Dynamics of small bubble interface perturbations in vertical hele‐shaw cell with magnetic liquid under the action of normal magnetic field

Abstract

A linearized problem of dynamics for small perturbations of the gas bubble rising in the Hele‐Shaw cell filled with magnetic liquid is considered. It is reduced to searching of eigenvalues and eigenfunctions for a linear operator with periodic boundary conditions. The obtained operator is presented as a sum of two linear operators: the second order differential operator with varying coefficients and the integro ‐ differential operator with the singularity of the Cauchy type. The spectral problem is solved by the Degenerate Matrices (DM) method using Chebyshev polynomials of the first and second kind.

Duju burbulo, judančio vertikalia Hele‐Shaw lastele užpildančiu magnetiniu skysčiu, paviršiaus dinamikos matematinis modelis yra suformuluotas kaip spektrinis uždavinys tam tikram tiesiniam operatoriui su periodinemis kraštinemis salygomis. Pastarasis uždavinys yra išsprestas skaitmeniškai išsigimstančiu matricu metodu.

Key words:

• Hele‐Shaw cell
• gas bubble
• magnetic liquid
• eigenvalues and eigenfunctions problem
• the Degenerate Matrices method
• Chebyshev polynomials