Freezing process of ferrofluid droplets: Numerical and scaling analyses

Abstract

In this study we present numerical and scaling analyses of deformation and freezing processes of ferrofluid droplets under magnetic field effects. A multiphase flow model coupled with an enthalpy-based lattice Boltzmann model is developed to directly simulate the deformation and subsequent freezing processes of a ferrofluid droplet with considerations of both volume expansion and magnetization effects. Meanwhile, analytical models and scaling analyses are derived to reveal how the morphology of the ferrofluid droplet responds to the magnetic field and how the morphology evolution affects the freezing time. We find that the magnetic force induced by magnetic field gradient is much larger than that induced by the magnetization effect, leading to the flattening or elongation of a ferrofluid droplet under magnetic squeeze or lift conditions. The height of the ferrofluid droplet almost linearly decreases (increases) in the low magnetic strength regime for magnetic squeeze (lift) cases, while it follows a nonlinear scaling law under high magnetic squeeze conditions. Besides, the propagation of the freezing front well obeys the scaling law h ∼ t 0.5 for high magnetic squeeze cases, but deviates much from that at the final freezing stage for both magnetic absence and lift cases.

Magnetowetting dynamics of sessile ferrofluid droplets: a review

“…The theory regarding the movement of MNPs towards the magnet was further confirmed by Fang et al 118 in a recent study. With the introduction of the magnet atop an FF droplet, the magnetic stress at the apex (Bo s ) due to FF magnetization further assists the attractive force due to the non-uniform magnetic field gradient (Bo m ), and hence the effective bond number can be represented as Bo e = Bo m + Bo s in this “lift configuration” (a term coined by Fang et al 118 ).…”

“…In recent years the influence of the magnetic field gradient and the substrate wettability has been explored further via numerical simulations. 107,118 Khan et al 107 complemented the previously discussed experimental observations of the flattening of water-based FF droplets in a non-uniform field through numerical simulations using a simplified lattice Boltzmann method. When the magnetic field strength was below the saturation magnetization, i.e., M o M s , the shape of the flattened drop perfectly matched with that obtained using spherical cap theory.…”

“…By contrast, for , the morphology of the elongated droplets was argued based on a non-dimensional traction number defined as S = μ 0 M 2 V 1/3 γ −1 (similar to Bo s in ref. 118), which compares the dominant role of the pressure jump at the droplet interface because of the normal magnetization vector with the capillary force. Fundamentally, the pressure jump at the apex is balanced by decreasing the curvature of the interface, which further favors the elongation of the FF droplets.…”

“…In the case of non-zero dipole-dipole interactions, the latter observations were further verified from various experimental studies, as discussed in the following paragraphs. Since the direction of the non-uniform magnetic field gradient or magnetic attraction force relies on the position (above or underneath) of the permanent magnet with respect to the droplet or substrate, thereby affecting the shape-evolution dynamics, 107,118 we shall be reviewing the relevant studies based on the configuration of the permanent magnet in the upcoming sections.…”

“…When a permanent magnet is placed underneath the droplet-substrate pair, the attractive force due to the magnetic field generally assists the gravitational force in flattening or squeezing the initially spherical droplet. 87,107,[118][119][120][121] As discussed previously, Nguyen et al 27 initially explored this configuration of the magnetic field wherein the flattening of the droplet and consequent decrease in the contact angle were seen. Later, Rigoni et al 87 presented a comprehensive report on the static magnetowetting behavior of sessile FF droplets on a non-uniform magnetic field by considering two crucial parameters, namely, the size and strength of the permanent magnet placed coaxially underneath the solid substrate, and the concentration of the nanoparticles.…”