A new boundary integral formulation to describe three-dimensional motions of interfaces between magnetic fluids

A new general three-dimensional hydrodynamic–magnetic boundary integral formulation for a magnetic free surfaces in viscous flows at low Reynolds numbers is developed. The formulation is based on an extension of the Lorentz reciprocal theorem for the incompressible flow of a magnetic fluid. Combining the reciprocal theorem and the fundamental solution of a creeping flow we obtain the integral representation of the flow in terms of hydrodynamic and magnetic potentials. According to this formulation, the magnetic and hydrodynamic quantities which are necessary for determination of the dynamics of a magnetic liquid are established by means of appropriate integral equations at the boundary of the region occupied by the magnetic liquid. The motion of a free surface with arbitrary magnetic properties and with the viscosity of the magnetic liquid and the surrounding fluid not identical may be explored with the present formulation. Two relevant physical parameters are revealed in the present hydrodynamic–magnetic boundary integral formulation: the ratio of the magnetic permeability and the magnetic capillary number. The proposed boundary integral equations has been developed in order to simulate the full time-dependent low Reynolds number distortion and orientation of a three-dimensional ferrofluid droplet under the action of shearing motions and magnetic fields.