Flow around a cylinder near a vibrating wall under a transverse magnetic field

The present work investigates the flow characteristics around a circular cylinder near a vibrating wall under a transverse magnetic field over the Reynolds number range of 200≤Re≤300, Hartmann number range of 0≤Ha≤8, and the gap ratio (the ratio of the gap size between the cylinder and the wall to the cylinder diameter) range of 0.25≤G/D≤0.5. The dimensionless amplitude and frequency of wall vibration range from 0.1 to 0.5 and 0 to 2, respectively. Three flow patterns are identified. The magnetic field's impact on stability is non-monotonic, initially promoting instability and later inhibiting it. Wall vibration has a promoting effect on flow instability, and this effect intensifies as the amplitude of wall vibration increases or the wall vibration frequency approaches the natural shedding frequency. As the wall vibration frequency approaches the natural shedding frequency, the lock-in phenomenon occurs. An increase in the amplitude of wall vibration expands the area of the lock-in region. Furthermore, the area of the lock-in region initially decreases and subsequently increases as the magnetic field strengthens. A sharp peak in the time average value of drag force and the root mean square of lift force fluctuation is observed when the wall vibration frequency equals the natural shedding frequency. Additionally, an analytical solution for the flow induced by the vibration of an infinite plate under a transverse magnetic field is derived.