Equations Governing the Flow of Ferrofluids

The equations for the ferrofluid are taken from Rosensweig [1985]. The linear momentum equation is augmented by a term due to the anti-symmetric term for the shear stress. The shear stress, of course, comes from forces applied on the surfaces of fluid elements.

The angular momentum equation involves couple stresses representing rotational forces applied on the surfaces of fluid elements.

The magnetization equation comes from Rosensweig [1985], Zahn and Pioch [1997], and Zahn and Greer [1995].

Here t is the time constant related to the speed at which the magnetization adjusts to a changing applied magnetic field.

These equations must be augmented by constitutive relations relating the shear stress and the couple stress to velocity, spin, and their derivatives. In a continuum approach, the most general, linear relation is

The symmetric portion of the shear stress is proportional to the velocity gradient with the proportionality coefficient being the shear viscosity. The anti-symmetric portion of the shear stress depends on the difference between the local rate of spin and one-half the vorticity, with a proportionality constant, z, called the vortex viscosity. The couple stress is proportional to the spin gradient, and the proportionality coefficient is called the shear coefficient of spin viscosity, or spin viscosity.

It can also be shown that the divergence of the anti-symmetric portion of the stress dyadic can be represented by the curl of a vector.


If the vector A is constant in space, then the non-symmetric portion of the stress tensor doesn't matter. When a magnetic field exists, the body force and torque terms are

When these equations are combined we get the linear and angular momentum equations, written here without the gravitational term, and the equation for magnetization.

Using the vector identity

the momentum equation can be written as

Thus we have several new parameters for a ferrofluid. In addition to the density and shear viscosity, we have a vortex viscosity, z, the spin viscosity, h', and the time constant, t, in the equation for magnetization.

Maxwell's equations for a non-conducting material are

The boundary conditions are derived as follows. Consider a region with half the volume inside and half the volume outside. Apply Maxwell's equations; use the divergence theory for the B equation and Stokes equation for the H equation. The resulting boundary conditions are jump conditions across the boundary.

The first equation says that the normal component of the magnetic flux is continuous across a bound


ary, and the second equation says that the tangential component of the magnetic field is continuous across a boundary.