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Non-Newtonian Fluid Simulator

Power-Law Fluid Simulator — Ostwald-de Waele Non-Newtonian Model

Real-time evaluation of the Ostwald-de Waele power law $\tau = K\,\dot{\gamma}^{n}$ for the shear stress $\tau$, apparent viscosity $\mu_{\text{app}}$ and pipe-flow apparent Reynolds number. Adjust the consistency index $K$, power-law index $n$, shear rate $\dot{\gamma}$ and fluid density $\rho$ to compare shear-thinning (pseudoplastic), Newtonian and shear-thickening (dilatant) regimes on the flow curve and inside the pipe velocity profile.

Parameters
Consistency index K
Pa·sⁿ
Power-law index n
Shear rate gamma-dot
1/s
Fluid density rho
kg/m³

The apparent Reynolds number assumes diameter $D = 10$ mm and mean speed $U = 1$ m/s. $\mu_{\text{app}} = K\,\dot{\gamma}^{\,n-1}$, $Re_{\text{app}} = \rho U D / \mu_{\text{app}}$.

Results
—
Shear stress τ
—
Apparent viscosity μ_app
—
Fluid type
—
Apparent Re (D=10mm, U=1m/s)
Flow curve τ vs γ̇ (log-log)

x-axis = $\log_{10}\dot{\gamma}$ (1/s) / y-axis = $\log_{10}\tau$ (Pa) / overlay of $\tau(\dot{\gamma})$ for six power-law indices $n$ (0.3, 0.5, 0.7, 1.0, 1.3, 1.7) / yellow dot = current $(\dot{\gamma},\tau)$. The slope is $n$ and the intercept is $\log K$.

Pipe velocity profile u(r)

Axisymmetric pipe cross section of radius $R$ / x-axis = radial location $r/R$ / y-axis = normalized speed $u/u_{\max}$ / $u(r) = u_{\max}\left[1 - (r/R)^{(n+1)/n}\right]$ / parabolic at n=1, flattened for n < 1, sharpened for n > 1.

Theory & Key Formulas

Ostwald-de Waele power-law constitutive equation ($K$ = consistency index in Pa s^n, $n$ = power-law index, $\dot{\gamma}$ = shear rate in 1/s):

$$\tau = K\,\dot{\gamma}^{\,n}$$

Because the apparent viscosity is $\mu_{\text{app}} = \tau/\dot{\gamma}$, it is constant only for a Newtonian fluid (n=1) and varies with shear rate otherwise:

$$\mu_{\text{app}} = K\,\dot{\gamma}^{\,n-1}$$

Laminar pipe-flow velocity profile ($R$ = inner radius, $u_{\max}$ = centerline speed):

$$u(r) = u_{\max}\left[1 - \left(\dfrac{r}{R}\right)^{(n+1)/n}\right]$$

Apparent pipe Reynolds number with diameter $D$, mean speed $U$ and density $\rho$:

$$Re_{\text{app}} = \dfrac{\rho\,U\,D}{\mu_{\text{app}}}$$

$n < 1$ = shear-thinning (pseudoplastic), $n = 1$ = Newtonian, $n > 1$ = shear-thickening (dilatant). At this tool's defaults ($K = 1$ Pa·sⁿ, $n = 0.5$, $\dot{\gamma} = 10$ 1/s) we get $\tau = \sqrt{10} \approx 3.16$ Pa, $\mu_{\text{app}} = 10^{-0.5} \approx 0.316$ Pa·s and $Re_{\text{app}} \approx 31.6$.

What is the Power-Law (Ostwald-de Waele) Fluid Model

🙋
My rheology textbook keeps referring to "power-law fluids." What does that actually mean in practice?
🎓
It is a fluid in which the shear stress and shear rate follow $\tau = K\,\dot{\gamma}^{n}$ instead of a straight line. $K$ is the consistency index that sets the viscosity scale, and $n$ is the power-law index that controls the rheology. $n = 1$ recovers a Newtonian fluid like water or air. $n < 1$ is shear-thinning (pseudoplastic), $n > 1$ is shear-thickening (dilatant). At this tool's defaults ($K = 1$ Pa·sⁿ, $n = 0.5$, $\dot{\gamma} = 10$ 1/s) you can read $\tau = \sqrt{10} \approx 3.16$ Pa and apparent viscosity $\mu_{\text{app}} \approx 0.316$ Pa·s in the Results card.
🙋
What is the practical benefit of being shear-thinning? Are there everyday examples?
🎓
Paint, blood, ketchup, shampoo and yoghurt are all shear-thinning. Brush paint slowly and it does not drip (high viscosity at low $\dot{\gamma}$); brush quickly and it spreads smoothly (low viscosity at high $\dot{\gamma}$). Without this behaviour paint would be unbrushable and blood would not flow through capillaries. Drop $n$ to 0.3 and watch the slope of the flow curve flatten, meaning shear rate barely increases stress, so the apparent viscosity falls.
🙋
And $n > 1$, the dilatant case?
🎓
Corn-starch slurry (oobleck) is the textbook demo. Push a finger in slowly and it sinks; punch it and it rebounds like a solid. As $\dot{\gamma}$ rises, colloidal particles cluster, and the apparent viscosity climbs steeply. Try $n = 1.7$ in this tool and the flow curve becomes nearly vertical. Engineers are now using the same physics in shear-thickening fluid armour and soft-body impact protection.
🙋
Why does the pipe velocity profile change shape so much with n?
🎓
The laminar solution is $u(r) = u_{\max}\left[1 - (r/R)^{(n+1)/n}\right]$. At $n = 1$ the exponent is 2, giving the classical Hagen-Poiseuille parabola. For $n < 1$ the exponent exceeds 2, so the centre flattens into a plug-like profile and the gradient steepens near the wall. For $n > 1$ the exponent drops below 2 and the centre sharpens. The lower canvas shows exactly that, and it is the key to understanding extrusion of polymer melts and blood flow through arteries.

