# Interdisciplinary Applied Mathematics

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Buoyancy force

2.06 x 10~14N

2.06 x 1017 N

Peclet number

Pe = Wsaj D

5

0.5 x 10~3

size, magnetic susceptibility, field strength, temperature, channel geometry and surface properties, and volume fraction of the particles. Structures rapidly disassemble once the magnetic field is removed due to the electrostatic repulsion at short-range and Brownian motion.

We can make an order of magnitude estimate for the attractive force between two paramagnetic beads by assigning representative values of H = 104 Am-1    for    the    magnetic    field    strength    and у = 1    for    the    volumetric

magnetic susceptibility. The particles respond to each other’s gradients of magnetic flux density, which decreases as r~4. At contact this force can be relatively large, 6 x 10-11 N, for a particle of radius 1 p,m or 6x 10~13 N for a particle of radius 0.1 p,m. These values may be compared to the parameters in Table 13.1. At a separation distance of r = 10a, characteristic of the interparticle distance in a random suspension at 0.5% volume fraction, the forces drop to 10~13 N and 10~15 N respectively for separations aligned with the dipoles. Actual field strengths may be lower than these levels, and the force estimates scale with H2. The general indications are that Brownian motion is significant for the larger particles only when they are dispersed, while for the submicron scales Brownian motion plays an increasing role. The characteristics of the magnetic field will change once chains of beads start to form.

Fluid forces have a central part in the dynamics of particles in MR flows. In the development of microdevices for pumping, mixing, or flow control in a microchannel, we need to determine flow characteristics around the individual particles and in the channel as a system. While the flow will be low Reynolds number in character, inertial effects will influence the large scale dynamics and will be appreciable where the geometry of the channel changes.

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