Abstract
This study investigates the linear stability of the Hartmann layers of an electrically conductive fluid between parallel plates under the impact of a transverse magnetic field. The corresponding Orr–Sommerfeld equations are numerically solved using Chebyshev’s pseudo-spectral method with Chebyshev polynomial expansion. The QZ algorithm is applied to find neutral linear instability curves. Details of the instability are evaluated by solving the generalized Orr–Sommerfeld system, allowing growth rates to be determined. The results confirm that a magnetic field provides a stabilizing impact to the flow, and the extent of this impact is demonstrated for a range of Reynolds numbers. From numerical simulations, it is observed that a magnetic field with a specific magnitude stabilizes the Hartmann flow. Further, the critical Reynolds number increases rapidly when the Hartmann number is greater than 0.7. Finally, it is shown that a transverse magnetic field overcomes the instability in the flow.
Highlights
Magnetohydrodynamics (MHD) plays a significant role in the Hartmann layer of electrically conductive fluids
The growth rate with respect to the Reynolds number is presented in Fig. 2(a) for Ha = 0.01–3
We can see that lower Hartmann numbers correspond to steady growth rates, whereas for Ha > 0.5, the fluid steadily destabilizes
Summary
Magnetohydrodynamics (MHD) plays a significant role in the Hartmann layer of electrically conductive fluids. It was found that the proportion of viscous and electromagnetic forces could be represented as the square of the Hartmann number Ha. Linear temporal stability analysis was used to find the time expansion of small two-dimensional disturbances in a basic flow, and QR/QZ methods have been employed to solve the complete eigenvalue problem for Hartmann flows.. A survey of the present literature suggests that numerical simulations of the stability of a basic flow and the growth of small disturbances in the Hartmann flow under an applied magnetic field are not exhaustive. By using the QZ algorithm, the eigenvalue problem is solved numerically, and details of linear stability can be investigated using an applied magnetic field. The linear instability of Hartmann flow in an electrically conductive fluid is analyzed in detail under this applied magnetic field.
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