The rotation of a suspended axisymmetric ellipsoid in a magnetic field

Under the influence of a uniform and parallel magnetic field, a ferromagnetic fiber suspended in a Newtonian fluid rotates to align with the field direction. This study examines the field-induced rotation process for an individual non-Brownian axisymmetric ellipsoid suspended in a stagnant Newtonian fluid. Theoretical predictions are derived by a perturbation analysis for the limiting case where the strength of the applied magnetic field far exceeds the saturation magnetization of the ellipsoid. Numerical calculations are performed for the more general problem of an ellipsoid with known isotropic, non-hysteretic magnetic properties, using nickel and a stainless steel as examples. The analysis encompasses materials with field-induced, nonlinear magnetic properties, distinguishing these results from the simpler cases where the particle magnetization is either independent of, or linearly dependent on, the strength of the applied external field. In this study, predictions indicate that when the ellipsoid is magnetically saturated, the particle rotation is governed by the magnetoviscous time constant, _MV = _s/₀ M _s ² . It is found that the rotation rate depends strongly on the aspect ratio,a/b, of the ellipsoid, but only weakly on the dimensionless magnetization,M _s/H ₀. A geometric parameter for an ellipsoid, defined in eq. (2.5) - a, b major, minor semi-axes of an axisymmetric ellipsoid - D demagnetization tensor for an ellipsoid - D ^M magnetometric demagnetization tensor, the volume-average ofD ^P (r) - D ^P (r) position dependent demagnetization tensor, implicitly defined in eq. (2.12) - D _xx,D _yy,D _zz demagnetization factors, the diagonal elements ofD. Values for ellipsoids are defined in eq. (2.15) - F ^(m) magnetic force exerted on a body in a magnetic field - H _i ;H _i magnetic field inside a ferromagnetic body; magnitude ofH _i - H ₀;H ₀ magnetic field applied by external sources; magnitude ofH ₀ - h _i ;h _ix,h _iy Cartesian components of dimensionless internal magnetic field,h _i =H _i /H ₀ - I moment of inertia tensor - k geometric parameter for hydrodynamic resistance of a body rotating in a Newtonian fluid given in eq. (2.3) - L ^(h);L _z ^(h) hydrodynamic torque exerted on a rotating body; thez-component of the hydrodynamic torque - L ^(m);L _z ^(m) magnetic torque exerted on a magnetic body in a magnetic field, eq. (2.10); thez-component of the magnetic torque - M;M the magnetization, or dipole moment density, of a magnetic material; the magnitude ofM - M _s the saturation value ofM, approached by all ferromagnetic materials asH _i becomes large (figure 3) - m _s the dimensionless saturation magnetization,M _s/H ₀ - r position vector of a point within a ferromagnetic body - s dummy integration variable in eq. (2.5) - t time - U magnetoquasistatic potential energy of a magnetic body in a magnetic field, given in eq. (2.8) - u curve-fitting variable in eq. (4.1);u = logH _i - V volume of a magnetic particle; for an axisymmetric ellipsoid,V = (4/3) ab ² - x, y, z rectangular coordinate axes fixed in the ellipsoid (figure 1) - angle of inclination of the major axis of the ellipsoid with respect toH ₀ - _s viscosity of the Newtonian suspending medium - µ ₀ the magnetic permeability of free space,µ ₀ =4 · 10^–7H/m - _MV the magnetoviscous time constant, a characteristic time for a process involving a competition of viscous and magnetic stresses - the magnetic susceptibility of a magnetic material, = M/H _i - ; _z angular velocity of a rotating body; angular velocity about thez-axis of an ellipsoid, _z=–d/dt