Smearing of a slug front

How much does the flow smear a sharp magnetised front by the time it reaches the pickup coil?

w = \left( 5 - \frac{1}{1.8} \right) \cdot x = 4.444 \cdot x

Method

1. Find the Reynolds number, ρ·v·D ÷ η, and name the regime: laminar below 2300, turbulent above 4000, transitional between.
2. Laminar: the Poiseuille profile puts the centre at 2v̄ and the wall at rest, so fluid at radius r reaches a plane x downstream at t = x ÷ u(r). The fraction of the cross-section the front has reached by time t is F = 1 − x ÷ (2·v̄·t). It passes a tenth at t = x ÷ (1.8·v̄) and nine tenths at t = 5x ÷ v̄, so the 10–90 rise takes (5 − 1/1.8)·x ÷ v̄ and the front is 4.444·x long. Molecular diffusion would only matter if it could cross the bore in the transit, and a micron particle's diffusivity of about 10⁻¹³ m²/s moves it a fraction of a micron in a second.
3. Turbulent: the profile is nearly flat and the front is spread by eddy dispersion instead. Taylor's result for a smooth pipe is an axial dispersion coefficient D_T = 10.1·a·u*, with a the radius and u* = v̄·√(f/8) the friction velocity; f is the Darcy friction factor, Blasius's 0.316·Re^−¼ up to Re = 10⁵ and Haaland's smooth-pipe form above. The front becomes an error function of standard deviation σ = √(2·D_T·x ÷ v̄), and the 10–90 width of an error-function edge is 2.563σ. The eddy turnover time is D ÷ u*.
4. Transitional: the width is a straight-line interpolation between the laminar formula and the turbulent one evaluated at Re = 4000, because the flow can be either and no formula is trustworthy there. The page says so.

Assumptions

• Fully developed flow in a straight smooth tube of uniform bore. The coil section, bends and the pump housing each disturb the profile, and a front written close to a bend is smeared by it as well.
• The magnetisation rides with the fluid — it is carried, not conducted. That is true for particles too large to diffuse and too small to slip, which is the micron slurry; a settling grit lags the fluid and a nanoparticle diffuses within it.
• The medium is Newtonian and its viscosity is not changed by the field. A magnetised slurry thickens along the field lines, which flattens the laminar profile somewhat — the smearing here is therefore an upper bound in the laminar case.
• Taylor's coefficient is for a smooth pipe at high Reynolds number; near the transition it is an estimate. The interpolation across the transition band is a convenience, not a law.
• Nothing here is Meyer's except the velocity the EPG #1 holding specifies. The shear picture is Poiseuille (1840), the dispersion Taylor (1954); the holding says how fast the medium moves and nothing about what happens to its magnetisation as it does.