A Level Set-Based Solver for Two-Phase Incompressible Flows: Extension to Magnetic Fluids

Abstract

Development of a two-phase incompressible solver for magnetic flows in the magnetostatic case is presented. The proposed numerical toolkit couples the Navier–Stokes equations of hydrodynamics with Maxwell's equations of electromagnetism to model the behaviour of magnetic flows in the presence of a magnetic field. To this end, a rigorous implementation of a second-order two-phase solver for incompressible nonmagnetic flows is introduced first. This solver is implemented in the finite-difference framework, where a fifth-order conservative level set method is employed to capture the evolution of the interface, along with an incompressible solver based on the projection scheme to model the fluids. The solver demonstrates excellent performance even with high density ratios across the interface (Atwood number $\approx 1$), while effectively preserving the mass conservation property. Subsequently, the numerical discretisation of Maxwell's equations under the magnetostatic assumption is described in detail, utilising the vector potential formulation. The primary second-order solver for two-phase flows is extended to the case of magnetic flows, by incorporating the Lorentz force into the momentum equation, accounting for high magnetic permeability ratios across the interface. The implemented solver is then utilised for examining the deformation of ferrofluid droplets in both quiescent and shear flow regimes across various susceptibility values of the droplets. The results suggest that increasing the susceptibility value of the ferrofluid droplet can affect its deformation and rotation in low capillary regimes. In higher capillary flows, increasing the magnetic permeability jump across the interface can further lead to droplet breakup as well. The effect of this property is also investigated for the Rayleigh–Taylor instability growth in magnetic fluids.

Keywords:

• Two-phase flows
• incompressible flow
• magnetostatic
• magnetic flows
• level set method
• ferrofluid droplets
• Rayleigh–Taylor instability

Acknowledgements

The authors would like to thank Victor Boniou for his valuable comments on the surface tension implementation and curvature calculations. Paria Makaremi-Esfarjani further acknowledges the helpful suggestions of Khashayar F. Kohan on the initial draft of the manuscript. The authors thank Mathias Larrouturou for useful feedback on the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The critical value of the transverse magnetic field, $H_{\mathrm{c}}$, is expected to be around 4.

Additional information

Funding

This study was supported by the Natural Science and Engineering Research Council of Canada (NSERC) through a NSERC Discovery Grant and Mitacs through the Mitacs Accelerate program.