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This research is focused on linear analysis of a plane-parallel flow
stability in a transverse magnetic field (Hartmann flow) within a convective
approximation. We derive and solve equations describing the perturbation
growth. Perturbation modes and their nonexcitation conditions have been
determined. An equation for the instability increment has b...
Context in source publication
Context 1
... of the roots (stable one) is immediately visible: iγ = −Ha 2 < 0. Another root corresponds a pure imaginary increment indicating the flow instability: iγ > 0 (see Fig. 2). Numerical solution of the equation (35) gives us other roots among which there are those corresponding to the instability. From (4) follows that Ha √ P r m ≪ 1, so in the convective approximation λ ≪ Ha −1 Re m −1 . Since the experiments were carried out at Ha ∼ 10 3 and Re m ∼ 10 −1 (mercury) it is possible to say that λ H ≪ 10 −2 . ...
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Citations
... The instability corresponds to unstable roots of the equations, and so we must show that such roots exist. The results should be in qualitative agreement with those from previous experiments, 15 which explored the high-frequency stability of nonlinear problems. ...
... The instability disappears as Re→∞ and becomes more extensive at low Reynolds numbers. 15 The Reynolds number changes more rapidly as Ha increases, indicating that magnetic energy dissipation becomes more effective. This is because, in an electrically conducting fluid, the Lorentz force absorbs the kinetic energy. ...
... We are going to determine the equations which define the disturbance growth using the plane waves perturbation approach. The results should be qualitatively agreed with the experimental results of some articles [10,11], where the stability of these nonlinear problems in high frequencies was explored. ...
The Hartmann layers of conducting fluid between parallel plates under the influence of the transvers magnetic field are considered. The standard boundary conditions with zero velocities on the plates are used. The perturbations are used to obtain the exact solution for the given system. It is shown explicitly that no instability near the stationary solution can be found by perturbations in 2-dim cases. In 3-dim case the instability arises just when quaternionic perturbations applied. We use analytical solutions to obtain explicit results and wave perturbations for considered system. Our results are agreed with experimental facts which was solved by other authors and confirms the well- known fact that traversal magnetic field overcomes the instability. Similarly we obtained the exact solution by using new quaternionic method and this way of nonlinear system consideration can deliver a variety of possibilities for finding new solutions.
... The instability corresponds to unstable roots of the equations, and so we must show that such roots exist. The results should be in qualitative agreement with those from previous experiments, 15 which explored the high-frequency stability of nonlinear problems. ...
... The instability disappears as Re→∞ and becomes more extensive at low Reynolds numbers. 15 The Reynolds number changes more rapidly as Ha increases, indicating that magnetic energy dissipation becomes more effective. This is because, in an electrically conducting fluid, the Lorentz force absorbs the kinetic energy. ...
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.
