Bingham line, a worked yield-stress example

A Bingham plastic is a fluid that will not move at all until the shear stress on it exceeds a yield stress, and then flows with a constant plastic viscosity above that point. Sludges, thick muds and many process pastes behave this way, and it is the reason a line that carries them cannot be sized off an ordinary friction chart.

This worked example pumps one of them, a yield-stress sludge, along a 100 m line. It is a built-in network in Fluid Network Studio. It shows how the solver handles a fluid with a yield stress, and how it decides the flow regime using a transition criterion built for yield-stress materials rather than the ordinary Reynolds number.

The scenario

A reservoir feeds a single line that delivers to a draw-off. The fluid is a Bingham plastic, so it carries an unsheared plug in the pipe core wherever the stress has not yet beaten the yield stress. The practical question is whether the line is running laminar or turbulent, because that decides the pressure gradient you need, and for a yield-stress fluid that boundary does not sit where a Newtonian one would.

The real setup

The example uses these parameters exactly as built:

• A source reservoir at a head of 30 m.
• A single line, 100 m long, 80 mm internal diameter, roughness 1.5e-4 m.
• A draw-off pulling a fixed flow of 0.002 m3/s, which is 2 litres per second.
• The fluid is a Bingham plastic with a yield stress of 8 Pa and a plastic viscosity of 0.03 Pa.s. Density 1150 kg/m3.

The physics and method

A Bingham plastic shears only in an annulus near the wall, around that central plug. Fluid Network Studio solves the laminar pressure gradient from the Buckingham-Reiner relation, which accounts for the plug, and uses the Darby-Melson correlation for turbulent friction. The crucial part is the regime decision. Rather than a plain Reynolds number, the solver uses the Hanks transition criterion, which is formulated for yield-stress fluids and so places the laminar to turbulent boundary correctly for a Bingham plastic. The line is solved through the same Global Gradient Algorithm used for water. The model is for homogeneous, non-settling fluids only, so no settling-slurry or deposition behaviour is modelled.

The solved result

The 100 m line costs 4.773 m of head to pass 2 L/s, and it is running laminar with an unsheared plug filling 74.3 per cent of the bore:

The wall shear stress is the number to hold on to. It comes out at 10.77 Pa against a yield stress of 8 Pa, so the fluid is only just being made to flow, and it is only shearing in a thin annulus at the wall. Three quarters of the bore, measured across, is travelling as a solid plug that never shears at all. That plug is the whole reason the friction factor lands at 0.473, nearly an order of magnitude above the 64/Re a Newtonian liquid of the same plastic viscosity would show at this Reynolds number.

The regime call matters just as much, and the two Reynolds numbers in the table show why. The line runs at 1 220, while the Hanks criterion puts the laminar-to-turbulent transition for this fluid at a Hedstrom number of 65 420 and a critical Reynolds number of 5 896, nearly three times the 2 100 a Newtonian liquid would use. The yield stress holds the flow laminar far past the point an ordinary friction chart would call it turbulent. Here both criteria happen to agree on laminar, but between 2 100 and 5 896 they would not, and that is the band where reading the regime off the wrong criterion sizes a yield-stress line on the wrong branch of the curve.

Every figure comes from the solver, from the Buckingham-Reiner relation inverted for the wall shear. Open the example, press Solve, and select the pipe to read the regime and the shear stress directly.

What the solver computes and what you learn

Solve it and you get the flow and the pressure along the line. Select the pipe and it reports whether the flow is laminar or turbulent according to the Hanks transition. The lesson is that yield stress changes both how much push you need and where the regime boundary sits. A fluid that is part-plug behaves differently from a thin liquid at the same nominal Reynolds number, and reading the regime from a criterion built for yield-stress fluids is what keeps the friction estimate honest.

Non-Newtonian fluids are part of the Advanced plan, so solving this example needs that plan.

Both Bingham correlations are checked in public: the verification cases solve a laminar line against the Buckingham-Reiner equation, inverted independently for the wall shear stress, and a turbulent one against the Darby-Melson correlation with the Hanks transition.

Take it further

The glossary explains yield stress and plastic viscosity, the power-law model for fluids without a yield stress, and the other non-Newtonian terms the solver uses. For a shear-thinning fluid in a whole network rather than one line, see the power-law fluid network example. A reminder on scope, Fluid Network Studio models homogeneous, non-settling fluids only, it does not predict settling or deposition. It supports your engineering work and does not replace a qualified engineer, and it makes no claim of compliance with any standard.

Open this example in FNS and raise the yield stress to see the regime respond.