Polynoid travelling-wave pump

Can a sequenced row of coils move the slurry without a mechanical pump, and how fast?

The formula

Δpm = n μ_{0} M G ℓg = (f \frac{L}{D} + K) \frac{ρ {(vf)}^{2}}{2}

Δpm: Driving pressure from n fringes, Pa
n: Number of drive coils
M: Medium magnetisation, A/m
G: Field gradient ∇H, A/m²
ℓg: Length of the gradient region, m
L: Loop length, m
D: Bore, m
K: Sum of minor-loss coefficients
ρ: Medium density, kg/m³
vf: Force-balance velocity, before the wave limit, m/s

LaTeX

Δpm = n \cdot \mu_0 \cdot M \cdot G \cdot ℓg = \left( f \cdot \frac{L}{D} + K \right) \cdot \frac{ρ \cdot \left( vf \right)^2}{2}

Work it out

M Medium magnetisation

kA/m

The medium's magnetisation in the coils' field, per unit volume of slurry. About 7.5 kA/m for 5 % micron iron at 50 kA/m; parts per million of that, and negative, for plain water.

G Field gradient

A/m²

How fast H falls off along the tube at each coil's end.

ℓg Length of the gradient region

How far along the tube each coil's fringe extends.

n Number of drive coils

Coils in the sequenced row. Each contributes one fringe's worth of pressure per pass.

s Coil pitch

Centre-to-centre spacing of the coils along the tube.

fs Sequence frequency

How many times a second the energised pattern steps one coil along. The wave speed is the pitch times this.

ρ Medium density

kg/m³

Density of the medium.

η Medium viscosity

mPa·s

Dynamic viscosity of the medium.

D Bore

Inside diameter of the tube.

L Loop length

Total length of tube in one lap.

K Minor-loss coefficients

The sum of the loss coefficients for bends and fittings.

or see a worked example with typical numbers

Method

One coil's fringe pulls with μ₀·M·∇H per unit volume over its gradient region, a pressure of μ₀·M·G·ℓ_g, as the body-force calculation gives. A row of n coils stepped in sequence hands the medium from one fringe to the next, so each contributes once per pass and the driving pressure is n times one fringe's.
The loop resists with Darcy–Weisbach: (f·L ÷ D + K)·ρ·v² ÷ 2, with f from the Reynolds number as in the loop pressure-drop calculation. Since f depends on v the balance is solved by bisection — the loss rises monotonically with v, so there is one velocity at which it equals the driving pressure.
The pattern advances one coil pitch every step: its speed is s·f_s. A medium pulled along by a moving field can at most keep up with it, so the reported velocity is the lower of the force-balance speed and the wave speed, and the slip is 1 − v ÷ v_w.
Compare the result with the 50 in/s the EPG #1 holding specifies for slurry.

Assumptions

Every fringe pulls at full strength on medium that is fully magnetised, and the sequencing is timed so that each coil is on exactly while the medium is in its fringe. That is the most the arrangement can do; a real sequence leaks some of each pulse into pushing the medium back at the trailing fringe, and the driving pressure is less than n fringes' worth.
M is induced by the same coils that pull. It is χ·H of the medium at the coil's field, so the force needs a susceptible medium: a loaded slurry, not a plain liquid. For water χ is −9 × 10⁻⁶ and the force is a repulsion of parts per million of what a 5 % iron slurry feels.
The medium is a continuum that the particles drag with them. In a settling grit the force acts on the particles alone and the carrier is left behind; in a micron slurry the viscous coupling is tight and the picture holds.
The loop loss is as the pressure-drop calculation gives it: smooth tube, Newtonian medium, handbook minor losses. A magnetised slurry inside the coils is thicker than this along the field.
Nothing here is Meyer's except the 50 in/s the EPG #1 holding specifies and the existence of the polynoid row in the EPG #3 holding. The force is the Kelvin force, the loss is Darcy's, the wave-speed limit is that of any induction machine; the estate gives no coil count, spacing, current or sequence rate.