| τy (Pa) | μ (Pa·s) | L (m) | V (m/s) | Bi |
|---|---|---|---|---|
| 10 | 0.10 | 0.01 | 1.0 | 1.0 |
| 50 | 0.20 | 0.02 | 0.5 | 10 |
| 5 | 0.005 | 0.005 | 2.0 | 2.5 |
| 200 | 1.00 | 0.05 | 0.1 | 1000 |
The Bingham number compares yield stress to viscous stress. Two common forms are:
- Bi = (τy · L) / (μ · V) using a velocity scale and length scale.
- Bi = τy / (μ · γ̇) using a shear-rate scale.
Larger values imply yield effects dominate, often creating plug regions.
- Select a method that matches your available measurements.
- Enter yield stress and plastic viscosity with correct units.
- Provide length and velocity, or enter shear rate instead.
- Press calculate to view Bi and an interpretation label.
- Use the export buttons to save CSV or PDF reports.
- For pipes, L is often the radius; for slots, half-gap is common.
- When shear rate is unknown, γ̇ ≈ V/L provides a quick estimate.
- If Bi varies widely, report ranges over expected operating conditions.
- Interpretation depends on geometry and boundary conditions.
1) Why the Bingham number matters
Many slurries and pastes do not move until a finite yield stress is exceeded. The Bingham number (Bi) is a compact way to quantify whether yield effects dominate the flow or whether viscous shear controls the response. In practice, Bi helps you anticipate plug formation, start-up pressure spikes, and sensitivity to operating speed.
2) What the ratio is comparing
A typical viscous stress scale is μV/L, where μ is the plastic viscosity, V is a representative speed, and L is a geometric length such as pipe radius or half-gap. The yield stress scale is τy. Their ratio, Bi = (τyL)/(μV), grows when yield stress or size increases, or when speed decreases.
3) Typical input ranges from real materials
Yield stress values vary widely: toothpaste and gels often fall near 20–200 Pa; ketchup may be 5–50 Pa; drilling muds can be 10–100 Pa; and cementitious pastes may exceed 50–300 Pa depending on water content. Plastic viscosity frequently ranges from 0.01–2 Pa·s in these applications, making Bi highly sensitive to both rheology and operating conditions.
4) Geometry choices for L in engineering
Picking L is not arbitrary. For pipe flow, L is commonly the pipe radius. For an annulus or slot, use the gap half-width. For a stirred vessel, a practical scale is the impeller clearance or blade width. Consistent geometry selection makes Bi comparable across tests and design iterations.
5) Velocity versus shear-rate formulations
When you can estimate a representative shear rate, the alternative form Bi = τy/(μγ̇) is convenient. In simple shear, γ̇ is measured directly. In internal flows, a quick estimate is γ̇ ≈ V/L, which makes both forms consistent. This calculator supports both approaches to match your data source.
6) Interpreting regimes with practical thresholds
As a working guide, Bi < 0.1 suggests viscous-dominated behavior where yield effects are minor. Values between 0.1 and 10 indicate a transitional regime where plug regions may appear but are sensitive to speed. Bi > 10 often signals yield-dominated flow requiring higher start-up pressure or careful control of low-speed operation.
7) Example calculations with numbers
Consider τy=50 Pa, μ=0.2 Pa·s, L=0.02 m, and V=0.5 m/s. Then Bi = (50×0.02)/(0.2×0.5)=10, which is yield-dominated. If you double the speed to 1.0 m/s, Bi drops to 5 and the flow becomes more transitional. Small speed changes can therefore strongly alter plug size and pressure drop.
8) Reporting and using Bi in projects
For reliable decisions, compute Bi across your expected temperature range and shear history, because both τy and μ can drift with time, mixing, and solids loading. Report the chosen geometry scale, units, and method used. Use the CSV and PDF exports to document design cases, lab runs, and sensitivity checks consistently.
1) Is the Bingham number dimensionless?
Yes. It is a ratio of yield stress to viscous stress scales, so the units cancel. A larger value means yield effects dominate relative to viscous shear for the selected geometry and operating point.
2) Which length should I use for L?
Use a length tied to the shear layer: pipe radius for pipes, half-gap for slots, and a representative clearance or blade scale for mixing. Consistency matters more than perfect uniqueness.
3) When should I use the shear-rate method?
Use Bi = τy/(μγ̇) when you measure shear rate directly in rheometry or when you can estimate a representative γ̇ for the process. It is often convenient for simple shear flows.
4) What does a very high Bi imply?
High Bi typically indicates plug-like regions, start-up thresholds, and strong sensitivity at low speeds. You may observe higher pressure requirements to initiate motion and a non-uniform velocity profile.
5) Does Bi replace pressure-drop calculations?
No. Bi helps classify whether yield or viscous effects dominate, but pressure drop still depends on geometry, flow rate, and constitutive model. Use Bi to guide which regime assumptions are reasonable.
6) How do temperature and mixing affect results?
Both τy and μ can change with temperature, shear history, and solids loading. Recalculate Bi for expected operating conditions, especially if the material thins, thickens, or structurally rebuilds over time.
7) Why do my Bi values vary across methods?
Differences usually come from how γ̇ is estimated or how L and V represent the process. Use γ̇≈V/L for a consistency check, and document the chosen method when reporting results.