Analysis of MHD stokes fluid flow in a cavity driven by moving parallel lid(s)

Controlling cavity flow through an effective magnetic field is highly desirable in many engineering applications. This work addresses the analytical solution for arbitrary depth cavity flow driven by two parallel lids under the influence of a uniform magnetic field acting along the x, y, or z axes, within the Stokes flow approximation. The formation of creeping flow and associated vortices is separated into symmetric and anti-symmetric modes, then combined to create the desired final cavity motion. The linear biharmonic equation of the stream function, modified by a Lorentz force term, is solved by constructing relevant real eigenvalues and eigenfunctions for both modes. This eigen-decomposition allows for the solution of algebraic linear equations for the coefficients in the series expansions, eliminating the need for numerical computations. This offers a significant advantage over the commonly used Papkovich-Faddle method. Our non-magnetic flow results precisely reproduce the dynamics available in the literature, primarily obtained through numerical simulations. Similarly, the MHD flow results derived from our analysis successfully replicate the numerical data found in the literature, with the exception of some ambiguous published data. These findings covering a range of Hartmann numbers between 0 and 80 valid for numerous cavity depths are further validated by finite element simulations conducted in Mathematica software, highlighting the value of the analytical solutions in discerning actual data from ambiguous information. The presented analytical solutions offer valuable physical insights into the vortical behavior of rectangular cavity motion under moderate and strong magnetic fields. The formulae clearly illustrate the breakup of the main recirculating zone, the centerline velocity structure, the core of the vortices, and the formation of boundary layers. These insights can be leveraged to determine the preferred magnetic field direction for optimal control of the cavity flow.