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Non-Newtonian Fluid Simulator

Bingham Plastic Fluid Simulator — Yield-Stress Non-Newtonian Fluid

Real-time evaluation of the Bingham plastic constitutive law $\tau = \tau_y + \mu_p\,\dot{\gamma}$ for shear stress $\tau$, apparent viscosity $\mu_{\text{app}}$, Bingham number $Bi$ and the BP pipe Reynolds number. Adjust the yield stress $\tau_y$, plastic viscosity $\mu_p$, shear rate $\dot{\gamma}$ and fluid density $\rho$ to visualise the flow curve and the pipe velocity profile of yield-stress fluids such as toothpaste, paint, fresh concrete and drilling mud, including the central rigid plug.

Parameters
Yield stress tau_y
Pa
Plastic viscosity mu_p
Pa·s
Shear rate gamma-dot
1/s
Fluid density rho
kg/m³

The BP Reynolds number assumes pipe diameter $D = 10$ mm and mean speed $U = 1$ m/s and uses the plastic viscosity: $Re_{BP} = \rho U D / \mu_p$. Bingham number $Bi = \tau_y / (\mu_p \dot{\gamma})$.

Results
—
Shear stress τ
—
Apparent viscosity μ_app
—
Bingham number Bi
—
BP Reynolds number
Flow curve τ vs γ̇

x-axis = shear rate $\dot{\gamma}$ (1/s) / y-axis = shear stress $\tau$ (Pa) / yellow line = Bingham fluid $\tau = \tau_y + \mu_p \dot{\gamma}$ with y-intercept at $\tau_y$ / dashed cyan = Newtonian reference $\tau = \mu_p \dot{\gamma}$ through the origin / yellow dot = current $(\dot{\gamma},\tau)$.

Pipe velocity profile u(r)

Axisymmetric pipe cross section of radius $R$ / central shaded band = plug flow (rigid core, $r < r_p$) / outer region = viscous shear flow. The plug fraction $r_p / R$ comes from the ratio of yield stress to wall shear stress; a higher Bi makes the plug thicker. Yellow curve = current profile.

Theory & Key Formulas

Bingham plastic constitutive equation ($\tau_y$ = yield stress in Pa, $\mu_p$ = plastic viscosity in Pa s, $\dot{\gamma}$ = shear rate in 1/s):

$$\tau = \tau_y + \mu_p\,\dot{\gamma}\quad(\tau \ge \tau_y)$$

For $\tau < \tau_y$ the material is rigid ($\dot{\gamma} = 0$). The apparent viscosity is $\mu_{\text{app}} = \tau / \dot{\gamma}$:

$$\mu_{\text{app}} = \dfrac{\tau_y}{\dot{\gamma}} + \mu_p$$

Bingham number (yield stress vs viscous stress) and BP Reynolds number (defined with $\mu_p$):

$$Bi = \dfrac{\tau_y}{\mu_p\,\dot{\gamma}},\qquad Re_{BP} = \dfrac{\rho\,U\,D}{\mu_p}$$

At this tool's defaults ($\tau_y = 20$ Pa, $\mu_p = 0.5$ Pa·s, $\dot{\gamma} = 10$ 1/s, $\rho = 1500$ kg/m³, $D = 10$ mm, $U = 1$ m/s) we get $\tau = 25.0$ Pa, $\mu_{\text{app}} = 2.50$ Pa·s, $Bi = 4.00$ and $Re_{BP} = 30.0$.

