Fluid Network Studio

Power-law fluid network, a worked non-Newtonian example

This worked example moves off water and onto a non-Newtonian fluid, a shear-thinning power-law liquid fed through a small network. It is a built-in network in Fluid Network Studio. It shows how the solver handles a fluid whose apparent viscosity changes with shear rate, and how it decides the flow regime when the usual Reynolds number no longer applies.

The scenario

Three reservoirs at different heads feed a common junction. The fluid is shear-thinning, meaning it gets less viscous the harder it is sheared, which is the behaviour of many polymer solutions, paints, foodstuffs and drilling fluids. The reservoirs sit at different levels, so the network has to work out how the flows divide and combine at the junction, and in which direction each pipe runs, all while the fluid's resistance depends on how fast it is moving in each pipe.

The real setup

The example uses these parameters exactly as built:

  • Three reservoirs at heads of 40 m, 28 m and 15 m, each feeding the junction through its own pipe.
  • Pipe A, 1000 m long, 300 mm diameter; pipe B, 800 m long, 200 mm diameter; pipe C, 1200 m long, 250 mm diameter. Roughness 1.5e-4 m on each.
  • The fluid is a power-law liquid with consistency index K = 0.5 and behaviour index n = 0.5, so it is strongly shear-thinning. Density 1000 kg/m3.

The physics and method

A power-law fluid has no single viscosity, so the ordinary Reynolds number does not exist for it. Fluid Network Studio uses the Metzner-Reed generalised Reynolds number, which folds the consistency index, the behaviour index and the local conditions into one figure that tells you whether the pipe is laminar or turbulent. Laminar friction follows from the power-law rheology directly. In turbulent flow the solver uses the Dodge-Metzner correlation, on smooth-pipe correlations. The whole network is solved through the same Global Gradient Algorithm used for water, so flows balance at the junction and heads are consistent around the network. The model is for homogeneous, non-settling fluids only.

What the solver computes and what you learn

Solve it and you get the flow and direction in each pipe and the head at the junction. Select any pipe and it reports its flow regime and its Metzner-Reed generalised Reynolds number, so you can see directly which pipes are laminar and which have tipped into turbulence. The lesson is that with a shear-thinning fluid the regime is not obvious by eye, because a pipe carrying more flow is also being sheared harder and so runs at a lower apparent viscosity. Seeing the generalised Reynolds number per pipe is the only honest way to know where you stand.

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

Take it further

The glossary explains the Metzner-Reed generalised Reynolds number, shear-thinning behaviour 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 model settling slurries 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 change the behaviour index to see the regime shift.