Generalized Reynolds Number (Non‑Newtonian) Calculator

Inputs (SI is used internally; choose your preferred units)

Density, ρ

Fluid mass density.

Mean velocity, v

Cross‑sectional average.

Pipe diameter, D

Hydraulic diameter.

Rheology model

Choose constitutive relation.

Dynamic viscosity, μ (Pa·s)

e.g., water ≈ 0.001 Pa·s at 20 °C.

Results

Generalized Re

—

(—)

Apparent viscosity, μ_app

—

Pa·s

Wall shear rate, γ_w

—

s⁻¹

Regime

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Example data

#	Model	ρ [kg/m³]	v [m/s]	D [m]	Params	μ_app [Pa·s]	Re (generalized)

Click a row to load its values into the form.

Formulas used

Wall shear rate: For laminar, circular pipe, Newtonian baseline uses \( \gamma_w \approx \dfrac{8v}{D} \). For power‑law and many shear‑dependent fluids, the Metzner–Reed correction gives \( \gamma_w \approx \dfrac{8v}{D}\,\dfrac{3n+1}{4n} \).

Generalized Reynolds number (apparent‑viscosity basis): \( \displaystyle \mathrm{Re}_g = \frac{\rho\,v\,D}{\mu_{\mathrm{app}}} \), where \( \mu_{\mathrm{app}} \) is the viscosity evaluated at the chosen shear rate.

Newtonian: \( \mu_{\mathrm{app}} = \mu \), so \( \mathrm{Re}=\rho v D/\mu \).
Power‑law: \( \mu_{\mathrm{app}} = K\,\gamma_w^{\,n-1} \).
Herschel–Bulkley: \( \mu_{\mathrm{app}} = \tau_y/\gamma_w + K\,\gamma_w^{\,n-1} \).
Carreau–Yasuda: \( \mu = \mu_\infty + (\mu_0-\mu_\infty)\,\big[1+(\lambda\gamma)^{a}\big]^{\frac{n-1}{a}} \).
Cross: \( \mu = \mu_\infty + \dfrac{\mu_0-\mu_\infty}{1+(\lambda\gamma)^{m}} \).

Metzner–Reed generalized Reynolds (power‑law): \( \displaystyle \mathrm{Re}_{g,\mathrm{MR}}= \frac{\rho\,v^{\,2-n} D^{\,n}}{K\,8^{\,n-1}\left(\frac{3n+1}{4n}\right)^{\,n-1}} \). This reduces to the classical definition at \(n=1\).

Regime hint: As a practical guide, laminar if \(\mathrm{Re}_g\lesssim 2100\), transitional for 2100–4000, turbulent if \(\mathrm{Re}_g\gtrsim 4000\). Thresholds vary slightly with rheology and entrance effects.

How to use

Enter density, mean velocity, and pipe diameter using convenient units.
Select a rheology model and provide the corresponding parameters.
Press Calculate to compute wall shear rate, apparent viscosity, and Reynolds.
Review the regime classification and, for power‑law, the Metzner–Reed value.
Use the example rows to explore typical fluids; click to auto‑fill inputs.
Export the example table as CSV or create a quick PDF summary.

All computations are performed in SI internally. Results are intended for steady, fully developed, incompressible, single‑phase pipe flow.

FAQs

It extends the classical Reynolds definition to non‑Newtonian fluids by using an appropriate apparent viscosity or, for power‑law, the Metzner–Reed expression that embeds shear‑rate corrections.

A common laminar estimate is \(8v/D\). For power‑law and related fluids, Metzner–Reed suggests multiplying by \((3n+1)/(4n)\). The tool applies these automatically for the supported models.

As a rule of thumb using the generalized Reynolds value: laminar < 2100, transitional 2100–4000, turbulent > 4000. Real thresholds depend on rheology, roughness, and entry length.

For power‑law fluids in pipes, Metzner–Reed is standard and aligns many friction‑factor correlations. The apparent‑viscosity form is more general for arbitrary viscosity models.

Yes. Set a flow index n > 1 for power‑law, or choose a Carreau/Cross parameter set that increases viscosity with shear rate.

Yes. Use Herschel–Bulkley and provide a nonzero yield stress. The tool treats yield via an apparent‑viscosity term \( \tau_y/\gamma_w \).

Assumes steady, fully‑developed, isothermal, single‑phase, incompressible pipe flow. Entrance effects, viscoelasticity, wall slip, and strong temperature variations are outside scope.

1) What is the generalized Reynolds number?

2) Which wall shear rate should I use?

3) What ranges indicate laminar or turbulent?

4) When should I use Metzner–Reed instead of μapp?

5) Can this handle shear‑thickening fluids?

6) Does it support yield‑stress fluids?

7) What are the limitations?

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