Level set method for two-phase incompressible flows under magnetic fields

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Abstract

This article presents a level set method for two-phase, incompressible flows under magnetic fields. The magnetic force caused by the difference in magnetic permeability of the fluids are derived and incorporated into the momentum equation using the level set function. The governing equation for magnetic fields (magnetostatic equation) is also obtained by considering the discontinuity in the spatial distribution of magnetic permeability using the level set function. To test the method, falling droplet, droplet oscillation and stretching, and rising bubble problems in the presence of a magnetic field are simulated.

Introduction

If two immiscible fluids of different magnetic permeability, such as air and a ferrofluid, are exposed to a magnetic field, the magnetic field deforms due to the change in magnetic permeability in space and a force is induced in the flow field. This force, which results from the coupling of matter and magnetic field, acts as an additional driving force for the flow. For this reason, magnetic fields have been utilized in many thermo-fluid applications, such as the flow control and heat transfer enhancement.

The two-phase flow in the presence of a magnetic field is a complex multiphysics problem, and there have been attempts to use computational methods to study this problem. Due to the complexity, however, only a small number of methods have been developed. Tagawa has derived a set of non-dimensional model equations for incompressible, immiscible two-phase flows in the presence of a magnetic field and solved them numerically with a finite-difference method using the HSMAC algorithm [1], [2]. He simulated a movement of a falling droplet of liquid metal into a liquid metal pool under a uniform magnetic field. He also simulated a rising air bubble in water in the presence of a non-uniform magnetic field. Huang et al. [3] have developed a model for 3D MHD free surface flows based on magnetic field induction equations. Gao et al. [4] have simulated a three-dimensional liquid metal droplet moving into magnetic field gradient regions in a vacuum space in the absence of gravity using the volume-of-fluid continuum-surface-force (VOF-CSF) method. They formulated the general one-fluid VOF model for tracking free surfaces and associated CSF model for applying surface tension to free surfaces. The motion of a moving liquid Lithium droplet in vacuum under a strong non-uniform magnetic field was simulated in their study.

In this article, a level set formulation for two-phase incompressible flows in the presence of magnetic fields is presented. Based on the work by Sussman et al. [5] and Smereka [6], a momentum equation and magnetostatic equation customized for two-phase magnetohydrodynamic flow problems are derived. In this study, electric current is not considered and the effect of magnetic field on the flow field is limited to the two-phase interface. However, electric current can be incorporated in the formulation. As test examples, a falling droplet in a uniform magnetic field, the responses of a liquid droplet to a uniform magnetic field, and a rising bubble in a uniform magnetic field are considered.

Section snippets

Momentum equation

In this section, the governing equations for two-phase flows under magnetic fields will be derived. Fig. 1 shows a discontinuous function f at the two-phase interface, where the x-axis is the direction normal to the interface represented by the level set function ϕ. Since f can be modeled at the interface using a Heaviside function asf=fl+(fgfl)H(ϕ) its normal derivative can be expressed as a delta function as follows:fn=(fgfl)δ(ϕ) Note that this result is valid for tensors of any order at

Numerical examples

In order to test the model, three numerical examples are considered. For all problems, the gas viscosity, liquid viscosity, gas density, and liquid density are 1.8×105kg/m/s, 1.0×103kg/m/s, 1.2kg/m3, and 1.0×103kg/m3, respectively. Surface tension is set to 0.073 N/m; the magnetic permeability of gas is assumed to be μgas=4π×107N/A2; the acceleration due to gravity is 9.81 m/s2.

As the first numerical example, a falling liquid droplet due to gravity is simulated with and without a

Conclusions

A level set formulation for computing two-phase incompressible flows in the presence of a magnetic field is presented with test examples. Using the properties of the level set function, a magnetostatic equation and a momentum equation are derived. Test examples demonstrate the validity and effectiveness of the approach.

Acknowledgement

This research was supported by the new faculty start-up grant from the Ulsan National Institute of Science and Technology, Ulsan, South Korea.

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