Calculator
Formula used
The elasticity number is a dimensionless measure comparing elastic effects to inertial effects in viscoelastic flow. Two common equivalent forms are:
- El = Wi / Re
- El = (λ · μ) / (ρ · L²) (using characteristic length L)
Here, λ is relaxation time, μ is viscosity, ρ is density, and L is the chosen length scale.
How to use this calculator
- Select a calculation mode based on your available data.
- Enter values and choose units; the tool converts them to SI.
- Click Calculate to display results above the form.
- Optionally enable Show steps for transparent computation.
- Download CSV or PDF to save the calculation record.
Example data table
| Case | λ (s) | μ (Pa·s) | ρ (kg/m³) | L (m) | El |
|---|---|---|---|---|---|
| A | 0.25 | 0.12 | 1000 | 0.010 | 0.000300 |
| B | 0.80 | 0.45 | 980 | 0.005 | 0.014694 |
| C | 0.05 | 0.01 | 1050 | 0.020 | 0.000001 |
Values are illustrative and depend on your selected length scale.
Professional notes on the elasticity number
1) What El measures
The elasticity number (El) is a dimensionless index that compares elastic stresses to inertial effects in a viscoelastic flow. It is often used to anticipate whether a flow will behave more like an elastic material response or a momentum‑dominated Newtonian response under the same geometry. It supports screening of candidate fluids before detailed rheological simulation or testing.
2) Relationship to Wi and Re
A convenient identity is El = Wi/Re. Because Wi increases with material time scale and deformation rate, while Re increases with inertia, El becomes a clean “elasticity versus inertia” ratio. When El is held fixed, changing velocity mainly shifts Wi and Re together.
3) Typical input ranges
For dilute polymer solutions, relaxation times commonly span about 0.01–5 s, while effective viscosities may range from roughly 0.001–5 Pa·s depending on concentration and temperature. Densities are frequently near 900–1200 kg/m³ for many liquids. A practical length scale L is often 0.1–10 mm in microchannels, or 1–50 mm in lab pipes.
4) Interpreting magnitudes
Values El ≪ 1 indicate that inertial transport is comparatively strong, so elastic effects are secondary unless Wi is very large. Values El ≳ 1 suggest elastic effects can dominate even at moderate Re, which is relevant in elastic turbulence, rod‑climbing, and strong normal‑stress behavior.
5) Choosing the length scale
El depends on L², so the choice of characteristic length must match the physics and the reporting standard: hydraulic diameter for ducts, gap width for Couette devices, or nozzle diameter for jets. Reporting El without stating L can make comparisons misleading.
6) Unit handling and consistency
This calculator converts common units into SI before evaluation. Because El uses λ·μ/(ρ·L²), errors in L have the largest impact. A 10% length error produces about a 20% El error, so measure geometry carefully and keep units explicit.
7) Reporting alongside context
In publications and lab notes, El is most informative when reported with Wi and Re, fluid temperature, and the constitutive model assumption used to estimate λ and μ. If μ is shear‑dependent, cite the shear rate or apparent viscosity definition used.
8) Quick example and takeaway
Example: λ = 0.25 s, μ = 0.12 Pa·s, ρ = 1000 kg/m³, L = 0.01 m gives El ≈ 3.0×10⁻4, indicating inertia dominates for that scale. Reducing L to 1 mm increases El by 100×, which can shift expected behavior substantially.
FAQs
1) Is El always equal to Wi/Re?
Yes, when Wi and Re are defined using the same characteristic length and velocity. If different conventions are used, the ratio can shift. Always document the definitions of Wi, Re, and the chosen length scale.
2) Which viscosity should I enter for non‑Newtonian fluids?
Use the viscosity that matches your operating shear rate, or a clearly stated apparent viscosity. For shear‑thinning fluids, reporting the shear rate and model used (for example, Carreau or power‑law) improves reproducibility.
3) How do I choose the best length scale L?
Select the geometric scale that governs deformation: hydraulic diameter for ducts, gap width for Couette flow, or nozzle diameter for jets. Because El depends on L squared, use the same convention when comparing experiments or papers.
4) What does a very small El mean in practice?
El much smaller than one suggests inertia dominates and elastic effects are less influential at that scale. However, very large Wi can still produce elastic instabilities, so interpret El alongside Wi and the flow configuration.
5) Can this calculator compute Wi and Re for me?
Yes. In the “From flow” mode, the tool computes Re from ρ, V, L, and μ, and computes Wi from λ, V, and L. It then reports El using El = Wi/Re.
6) Why do my results change a lot when I change L?
El scales with 1/L². Halving L increases El by four. This sensitivity is physical, not a software issue, so ensure the length reflects the actual region where deformation and inertia compete.
7) What should I save in a lab notebook with El?
Record λ, μ, ρ, L, temperature, and the definitions used for Wi and Re. Also note the measurement method for λ and μ. Saving the exported CSV or PDF keeps a consistent snapshot.