent thermal conductivity, the temperature field escalates. The velocity field in radial and azimuthal direction decays for up surging the Casson fluid and magnetic parameters. A comparative analysis of the present investigation with an already published work is also added to substantiate the envisioned problem.

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Abbreviations

\(b\) :

Rate of stretching

\(B_{0}\) :

Magnetic field strength

\(C_{w}\) :

Concentration at the surface of the disk

\(C_{\infty }\) :

Ambient concentration

\(c_{p}\) :

Specific heat

\(c_{b}^{*}\) :

Drag factor

\(D_{{\text{B}}}\) :

Brownian diffusion coefficient

\(D_{{{\text{B}}_{\infty } }}\) :

Ambient diffusion coefficient

\(D_{{\text{T}}}\) :

Thermophoretic diffusion coefficient

\(d\) :

Variable thermal conductivity parameter

\(E_{a}\) :

Coefficient of activation energy

\(E = \frac{{E_{a} }}{\kappa T}\) :

Activation energy parameter

\(e\) :

Variable molecular diffusivity parameter

\(f\) :

Radial velocity profile

\(F = \frac{{c_{b}^{*} }}{{r\sqrt {K^{*} } }}\) :

Non-uniform inertia coefficient

\(F_{r} = \frac{{c_{b}^{*} }}{{\sqrt {K^{*} } }}\) :

Darcy–Forchheimer number

\(h\) :

Axial velocity profile

\(h_{1}\) :

Convective heat transfer coefficient

\(h_{2}\) :

Convective mass transfer coefficient

\(H_{1} = \frac{{h_{1} }}{{k_{\infty } }}\sqrt {\frac{\nu }{\Omega }}\) :

Heat transfer Biot number

\(H_{2} = \frac{{h_{2} }}{{D_{{{\text{B}}_{\infty } }} }}\sqrt {\frac{\nu }{\Omega }}\) :

Mass transfer Biot number

\(Ha = \frac{{\sigma B_{0}^{2} }}{\rho \Omega }\) :

Magnetic parameter

\(j\) :

Tangential velocity profile

\(k\left( T \right)\) :

Variable thermal conductivity

\(K^{*}\) :

Permeability of porous medium

\(K_{1} = \lambda_{1} \Omega\) :

Thermal relaxation parameter

\(K_{2} = \lambda_{2} \Omega\) :

Concentration relaxation parameter

\(k_{r}^{2}\) :

Chemical reaction rate

\(L = S\sqrt {\frac{\Omega }{\nu }}\) :

Velocity slip parameter

\(m\) :

Hall current parameter

\(n\) :

Fitted rate constant

\(N_{b} = \frac{{\tau D_{{\text{B}}} \left( {C_{w} - C_{\infty } } \right)}}{\nu }\) :

Brownian motion parameter

\(N_{t} = \frac{{\tau D_{{\text{T}}} \left( {T_{w} - T_{\infty } } \right)}}{{\nu T_{\infty } }}\) :

Thermophoretic parameter

\(P = \frac{b}{\Omega }\) :

Stretching rate to angular frequency scaled stretching parameter

\(\Pr = \frac{{\mu c_{p} }}{{k_{\infty } }}\) :

Prandtl number

\(Q_{1}\) :

Heat absorption and generation coefficient

\(Q^{*} = \frac{{Q_{1} }}{{\rho c_{p} \Omega }}\) :

Heat source parameter to the angular velocity

\(Q_{w}\) :

Heat flux

\(Q_{{\text{m}}}\) :

Mass flux

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

Radiative heat flux

\({\text{Re}} = r^{2} \frac{\Omega }{\nu }\) :

Reynold number

\(Rd = \frac{{4\overline{\sigma }T_{\infty }^{3} }}{{3\overline{k} \, k}}\) :

Radiation parameter

\(S\) :

First-order velocity slip coefficient

\(S_{c} = \frac{\nu }{{D_{{{\text{B}}_{\infty } }} }}\) :

Schmidt number

\(T\) :

Temperature

\(T_{w}\) :

The temperature at the surface of the disk

\(T_{\infty }\) :

Ambient temperature

\(\left( {u,v,w} \right)\) :

Velocity components in the direction of \(\left( {r,\theta ,z} \right)\)

\(\beta\) :

Casson parameter

\(\sigma\) :

Electrical conductivity

\(\nu\) :

Kinematic viscosity

\(\rho\) :

Density

\(\lambda_{1}\) :

The relaxation time of heat flux

\(\lambda_{2}\) :

The relaxation time of mass flux

\(\lambda = \frac{\nu }{{K^{*} \Omega }}\) :

Local porosity parameter

\(\delta = \frac{{k_{r}^{2} }}{\Omega }\) :

Chemical reaction parameter

\(\alpha = \frac{{T_{w} - T_{\infty } }}{{T_{\infty } }}\) :

Temperature ratio parameter

\(\tau\) :

The ratio of capacities or heat capacity of nanofluid and solid particles

\(\tau_{w}\) :

Total shear stress

\(\tau_{zr}\) :

Shear stress in the radial direction

\(\tau_{z\theta }\) :

Shear stress in the tangential direction

\(\Omega\) :

Angular velocity

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Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through research groups program under Grant Number R.G.P-1/192/42.

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Authors and Affiliations

  1. Department of Computer Science, Bahria University, Islamabad, 44000, Pakistan

    Naila Shaheen & Muhammad Ramzan

  2. Department of Mathematics, College of Sciences, King Khalid University, Abha, 61413, Saudi Arabia

    M. Kbiri Alaoui

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  1. Naila Shaheen
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  2. Muhammad Ramzan
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  3. M. Kbiri Alaoui
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Contributions

MR did conceptualization, NS worked on methodology and wrote the original draft, and MKA did validation and helped in the revised draft.

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Correspondence to Muhammad Ramzan.

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Shaheen, N., Ramzan, M. & Alaoui, M.K. Impact of Hall Current on a 3D Casson Nanofluid Flow Past a Rotating Deformable Disk with Variable Characteristics. Arab J Sci Eng 46, 12653–12666 (2021). https://doi.org/10.1007/s13369-021-06060-1

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  • DOI: https://doi.org/10.1007/s13369-021-06060-1

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