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

A power-law fluid is one with no single viscosity: its apparent viscosity changes with how hard it is being sheared, set by a consistency index K and a behaviour index n. Below n = 1 it is shear-thinning, above n = 1 shear-thickening, and at n = 1 it is an ordinary Newtonian liquid. Polymer solutions, paints, foodstuffs, pulps and drilling fluids are the everyday cases.

This worked example moves off water and onto one of them, a strongly shear-thinning liquid fed through a small network. It is a built-in network in Fluid Network Studio. It shows how the solver 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, so it gets less viscous the harder it is sheared. 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

Because a power-law fluid has no single viscosity, 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.

The solved result

The junction settles at 31.79 m, above the middle reservoir, so only the top reservoir supplies and the other two receive:

The regime is different in each pipe, and the generalised Reynolds number is the only way to see it:

Junction head is 31.79 m, which is 311.8 kPa gauge at ground level, and the three flows balance to zero there to within 1 x 10^-16 cubic metres per second.

The direction result comes first. Nobody told the solver that the 28 m reservoir would end up receiving rather than feeding, and by eye it looks like a supply: it is the middle tank of three. But the junction settles at 31.79 m, nearly four metres above it, so pipe B reverses and the middle reservoir fills. Only the top tank is actually supplying this network.

The regime result is the one specific to a shear-thinning fluid. Pipe B, at 0.432 m/s, sits at a generalised Reynolds number of 643 and is firmly laminar, while A and C at 4 495 and 5 663 are turbulent, and their friction factors differ by a factor of four as a result. You cannot pick that apart by looking at velocities, because with a power-law fluid the pipe carrying more flow is also being sheared harder and so runs at a lower apparent viscosity. The pipes that move fastest are the ones that behave most like water. That feedback is exactly what makes a non-Newtonian network worth solving rather than estimating.

Every figure here comes from the solver. Open the example, press Solve, and select any pipe to read its regime and generalised Reynolds number directly.

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.

The power-law path is checked in public: the verification cases solve a laminar line against the exact analytic solution and a turbulent one against the Dodge-Metzner correlation, implemented independently from its published form, alongside this network itself.

Take it further

The glossary explains the Metzner-Reed generalised Reynolds number, shear-thinning behaviour and the Bingham-plastic model for fluids that also carry a yield stress. For a yield-stress fluid in a single line, see the Bingham sludge line example. 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.