Stability analysis of polarized domains

Polarized ferrofluids, lipid monolayers and magnetic bubbles form domains with deformable boundaries. Stability analysis of these domains depends on a family of nontrivial integrals. We present a closed form evaluation of these integrals as a combination of Legendre functions. This result allows exact and explicit formulae for stability thresholds and growth rates of individual modes. We also evaluate asymptotic behavior in several interesting limits. PACS numbers: 75.50.Mm, 75.70.Kw, 68.18.+p, 02.30.Gp Stability analysis predicts the evolution of cylindrical polarized domains in many systems, including magnetic bubbles [1], ferrofluid labyrinths [2, 3] and amphiphilic Langmuir monolayers [4, 5]. In all these contexts the same family of nontrivial integrals arises. We evaluate these integrals in closed form in terms of Legendre functions. Our closed form solution allows explicit analytic expressions for stability thresholds and individual mode growth and decay rates. We also present asymptotic behavior of the integrals in interesting limits such as large aspect ratio, or high spatial frequency. Let us briefly describe one particular physical system. Consider an initially circular (radius R) droplet of ferrofluid [6] trapped between two parallel glass plates, separated by a distance h. A convenient measure of the droplet shape is given by the aspect ratio p = 2R h . (1) Apply a uniform magnetic field perpendicular to the plates. Varying the magnetic field alters the balance between magnetic interactions and surface forces. We measure the relative strength of magnetic and surface forces