Freezing process of ferrofluid droplets: Numerical and scaling analyses

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 $h\sim t^{0.5}$ for high magnetic squeeze cases, but deviates much from that at the final freezing stage for both magnetic absence and lift cases.

Figure 1

Schematic of the computational domain and external forces acting on a ferrofluid droplet: (a) the magnetic squeeze condition; (b) the magnetic lift condition.

Figure 2

Validations of the accuracy of the multiphase model and the solid-liquid phase change model. (a) Suspension droplet test for the Laplace equation. (b) Solidification in a one-dimensional semi-infinite system. (Here, symbol “Eq. *” denotes Eq. (26). $T_i$ is the initial temperature in liquid phase. $T_i=T_m$ is the one-phase Stefan problem, while $T_i>T_m$ is the two-phase Stefan problem.)

Figure 3

Variation of the normalized height of frozen droplets ($h_f/h_d$) with volume expansion (${\rho }_l/{\rho }_s$). $h_d$ and $h_f$ are the droplet heights before freezing and after frozen, respectively. All the data are calculated without an external magnetic field.

Figure 4

Time-lapsed freezing front propagation ($h/R_0$) during freezing of ferrofluid droplets without an external magnetic field. $R_0$ is the radius of the undeformed droplet and $T_{i,d}$ is the initial temperature inside the droplet before freezing. The substrate temperature and environmental temperature are set to be $-6{}^{\circ }\mathrm{C}$ and $22{}^{\circ }\mathrm{C}$, respectively. Inset shows the temperature variation inside the droplet at $t=15.5$ s. Symbols “Eq. *” and “Eq. #” denote Eqs. (34) and (38), respectively.

Figure 5

Comparisons of droplet morphology (solid lines) and freezing front location (dashed lines) between the experimental results (black) and the numerical results (green) during the freezing of ferrofluid droplets at different time moments without an external magnetic field. See the movie in the Supplemental Material [33] for the dynamic freezing front propagation recorded in experiments.

Figure 6

(a) Temperature distributions along the vertical centerline of the domain. (b) Heat flux at the substrate and the freezing front. ($q_s$ and $q_l$ are the heat flux at the freezing front computed from the solid side and the liquid side, respectively.)

Figure 7

Comparisons of the shapes of frozen ferrofluid droplets between the experimental (left side of images) and the numerical (right side of images) results under different magnetic squeeze strengths.

Figure 8

Effects of magnetic field strength on the morphology of ferrofluid droplets and freezing time under the magnetic squeeze conditions. (a) Deformation of ferrofluid droplets $h_d/h_{d0}$ before the onset of freezing. (b) Dimensionless height ($h_f/h_{f0}$) of frozen droplets (left axis) and completely freezing time ($t_f/t_0$) (right axis) versus the effective magnetic Bond number ${\mathrm{Bo}}_e$. (Symbols “Eq. *”, “Eq. #”, “Eq. &” denote Eqs. (42), (44), and (43), respectively; $h_{d0}, h_{f0}$, and $t_0$ are, respectively, the initial droplet height, frozen droplet height, and freezing time without an external magnetic field.)

Figure 9

(a) Temperature distributions along the vertical centerline of domain at different moments (b) Time evolution of the freezing front under different magnetic squeeze conditions. Symbol “Eq. *” denotes Eq. (34).

Figure 10

Comparisons of the shapes of ferrofluid droplets before freezing (a), and completely frozen (b), between the experimental (left side of the images) and the numerical (right side of the images) results under different magnetic lift conditions.

Figure 11

Effects of magnetic field strength on the morphology of ferrofluid droplets and freezing time under the magnetic lift conditions. (a) Deformation of ferrofluid droplet $h_d/h_{d0}$ before the onset of freezing. (b) Dimensionless height ($h_f/h_{f0}$) of frozen ferrofluid droplets (left axis) and completely freezing time ($t_f/t_0$) (right axis) versus the effective magnetic Bond number ${\mathrm{Bo}}_e$. Symbols “Eq. *” and “Eq. #” denote Eqs. (47) and (46), respectively.

Figure 12

(a) Temperature distributions along the vertical centerline of thedomain at different time instants. (b) Propagations of the freezing front under different magnetic lift conditions. Symbol “Eq. *” denotes Eq. (34).