Computational Analysis and Heat Transfer of MHD Transient Free Convection Flow in a Vertical Microchannel in Presence of Hall and Ion Slip Effects

In the present investigation, transient magnetohydrodynamic flow with Hall and ion slip current in a microchannel with the consideration of induced magnetic field is investigated. The microchannel walls are assumed to be electrically insulated and heated asymmetrically. Obtained dimensional partial differential equation are rendered dimensionless by employing suitable parameters and thereafter solved numerically in Matlab. Relevant actions of parameters on dimensionless velocity and induced magnetic field are depicted graphically, also the shear stress, rate of heat transfer, volume flow rate and induced current density are tabulated for various applicable parameters. Numerical result obtained in this work was compared independently with steady state existing benchmark at large value of time. During the course of the analysis, results obtained shows that at the early stages of time and in the simultaneous occurrence of Hall and ion slip effects, velocity and induced magnetic field behavior are found to be oscillatory all through the microchannel domain. Higher values of dimensionless time and Hartmann number however cancels out the oscillation and improves the flow. Furthermore, result also show that the rate of heat transfer at the transient time could be optimized by changing the thermal surrounding condition of the micochannel.

\(C_{p}\), \(C_{v}\) :

Specific heat at constant pressure and volume respect \(\left( {Joule\left( J \right)} \right)kg^{ - 1} \left( {Kelvin\left( K \right)^{ - 1} } \right)\)

\(g\) :

Acceleration due to gravity \(\left( {meter\left( m \right)\left( {\sec \left( s \right)} \right)^{ - 2} } \right)\)

\(\vec{H}\) :

Magnetic field vector \(\left( {Tesla\left( T \right)} \right)\)

\(k\) :

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

\(t^{\prime}\) :

Time (dimensionless)

\(t\) :

Time (dimensional)

M :

Hartmann number (dimensionless)

\(\Pr\) :

Prandtl number (dimensionless)

\(T\) :

Dimensional temperature \(\left( K \right)\)

\(T_{1}\) :

Temperature at right wall \(\left( K \right)\)

\(T_{2}\) :

Temperature at left wall \(\left( K \right)\)

\(\left( {u^{\prime},w^{\prime}} \right)\) :

Component of velocity along primary and secondary directions \(\left( {ms^{ - 1} } \right)\)

\(\left( {H_{x}^{\prime } ,H_{z}^{\prime } } \right)\) :

Component of induced magnetic field along primary and secondary directions \(\left( {Ampere\left( A \right)m^{ - 2} } \right)\)

\(b\) :

Microchannel width \(\left( m \right)\)

\(Pm\) :

Magnetic Prandtl number (dimensionless)

\(Kn\) :

Knudsen number (dimensionless variable)

\(\ln\) :

Fluid wall interaction parameter (dimensionless variable)

\(\beta\) :

Volumetric thermal expansion coefficient \(\left( {K^{ - 1} } \right)\)

\(\mu\) :

Dynamic viscosity \(\left( {kgm^{ - 1} s^{ - 1} } \right)\)

\(\mu_{e}\) :

Magnetic permeability \(\left( {Newton\left( N \right)A^{ - 2} } \right)\)

\(\sigma\) :

Electrical conductivity of the fluid \(\left( {ohm^{ - 1} m^{ - 1} } \right)\)

\(\rho\) :

Fluid density \(\left( {kgm^{ - 3} } \right)\)

\(\beta_{h}\) :

Hall parameter (dimensionless variable)

\(\beta_{i}\) :

Ion slip current (dimensionless variable)

\(\beta_{v}\), \(\beta_{t}\) :

Velocity slip, temperature jump (dimensionless variable)

\(\theta\) :

Temperature (dimensionless)

Authors received no funding for this manuscript.

Authors and Affiliations

1. Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

   Basant K. Jha

2. Department of Mathematics, Air Force Institute of Technology, Kaduna, Nigeria

   Peter B. Malgwi

Contributions

BKJ, PBM formulated the model. PBM analyzed and interpreted results. Write up by PBM. All authors have read and accepted the manuscript.

Corresponding author

Correspondence to Peter B. Malgwi.

Conflict of interest

The authors declare that they have no conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Jha and Malgwi [29]

\(C_{1} = \xi + \frac{{\beta_{v} Kn(1 - \xi )}}{{1 + 2\beta_{v} Kn\ln }}\), \(C_{2} = \frac{1 - \xi }{{1 + 2\beta_{v} Kn\ln }}\), \(M_{1}^{2} = \frac{{MP_{m} }}{{1 + \beta_{e} \beta_{i} + i\beta_{e} }}\), \(k_{1} = \frac{{C_{1} }}{M}\), \(k_{2} = \frac{{C_{2} }}{{M^{2} M_{1}^{2} }}\), \(k_{3} = \frac{{C_{2} }}{2M}\), \(k_{4} = \frac{1}{{M_{1}^{2} M}}\), \(k_{5} = k_{1} + k_{2} + k_{3}\), \(k_{6} = \frac{{k_{5} - k_{2} \cosh \left( {M_{1} \sqrt M } \right)}}{{\sinh \left( {M_{1} \sqrt M } \right)}}\), \(k_{7} = \frac{{k_{4} - k_{4} \cosh \left( {M_{1} \sqrt M } \right)}}{{\sinh \left( {M_{1} \sqrt M } \right)}}\), \(A = M_{1} \sqrt M\), \(k_{8} = \frac{A}{{M_{1}^{2} }}\), \(k_{9} = \frac{{k_{1} }}{{M_{1}^{2} }}\), \(k_{10} = Ak_{8}\), \(k_{11} = \frac{{2k_{3} }}{{M_{1}^{2} }}\), \(k_{12} = \beta_{v} Knk_{4} k_{10} - k_{7} k_{8}\), \(k_{13} = k_{6} k_{8} - \beta_{v} Knk_{2} k_{10} + \beta_{v} Knk_{11} - k_{9}\), \(k_{14} = \frac{{A\sinh \left( A \right) + \beta_{v} KnA^{2} \cosh \left( A \right)}}{{M_{1}^{2} }}\), \(k_{15} = \frac{{A\cosh \left( A \right) + \beta_{v} KnA^{2} \sinh \left( A \right)}}{{M_{1}^{2} }}\), \(k_{16} = \frac{1}{{M_{1}^{2} }}\left( { - k_{1} - 2k_{3} } \right)\), \(k_{17} = \frac{{\beta_{v} Kn2k_{3} }}{{M_{1}^{2} }}\), \(k_{18} = k_{16} - k_{17}\), \(k_{19} = k_{4} k_{14} + k_{7} k_{15}\), \(k_{20} = k_{18} + k_{2} k_{14} + k_{6} k_{15}\), \(C_{3} = \frac{{k_{13} - k_{20} }}{{k_{12} + k_{19} }}\), \(C_{4} = k_{2} + C_{3} k_{4}\), \(C_{5} = k_{6} + C_{3} k_{7}\), \(C_{6} = \frac{{k_{13} k_{19} + k_{20} k_{12} }}{{k_{19} + k_{12} }}\).

Reprints and permissions