What is a Bingham Plastic Fluid

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I keep seeing "Bingham plastic" in rheology texts. How does it differ from a Newtonian or a power-law fluid?
🎓
The key feature is a yield stress $\tau_y$. A Newtonian fluid flows for any positive stress, and so does a power-law fluid. A Bingham plastic refuses to deform while $\tau < \tau_y$ — it just sits there like a solid. Once the stress exceeds $\tau_y$ the material starts flowing and obeys a linear law $\tau = \tau_y + \mu_p \dot{\gamma}$, where $\mu_p$ is the plastic viscosity. With this tool's defaults ($\tau_y = 20$ Pa, $\mu_p = 0.5$ Pa·s, $\dot{\gamma} = 10$ 1/s) you can read $\tau = 25.0$ Pa, apparent viscosity $\mu_{\text{app}} = 2.50$ Pa·s, Bingham number $Bi = 4.00$ and $Re_{BP} = 30.0$ directly from the Results card.
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What everyday fluids are Bingham plastics?
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The classic example is toothpaste. It will not pour by itself out of the tube; you have to squeeze it. Once squeezed it holds its shape on the brush. Unstirred paint, fresh concrete, drilling mud, mayonnaise, certain ketchups and most greases behave the same way. In construction, the fact that fresh concrete does not slump under its own weight is exactly Bingham behaviour, and the slump test is essentially a rough measurement of $\tau_y$.
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Plug flow is the boxy-centre velocity profile, right? Why does it appear?
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Exactly — the rigid core moves as one block. Inside a pipe the shear stress varies linearly as $\tau(r) = (\Delta p / 2L)\, r$, zero on the axis and maximal at the wall. In the central region where $\tau(r) < \tau_y$ the yield criterion fails, so the fluid cannot deform; it just rides along at uniform velocity. That central band is labelled "Plug flow (rigid core)" on the lower canvas. Push the $\tau_y$ slider up or pull $\dot{\gamma}$ down and you see the plug fraction $r_p / R$ grow. As $Bi$ tends to infinity the whole pipe locks up.
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How should I read the Bingham number $Bi$ in practice?
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$Bi = \tau_y / (\mu_p \dot{\gamma})$ is the ratio of yield stress to viscous stress. Roughly, $Bi < 1$ means almost Newtonian, $1 \le Bi \le 10$ is a mixed regime with a visible plug, and $Bi > 10$ is dominated by the yield stress. The default $Bi = 4$ sits in the textbook sweet spot: clearly a Bingham material yet still shearing strongly near the wall. Process engineers estimate $Bi$ from the typical shear rate in mixing, conveying or extrusion to pick the right scale of behaviour.

Frequently Asked Questions

All three describe fluids with a yield stress but differ in the post-yield law. Bingham is linear $\tau = \tau_y + \mu_p \dot{\gamma}$ with two parameters and is the simplest. Herschel-Bulkley is a power law $\tau = \tau_y + K \dot{\gamma}^n$ with three parameters, capturing post-yield non-linearity in paints, drilling muds and yoghurt. Casson uses $\sqrt{\tau} = \sqrt{\tau_y} + \sqrt{\mu_C \dot{\gamma}}$ and is the textbook choice for blood and many printing inks. When the data span a wide shear-rate range and curve away from a straight line above yield, Herschel-Bulkley is usually the right next step.
Three approaches dominate. A stress sweep ramps the stress and identifies the value at which the shear rate suddenly jumps. A shear-rate plot fits a straight line in the low-$\dot{\gamma}$ region and extrapolates to $\dot{\gamma} = 0$; the intercept is $\tau_y$. A static method, such as a vane rheometer or the slump test, measures the stress at which the material physically starts to move. The three numbers usually disagree by 20 to 30 % because they distinguish dynamic and static yield stresses, so reporting always notes which definition was used.
The strict Bingham model has infinite apparent viscosity at $\dot{\gamma} = 0$, which makes naive solvers fail. Industrial CFD codes use the Papanastasiou regularisation $\mu_{\text{eff}} = \mu_p + \tau_y (1 - e^{-m \dot{\gamma}}) / \dot{\gamma}$ with $m$ between 100 and 1000 s, replacing the discontinuity with a smooth viscosity. OpenFOAM's BinghamPlastic class and ANSYS Fluent's Bingham option implement variants. Too large an $m$ causes oscillations, too small misses the physics; starting at $m = 500$ s and monitoring residuals is a safe recipe.
The BP Reynolds number $Re_{BP} = \rho U D / \mu_p$ shown by this tool is the simplest definition and is fine for laminar back-of-the-envelope work. For transition prediction you also need the Hedstrom number $He = \rho D^2 \tau_y / \mu_p^2$ and a correlation such as Hanks's $Re_c = 4 \alpha (1 - 4\alpha/3 + (4\alpha/3)^4/3)$ where $\alpha$ is a function of $He$. Large $He$ pushes the transition Reynolds number upward, so fluids with strong yield stress stay laminar at much higher flow rates than Newtonian fluids would.

