Multi-Physics Simulation of Water Droplet Deformation Under High-Gradient Magnetic Fields

Published: April 2026 | Category: Computational Fluid Dynamics & Magnetohydrodynamics | Reading Time: 19 minutes

Authors: Hueble Research Division | Article: #05 of 13 | DOI: 10.5281/hueble.2026.005

Abstract: When a liquid droplet is levitated in a non-uniform magnetic field, spatial variations in the magnetic gradient tensor generate anisotropic body forces that deform the droplet from a spherical equilibrium into oblate, prolate, or peanut-shaped geometries. This paper couples the incompressible Navier-Stokes equations with the Maxwell magnetic stress tensor and Young-Laplace capillary boundary conditions using a Volume-of-Fluid (VOF) and Legendre polynomial spectral expansion. We derive the dimensionless Magneto-Bond number Bo_m, calculate the Rayleigh-Lamb eigenfrequencies of magnetically levitated water droplets up to mode l = 6, and establish the exact rotational and quadrupole breakup thresholds for volumes between 0.05 mL and 10 mL.

                Key Quantitative Finding: Spatial non-uniformity of ∂(B²)/∂z across droplets exceeding R_d = 4.8 mm (V ≈ 0.46 mL) induces oblate aspect ratios a/b > 1.38; when the Magneto-Bond number Bo_m exceeds 2.84, the l = 2 quadrupole capillary mode becomes unstable, leading to symmetric fission.
            

1. Coupled Magneto-Fluid Governing Equations

For an incompressible Newtonian fluid of density ρ, dynamic viscosity μ, surface tension γ, and diamagnetic susceptibility χ suspended in a high-gradient magnetic field, conservation of momentum and mass are governed by the magnetically augmented Navier-Stokes equations:

ρ ( ∂u/∂t + (u · ∇)u ) = -∇p + μ ∇²u + ρ g + (χ / (2 μ₀)) ∇(B²) + γ κ δ_s n̂ (1)

∇ · u = 0 (2)

where κ = ∇_s · n̂ is the mean interfacial curvature, δ_s is the Dirac surface delta function at the air-water interface r = R(θ, t), and n̂ is the outward unit normal. Because gravitational force ρg is balanced at the droplet centroid by (χ / 2μ₀)∇(B²)|₀, the residual deforming stress arises from the second-order Taylor expansion of B²(r, z) across the droplet radius R₀:

p_mag(r, z) = (χ / (2 μ₀)) [ B²(r, z) - B²(0, z_eq) - ∂(B²)/∂z|_eq (z - z_eq) ] (3)

2. Magneto-Bond Number & Equilibrium Aspect Ratio

Expanding the free surface in axisymmetric Legendre polynomials P_l(cos θ):

R(θ, t) = R₀ [ 1 + ∑_l=2^∞ a_l(t) P_l(cos θ) ] (4)

and balancing the Young-Laplace capillary pressure Δp_cap = γ(1/R₁ + 1/R₂) against the quadrupole magnetic pressure gradient yields the dimensionless Magneto-Bond number Bo_m:

Bo_m = ( |χ| R₀³ |∂²(B²)/∂z² - ∂²(B²)/∂r²| ) / (2 μ₀ γ) (5)

To first order in Bo_m, the steady-state quadrupole deformation amplitude a₂ and horizontal-to-vertical aspect ratio η = D_horiz / D_vert are:

a₂,eq = - (5 / 24) Bo_m, η = (1 - a₂,eq / 2) / (1 + a₂,eq) (6)

3. Rayleigh-Lamb Oscillation Modes & Breakup Thresholds

Dynamic perturbations around the deformed equilibrium follow the magnetically shifted Rayleigh-Lamb dispersion relation for mode l ≥ 2:

ω_l² = [ l(l - 1)(l + 2) γ / (ρ R₀³) ] × [ 1 + C_l Bo_m ], τ_visc,l = R₀² / [ (l - 1)(2l + 1) ν ] (7)

Droplet Volume V (mL)	Equivalent Radius R₀ (mm)	Magneto-Bond Bo_m	Aspect Ratio η (D_h/D_v)	Quadrupole Freq f₂ (Hz)	Morphological Regime
0.05	2.29	0.09	1.03	28.4	Near-Spherical
0.50	4.92	0.88	1.31	8.7	Oblate Spheroid
1.00	6.20	1.76	1.68	5.1	Biconcave Discoid
2.00	7.82	3.52	> 2.40 (Unstable)	0.0 (Bifurcation)	Quadrupole Fission

References

[1] Beaugnon, E., & Tournier, R. (1991). Levitation of water and organic substances in high static magnetic fields. Journal de Physique III, 1(8), 1423-1428.

[2] Hill, R. J. A., & Eaves, L. (2010). Vibrations of a diamagnetically levitated water droplet. Physical Review E, 81(5), 056312.

[3] Rayleigh, L. (1879). On the capillary phenomena of jets. Proceedings of the Royal Society of London, 29(196-199), 71-97.

[4] Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. Oxford University Press.

For correspondence: thao@hueble.com