Hydromagnetic Stability of a Liquid Film Flowing down an Inclined Conducting Plane

Gupta, Ajay; Rai, Lajpat

doi:10.1143/jpsj.24.626

Cited by 6 publications

(11 citation statements)

References 4 publications

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“…However, in free-surface flow these calculations are typically performed at fixed inclination angle (Yih 1963;Hsieh 1965;Lin 1967;Gupta & Rai 1968;Yih 1969;De Bruin 1974;Lam & Bayazitoglu 1986;Floryan et al 1987;Kelly et al 1989), rather than fixed Ga. Yet, as follows from (2.41), as long as ReHa/ tanh(Ha/2 ) Ga or ReHa 2 /(sech(Ha) − 1) Ga, respectively for insulating and conducting lower wall, θ remains small, which is the case for several of the calculations presented here.…”

Section: Resultsmentioning

confidence: 90%

“…The fact that all known conducting laboratory fluids have small magnetic Prandtl numbers (Pm . 10 −5 ) has led to a widespread adoption of the inductionless approximation (Hsieh 1965;Gupta & Rai 1968;Gordeev & Murzenko 1990;Korsunsky 1999;Mukhopadhyay et al 2008). However, the smallb assumption is not guaranteed to hold a priori, and the full problem must be solved to confirm that the scheme is valid in the parameter regime of interest (Takashima 1996).…”

Section: Problem Formulationmentioning

confidence: 99%

“…Moreover, for Pm 6 10 −4 (an interval encompassing all known conducting laboratory fluids) Re c was found to experience a mild, O(10 −3 ), decrease compared to its value in the inductionless limit Pm 0, where the magnetic field becomes a background variable decoupled from the flow. The free-surface version of the problem, hereafter referred to as 'free-surface Hartmann flow', has mainly been studied via weakly nonlinear analysis (Gordeev & Murzenko 1990;Korsunsky 1999;Mukhopadhyay, Dandapat & Mukhopadhyay 2008) and using asymptotic approximations of the linearized stability equations for large-wavelength gravity waves (Hsieh 1965;Ladikov 1966;Gupta & Rai 1968) or gravity-capillary waves (Lu & Sarma 1967). However, to the best of our knowledge, the degree of applicability of the inductionless approximation has not been discussed in the context of free-surface MHD stability problems (cf.…”

Section: Introductionmentioning

confidence: 99%

“…The first of these modes is related to the unstable gravity wave, oftentimes referred to as the 'soft instability mode', encountered in non-MHD parallel flow down an inclined plane (Yih 1963(Yih , 1969Lam & Bayazitoglu 1986;Floryan, Davis & Kelly 1987;Kelly et al 1989). In inductionless problems, the gravity wave becomes stabilized when Ha is increased (Hsieh 1965;Gupta & Rai 1968), but in the strong-field limit its growth rate Γ does not follow the characteristic |Γ | ∝ Ha 2 Lorentz damping of the shear modes in the spectrum. Instead, it transitions to an asymptotically neutral phase, where the Lorentz force nearly balances gravity, and the decay rate |Γ | ∝ Ha −2 decreases towards zero.…”

Section: Introductionmentioning

confidence: 99%

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Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

2009

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We study the linear stability of the flow of a viscous electrically conducting capillary fluid on a planar fixed plate in the presence of gravity and a uniform magnetic field, assuming that the plate is either a perfect electrical insulator or a perfect conductor. We first confirm that the Squire transformation for magnetohydrodynamics is compatible with the stress and insulating boundary conditions at the free surface, but argue that unless the flow is driven at fixed Galilei and capillary numbers, respectively parameterising gravity and surface tension, the critical mode is not necessarily two-dimensional. We then investigate numerically how a flow-normal magnetic field, and the associated Hartmann steady state, affect the soft and hard instability modes of free-surface flow, working in the low-magnetic-Prandtl-number regime of laboratory fluids (Pm 10 −4 ). Because it is a critical-layer instability (moderately modified by the presence of the free surface), the hard mode is found to exhibit similar behaviour to the even unstable mode in channel Hartmann flow, in terms of both the weak influence of Pm on its neutral-stability curve, and the dependence of its critical Reynolds number Re c on the Hartmann number Ha. In contrast, the structure of the soft mode's growth-rate contours in the (Re, α) plane, where α is the wavenumber, differs markedly between problems with small, but nonzero, Pm, and their counterparts in the inductionless limit. As derived from large-wavelength approximations, and confirmed numerically, the soft mode's critical Reynolds number grows exponentially with Ha in inductionless problems. However, when Pm is nonzero the Lorentz force originating from the steady-state current leads to a modification of Re c (Ha) to either a sublinearly increasing, or decreasing function of Ha, respectively for problems with insulating and conducting walls. In the former, we also observe pairs of counter-propagating Alfvén waves, the upstream-propagating wave undergoing an instability driven by energy transfered from the steady-state shear to both of the velocity and magnetic degrees of freedom. Movies are available with the online version of the paper.

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Section: Resultsmentioning

confidence: 90%

Section: Problem Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

2009

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“…Their results indicated that the micropolar parameters of the fluid play an important role in stabilizing the film flow. Gupta [10] conducted a linear stability analysis of a viscoelastic fluid film flowing along an inclined plane and subject to a small-amplitude and long-wavelength disturbance. The solutions obtained by deriving the viscoelastic analog of the Orr-Sommerfeld equation with the requisite boundary conditions revealed that viscoelastic effects tend to destabilize the film flow.…”

Section: Introductionmentioning

confidence: 99%