Introduction & Context

Bingham fluids exhibit a finite yield stress τ₀ that must be exceeded before flow begins. Once yielded, the shear stress increases linearly with shear rate, giving a constant plastic viscosity μₚ. Accurate determination of τ₀ and μₚ is essential for sizing pumps, pipes, mixers, and extruders in food, pharmaceutical, mineral, and wastewater processes, and it also provides the basis to calculate dynamic viscosity from shear stress for related rheological analyses. The simplest and most widely accepted method is linear regression of steady‑state rheogram data collected in the laminar regime.

Methodology & Formulas

Data acquisition
Measure steady shear stress τ at several shear rates γ̇ in the laminar range. Ensure the instrument gap and geometry satisfy the following limits:

Regime	Condition	Typical threshold
Laminar (Couette)	Re = ρ ω R δ / μ_p	Re < 1700
No slip at wall	roughness < 5 % gap	—

Linearisation
The Bingham model is
\[ \tau = \tau_0 + \mu_{\text{p}} \dot{\gamma} \]
which is linear in γ̇. Treat τ as the dependent variable and γ̇ as the independent variable.
Regression coefficients
For n data pairs (γ̇_i, τ_i) compute the sums \[ S_1 = \sum_{i=1}^{n} \dot{\gamma}_i,\quad S_2 = \sum_{i=1}^{n} \dot{\gamma}_i^{2},\quad S_3 = \sum_{i=1}^{n} \tau_i,\quad S_4 = \sum_{i=1}^{n} \dot{\gamma}_i \tau_i \] The slope (plastic viscosity) and intercept (yield stress) are \[ \mu_{\text{p}} = \frac{n S_4 - S_1 S_3}{n S_2 - S_1^{2}},\qquad \tau_0 = \frac{S_3 - \mu_{\text{p}} S_1}{n} \]
Unit conversion
Plastic viscosity is often reported in centipoise:
μ_p,cP = μ_p / 10⁻³
Yield stress may be quoted in bar:
τ_0,bar = τ₀ / 10⁵

Collect steady-state shear-stress (τ) data at increasing shear rates (γ̇).
Fit the linear portion of the curve (high-γ̇ region) to the Bingham model: τ = τ_y + μ_p γ̇.
Extrapolate the straight line to γ̇ = 0; the intercept is the yield stress τ_y.
Discard data below the transition shear rate where the curve deviates from linearity; these points represent material creep before yielding.

Controlled-stress rheometer with vane geometry: minimizes wall slip and measures true τ_y within ±2 %.
Stress-growth test: apply a constant low stress and record the strain; the stress at which strain suddenly increases is τ_y.
Slump test (for high-concentration pastes): correlate slump height to τ_y via calibrated empirical equations; accuracy ±5 %.

No. Single-point instruments assume a Newtonian response; Bingham fluids require at least two data points to separate τ_y and plastic viscosity μ_p. Without the intercept, the yield stress is indeterminate.

Yield stress decreases exponentially with temperature: τ_y(T) = τ_y,ref exp[−k(T − T_ref)].
Typical k values for mineral slurries: 0.02–0.04 °C⁻¹; a 20 °C rise can halve τ_y.
Always measure τ_y at the process temperature or apply the Arrhenius correction before scale-up.

Worked Example – Yield Stress of a Drilling Mud

A process engineer is characterising a new water-based drilling mud that must circulate through a 150 mm ID riser. Laboratory rheometer data (shear rate \(\gamma\) vs. shear stress \(\tau\)) were collected at 25 °C and need to be fitted to the Bingham model so that the pump specification can be completed.

Knowns

Number of data points, n = 4
\(\sum\gamma\) = 65 s⁻¹
\(\sum\tau\) = 406 Pa
\(\sum\gamma^2\) = 1425 s⁻²
\(\sum\gamma\tau\) = 8220 Pa·s⁻¹

Step-by-Step Calculation

Compute the denominator for the linear-regression slope: \[ \text{denom} = n\sum\gamma^2 - (\sum\gamma)^2 = 4(1425) - (65)^2 = 1475 \]
Calculate the plastic viscosity \(\mu_\text{p}\): \[ \mu_\text{p} = \frac{n\sum\gamma\tau - \sum\gamma\sum\tau}{\text{denom}} = \frac{4(8220) - (65)(406)}{1475} = 4.4\ \text{Pa·s} \]
Convert \(\mu_\text{p}\) to centipoise for field reporting: \[ \mu_\text{p} = 4.4\ \text{Pa·s} \times 1000 = 4400\ \text{cP} \]
Determine the yield stress \(\tau_0\): \[ \tau_0 = \frac{\sum\tau - \mu_\text{p}\sum\gamma}{n} = \frac{406 - 4.4(65)}{4} = 30\ \text{Pa} \]
Express \(\tau_0\) in bar for hydraulic calculations: \[ \tau_0 = 30\ \text{Pa} \div 100\,000 = 0.0003\ \text{bar} \]

Final Answer

Plastic viscosity μ_p = 4.4 Pa·s (4400 cP) and yield stress τ₀ = 30 Pa (0.0003 bar). The mud behaves as a Bingham fluid with these parameters.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle

Calculation of Yield Stress for Bingham Fluids

Introduction & Context

🚀 Skip the Manual Math!

Methodology & Formulas

Worked Example – Yield Stress of a Drilling Mud