Fluid Network Studio

Bingham line, a worked yield-stress example

This worked example is a yield-stress fluid, a Bingham-plastic sludge pumped along a line. It is a built-in network in Fluid Network Studio. It shows how the solver handles a fluid that will not move at all until the stress on it passes a threshold, and how it decides the flow regime for such a fluid 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, which behaves like a solid until the applied shear stress exceeds its yield stress, and then flows with a constant plastic viscosity above that point. Sludges, thick muds and many process pastes behave this way. 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 carries an unsheared plug in the pipe core where the stress has not yet beaten the yield stress, and shears only in an annulus near the wall. Fluid Network Studio solves the laminar pressure gradient from the Buckingham-Reiner relation, which accounts for that 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 network kernel used for water. The model is for homogeneous, non-settling fluids only, so no settling-slurry or deposition behaviour is modelled.

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.

Take it further

The glossary explains yield stress, plastic viscosity, the Hanks transition and the other non-Newtonian terms the solver uses. 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.