Abstract
This study is concerned with the numerical linear stability analysis of liquidmetal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios c≪1, an extremely strong magnetic field with Hartmann number Ha∼c−4 is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for threedimensional smallamplitude perturbations in ducts with realistic wall conductance ratios c=1, 0.1 and 0.01 and Hartmann numbers up to 104. As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness δ∼Ha−1/2. This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as Rec∼110Ha1/2 and kc∼0.5Ha1/2. The respective critical Reynolds number based on the total volume flux in a square duct with c≪1 is Re¯¯¯¯c≈520. Although this value is somewhat larger than Re¯¯¯¯c≈313 found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.
Original language  English 

Pages (fromto)  129146 
Journal  Journal of Fluid Mechanics 
Volume  788 
Early online date  22 Dec 2015 
DOIs  
Publication status  Published  10 Feb 2016 
Keywords
 highHartmannnumber flows
 instability
 MHD and electrohydrodynamics
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Profiles

Janis Priede
 Faculty Research Centre in Fluid and Complex Systems  Associate Professor Academic
Person: Teaching and Research