Directions

1. Pick an example fluid, or set the parameters of each fluid with the sliders.
2. Each example fluid button loads parameter values and plots only the fluid model that matches the example, except default, which plots all three. The text under the plots discusses only the curves that are shown. The values are illustrative, not measurements of real fluids. To compare with another fluid model, move one of its sliders. That curve is added to the plots and to the text, and the example button stays selected. Changing a parameter of the example's own fluid model deselects the button. The examples are:
   • default: μ = 0.5 Pa·s, K = 1.5 Pa·sⁿ with n = 0.5, and τ₀ = 15 Pa with μp = 0.4 Pa·s. The three curves cross near 10 s⁻¹, so the differences are easy to compare.
   • polymer solution: strongly shear-thinning (n = 0.35) with almost no yield stress. Its viscosity drops steadily as the shear rate increases.
   • paste (yield stress): a large yield stress (τ₀ = 45 Pa), like toothpaste. It does not flow in the pipe unless τw exceeds τ₀.
   • cornstarch slurry: shear-thickening (n = 1.5). Its viscosity increases as the shear rate increases.
3. Read the flow curve and the apparent-viscosity plot. Switch the flow curve between linear and log–log axes.
4. Hover over or click either plot to move the shear-rate probe, or use the γ̇ slider. The table lists τ, η and pipe velocity at that shear rate.
5. Change the wall shear stress τw to see how the velocity profiles in the pipe respond.
6. Try n < 1 for shear-thinning and n > 1 for shear-thickening, and raise τ₀ until the Bingham fluid stops flowing.

Details

Shear stress τ is related to shear rate γ̇ by a constitutive equation. The apparent viscosity is η = τ / γ̇.

• Newtonian: τ = μ γ̇, so η = μ is constant.
• power-law (Ostwald–de Waele): τ = K γ̇ⁿ, so η = K γ̇ⁿ⁻¹. n < 1 is shear-thinning, n = 1 is Newtonian and n > 1 is shear-thickening.
• Bingham plastic: τ = τ₀ + μp γ̇ for τ > τ₀, and no deformation for τ ≤ τ₀. The apparent viscosity η = τ₀ / γ̇ + μp decreases with shear rate.

Pipe flow is steady, laminar and fully developed in a pipe of radius R = 1 cm. The shear stress varies linearly, τ = τw r / R, and the velocity profile comes from integrating the constitutive equation with v = 0 at the wall:

• Newtonian: v = τw (R² − r²) / (2 μ R)
• power-law: v = (τw / K R)^1/n · n/(n+1) · [R^(n+1)/n − r^(n+1)/n]
• Bingham: for r ≥ r₀ = τ₀ R / τw, v = τw (R² − r²) / (2 μp R) − τ₀ (R − r) / μp; the core r < r₀ moves as a plug at the velocity at r₀.

Definition of terms:

• τ: shear stress (Pa)
• γ̇: shear rate (s⁻¹)
• η: apparent viscosity, η = τ / γ̇ (Pa·s)
• μ: viscosity of the Newtonian fluid (Pa·s)
• K: consistency index of the power-law fluid (Pa·sⁿ)
• n: flow behavior index of the power-law fluid (dimensionless)
• τ₀: yield stress of the Bingham plastic (Pa)
• μp: plastic viscosity of the Bingham plastic (Pa·s)
• τw: shear stress at the pipe wall (Pa), τw = ΔP R / 2L
• ΔP: pressure drop over the pipe length (Pa)
• L: pipe length (m)
• R: pipe radius (m), fixed at 1 cm
• r: radial position measured from the pipe centerline (m)
• r₀: radius of the rigid plug in the Bingham plastic (m), r₀ = τ₀ R / τw
• v: local fluid velocity (m/s)

Mean velocity is found by numerical integration of v over the cross-section. The example fluids use illustrative parameter values, not measured data.

About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.