Magnetic (electric) drop deformation in uniform external fields: Volume averaged methods and formation of static and dynamic conical tips

Abstract

Analytical relationships describing droplet deformation in external magnetic (electric) fields rely on spheroidal and ellipsoidal shape approximations. We show that the ellipsoidal shape approximations that assume a uniform internal magnetic field are only valid for small deformations (aspect ratio a/b ≈ 4). For large droplet deformations, the non-uniformity in the field within the droplet becomes substantial, rendering such approximations to be invalid. To overcome the limitations of ellipsoidal theory, we perform numerical simulations to determine volume averaged demagnetization factor and fields. Based on the numerical simulations, we propose semi-analytical relationships to describe small and large deformations for magnetic droplets using volume averaged methods. We test and validate our results with the existing experimental results and find an excellent agreement between our model and experimental studies. We extend our analysis and investigate static and dynamic droplets with conical tips. We show that droplets with conical tips could be defined solely by the characteristic half cone angle. We analyze unstable droplets with extremely high susceptibility χ → ∞ and find that conical tips with a half cone angle of θc ≈ 30° and an aspect ratio of ≈3.7 are formed prior to breakup, in agreement with the prior experimental studies of charged electric droplet breakup. We show that the volume averaged methods derived for droplets with finite tip curvature are also valid and in good agreement with the computational and previous experimental studies of magnetic droplets with conical tips.