Journal of Magnetism and Magnetic Materials 272–276 (2004) 657–658 A new approach to computations of forces in magnetic fluids W. Zamboni a, *, G. Coppola b , M. d’Aquino a , G. Miano a , C. Serpico c,1 a DIEL - University of Napoli ‘‘Federico II’’, Via Claudio, 21, 80125 Napoli, Italy b DETEC, Universita degli Studi di Napoli ‘‘Federico II’’, P.le Tecchio, 80, 80125 Napoli, Italy c INFM Napoli and DIEL, Universita degli Studi di Napoli ‘‘Federico II’’, Via Claudio, 21, 80125 Napoli, Italy Abstract The hydrostatic equilibrium of a linear magnetic fluid subject to an applied stationary magnetic field, is studied in a 2D geometry. The free boundary mechanical equilibrium equation has been solved by using a new method based on a pseudo-unsteady procedure which dynamically converge toward the equilibrium. The position of the fluid is modeled by defining a phase characteristic function w; with w ¼ 1 in liquid and w ¼ 0 in air. This function is numerically computed by associating at each cell of the mesh a scalar field 0pCp1 equal to the portion of the cell area occupied by the fluid. r 2003 Elsevier B.V. All rights reserved. PACS: 41.20.Gz; 47.11.þj Keywords: Magnetic fluid; Magnetic force; Helmholtz formula; Volume of fluid method; Free boundary We consider the problem of hydrostatic equilibrium of a linear magnetic fluid subject to an applied stationary magnetic field. The solution of this problem can be found by self-consistently solving the magneto- static field equations r B ¼ 0; r H ¼ J; along with the equation of mechanical equilibrium rp ¼ r m g þ f mag ; where f mag is the force due to the magnetic field, p is the appropriately defined fluid pressure, r m the fluid mass density and g the acceleration of gravity. The issue of self-consistently defining f mag and p is rather controversial and in this paper we adopted the treatment proposed in Ref. [1]. In particular, we followed the so- called Couloumb approach (see Ref. [2]) which leads to the following formula of the total volume force density f tot V ¼ r m g rp 0  1 2 H 2 rm þr 1 2 r m H 2 @m @r m   ; ð1Þ where p 0 ¼ p 0 ðT ; r m Þ is the pressure of the fluid at given T and r m for M ¼ 0; and m is the magnetic permeability of the fluid. The last two terms in Eq. (1), f mag ¼  1 2 H 2 rm þ r½ 1 2 r m H 2 @m=@r m ; constitute what is com- monly referred as Helmholtz formula for magnetic forces. The above theoretical framework is used for the computation of the magneto-mechanical equilibrium of a magnetic linear fluid subject to a stationary applied magnetic field produced by two sheet-shaped coils (see Fig. 1). The geometry of the problem considered is 2D and the applied field is almost tangent to the liquid free surface. The magnetic flux density is expressed in terms of the stream function c and this leads to the Poisson equation rð1=mÞrc ¼ J : The free-boundary mechan- ical equilibrium configuration has been obtained by simulating the unsteady evolution of the magnetic fluid under the action of both magnetic and gravitational fields. The problem is modeled as the incompressible flow of two immiscible phases, liquid and air, with different densities. The equations solved are thus the momentum balance equation, modeled by the variable density Navier–Stokes equation in whole domain for- mulation, together with the incompressibility constraint for the velocity field: @V=@t þrðVVÞ¼ 1=rðr  ð2ZDÞrp 0 þ FÞ and r V ¼ 0; where V is the velocity field, Z is the dynamic viscosity, D the viscous stress tensor D ¼ 1 2 ðrV þrV T Þ and F the body force rg þ f mag : The methodology here used in order to ARTICLE IN PRESS *Corresponding author. Tel.: +39-081-7683505; fax: +39- 081-7683180. E-mail addresses: zamboni@unina.it (W. Zamboni), serpico@unina.it (C. Serpico). 1 Also can be corresponded to Tel.: +39-081-768-3503. 0304-8853/$-see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.646