Real-World Applications

Fresh concrete rheology: Fresh concrete is a mix of cement, water, aggregate and admixtures with typical values $\tau_y = 100$ to $2000$ Pa and $\mu_p = 10$ to $200$ Pa·s. Pumpability, formwork filling and slump flow are essentially determined by this pair. The slump value correlates directly with the yield stress, and self-compacting-concrete design tunes $\tau_y$ down for flowability while preventing segregation. Sweeping $\tau_y$ between 100 and 500 Pa in this tool reveals strong changes in plug radius and apparent viscosity.

Drilling fluids and oilfield flow: Bentonite drilling muds typically have $\tau_y = 5$ to $50$ Pa and $\mu_p = 0.01$ to $0.1$ Pa·s. Cuttings transport relies on a high enough yield stress to prevent the cuttings from settling when circulation stops, while pump power scales with $\mu_p$. Balancing both is a routine optimisation problem. In this tool, raising $\tau_y$ and lowering $\dot{\gamma}$ thickens the rigid plug and demonstrates how cuttings transport improves at the cost of pressure loss.

Toothpaste, cosmetics and food pastes: Toothpaste sits around $\tau_y = 100$ to $300$ Pa and $\mu_p = 1$ to $10$ Pa·s. It needs squeezing to start flowing, holds shape on a brush, then thins under the high shear of saliva — all consequences of a finite yield stress. Mayonnaise, ketchup, face cream and lipstick follow the same design rules: too low a $\tau_y$ and the product slumps in its container, too high and it feels unpleasant to apply. Product rheology is tightly tied to sensory evaluation.

Blood viscoelasticity and biomechanics: Blood has a tiny yield stress ($\tau_y \approx 0.005$ to $0.05$ Pa) and is more accurately described by the Casson model, but a Bingham approximation is sometimes used in simplified analyses. In low-shear regions like capillaries, red-cell aggregation (rouleaux) creates an effective yield stress that raises the risk of flow stagnation. CFD models of aneurysms and stenoses match measurements better when a yield stress is included. Set $\tau_y$ to 0.1 Pa in this tool and watch how the apparent viscosity changes with shear rate.

Common Misconceptions and Pitfalls

The first misconception is to "treat a Bingham material as perfectly rigid below the yield stress." Real materials always exhibit some creep at $\tau < \tau_y$, deforming slowly on long time scales. That is why toothpaste does not freeze inside the tube and freshly cast concrete still levels out a little. CFD codes that try to enforce true rigidity tend to fail, so production solvers use Papanastasiou or Bercovier-Engelman regularisation to treat the unyielded zone as a very viscous fluid. This tool follows the same educational simplification and does not pretend to capture the strict low-shear physics.

The second misconception is to "identify the plastic viscosity $\mu_p$ with the Newtonian viscosity." $\mu_p$ is the slope of the stress versus shear-rate curve after the yield stress has been removed, not the apparent viscosity. The latter is $\mu_{\text{app}} = \tau_y / \dot{\gamma} + \mu_p$ and grows rapidly as $\dot{\gamma}$ falls. At this tool's defaults $\mu_p = 0.5$ Pa·s but $\mu_{\text{app}} = 2.5$ Pa·s — a factor of five. Using $\mu_p$ directly in a pipe pressure-drop calculation underestimates the loss; use $\mu_{\text{app}}$ at the process shear rate or solve the Buckingham-Reiner equation directly.

Finally, do not "assume the linear Bingham law fits every yield-stress fluid." Many real materials such as paint, drilling mud and yoghurt show clear post-yield shear thinning, and the linear law overestimates the stress at high $\dot{\gamma}$. When the data span a wide shear-rate range the Herschel-Bulkley model ($\tau = \tau_y + K \dot{\gamma}^n$) is the standard upgrade. The Bingham law is appropriate when the operating shear-rate window is narrow or as a first estimate. Sweeping $\dot{\gamma}$ in this tool makes the linear assumption visible — real materials curve away from this straight line.

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