Magnetoviscous rise

How much does the field thicken the slurry?

The formula

r = 1.5 φ \frac{ξ - \tanh (ξ)}{ξ + \tanh (ξ)}, ηB = ηE (1 + r)

LaTeX

r = 1.5 \cdot φ \cdot \frac{ξ - \tanh(ξ)}{ξ + \tanh(ξ)}, \qquad ηB = ηE \cdot (1 + r)

Work it out

c Carrier fluid

The liquid whose viscosity is being raised.

m Particulate

The magnetic solid. Its saturation magnetisation sets each particle's moment.

φ Particulate loading

The solid's share of the volume. The rise is proportional to it, in the dilute limit the formula is good for.

d Particle diameter

µm

Sets the moment, and with it how completely the field holds the particle.

B Applied flux density

The field in the tube, taken across the flow's vorticity — the orientation that gives the full effect.

T Temperature

°C

Thermal energy is what the field's hold on the particle is measured against.

or see a worked example with typical numbers

Find the particle's moment: the material's saturation magnetisation times the sphere's volume, πd³ ÷ 6.
Multiply by the flux density for the magnetic energy, and divide by k_BT for ξ. A micron iron particle in 50 mT has ξ in the millions; a ten-nanometre magnetite grain has ξ of about 3.
Shliomis' rotational viscosity: the rise is 1.5 φ times (ξ − tanh ξ) ÷ (ξ + tanh ξ). The bracket is ξ²/6 for small ξ and 1 for large, so the rise is at most 1.5 φ — 7.5 % for a 5 % slurry.
The field-free viscosity of the suspension is the carrier's times Einstein and Batchelor's bracket, 1 + 2.5φ + 6.2φ², as the slurry-viscosity calculation gives it. The viscosity in the field is that, times one plus the rise.

A particle whose moment is rigidly fixed to its body, so that holding the moment holds the particle — the Brownian relaxation case. A particle that relaxes by the Néel mechanism lets its moment follow the field while the body rolls freely, and shows no rise; the magnetic-relaxation calculation decides which case a given size and material is, and this page does not check.
Dilute, non-interacting spheres, with the field across the vorticity of a simple shear. Along the vorticity the effect vanishes; in a pipe the field of a solenoid is along the flow, and the vorticity is around it, so the geometry is the favourable one.
Shliomis' 1972 result is good to a few per cent loading. Above that, and in any field strong enough to chain the particles, the real thickening is larger by a great deal — a magnetorheological fluid stiffens to a paste at a few hundred millitesla — and this formula does not describe it.
The carrier's viscosity is the 20 °C handbook value.
Nothing here is Meyer's. The estate material describes a slurry moving through a coil and says nothing about what the coil does to the slurry's flow; the calculation exists because the answer is "a few per cent" for separate particles and "it stops" for chained ones, and a reader should know which regime they are in.