Frequently Asked Questions

Because it breaks down at both ends of the shear-rate window. At low $\dot{\gamma}$ the prediction $\mu_{\text{app}} = K \dot{\gamma}^{n-1}$ diverges for $n < 1$, but real polymer solutions plateau at a finite zero-shear viscosity $\mu_0$. At high $\dot{\gamma}$ the apparent viscosity collapses to zero, missing the infinite-shear plateau $\mu_\infty$. Within the process shear-rate range (typically 1 to 1000 1/s) the power law is a good local fit, but for a span that crosses both plateaus you need Carreau ($\mu = \mu_\infty + (\mu_0 - \mu_\infty)[1 + (\lambda\dot{\gamma})^2]^{(n-1)/2}$) or Cross models.
The apparent Re shown in this tool divides the convective inertia by the local $\mu_{\text{app}}$ at the wall shear rate, which is good enough for laminar back-of-the-envelope estimates but misses the transition point. The Metzner-Reed generalized Reynolds number $Re_{\text{gen}} = \rho U^{2-n} D^n / [K\,((3n+1)/(4n))^n\,8^{n-1}]$ is derived from the laminar Hagen-Poiseuille solution for a power-law fluid and recovers the Newtonian transition value around 2100. Production CFD codes such as Fluent and OpenFOAM compute the generalized Re internally; treat the value shown here as a sanity check.
$K$ usually follows an Arrhenius relation $K(T) = K_0 \exp(E_a / RT)$ with activation energy $E_a$ around 20 to 50 kJ/mol, so a 10 K increase typically lowers viscosity by 20 to 30 %. The power-law index $n$ is much less sensitive to temperature, drifting by only $\pm 0.05$ across normal processing windows. Injection-moulding simulations therefore use $K(T)$ together with a cooling model to predict freeze-off at the flow front, while $n$ is kept constant.
Ostwald-de Waele is a two-parameter model without a yield stress. Fluids that do not flow until $\tau$ exceeds a yield stress $\tau_y$ need either Bingham ($\tau = \tau_y + \mu_p \dot{\gamma}$, linear above yield) or Herschel-Bulkley ($\tau = \tau_y + K \dot{\gamma}^n$, power-law above yield). Fresh concrete, toothpaste and drilling muds all carry yield stress, so the power law alone misses the low-shear regime. Use Ostwald-de Waele only when the yield stress is negligible or the process operates at high enough shear that it is irrelevant.

Real-World Applications

Polymer and plastics processing: Polyethylene and polypropylene melts have $K = 10^3$ to $10^5$ Pa·sⁿ and $n = 0.3$ to $0.6$, strongly shear-thinning. Inside an injection-mould gate the local shear rate reaches $10^3$ to $10^4$ 1/s and the apparent viscosity drops by two orders of magnitude, which is what makes filling possible. Setting $K = 100$, $n = 0.4$ and $\dot{\gamma} = 1000$ 1/s in this tool reproduces that decade-scale drop in $\mu_{\text{app}}$. Mould-flow codes such as Moldflow and Moldex3D rely on exactly this constitutive equation.

Food and cosmetics: Ketchup typically has $n \approx 0.3$ to $0.5$, yoghurt $n \approx 0.5$ to $0.7$ and shampoo $n \approx 0.5$ to $0.8$. Squeezability, brush feel, sensory mouthfeel and gel coating thickness all hinge on the power-law index. Product designers measure $K$ and $n$ on a rheometer and adjust thickeners or emulsifiers to hit a target. Toggling $n$ between 0.3 and 0.8 in this tool makes the contrast in velocity profiles immediately visible.

Blood rheology and biomechanics: Blood is mildly shear-thinning ($K \approx 0.01$ to $0.1$ Pa·sⁿ, $n \approx 0.7$ to $0.9$) because red cells aggregate at low shear and deform at high shear. In CFD studies of aneurysms or stenoses, the wall shear stress distribution computed with this constitutive law is used to assess thrombus formation risk. The pipe-profile canvas at $n = 0.8$ versus $n = 1.0$ highlights the difference from a purely Newtonian approximation.

Drilling fluids and geothermal flow: Bentonite-based drilling muds typically have $n \approx 0.4$ to $0.7$. The ability to lift cuttings to surface (hole-cleaning capability) depends on the velocity profile flatness; a lower $n$ flattens the centre and improves transport. Pump power scales with the high-shear apparent viscosity, so balancing $K$ and $n$ is a routine optimisation problem at the well-site.

Common Misconceptions and Pitfalls

The most common pitfall is to "assume that $n < 1$ guarantees shear thinning at every shear rate, no matter how small." Real polymer solutions exhibit a zero-shear plateau and the viscosity flattens to $\mu_0$ for $\dot{\gamma}$ below about 0.01 1/s. Because Ostwald-de Waele predicts $\mu_{\text{app}} \to \infty$ at $\dot{\gamma} \to 0$ for $n < 1$, CFD solvers can diverge at startup. OpenFOAM's powerLaw class therefore exposes nuMin and nuMax for clipping; this tool sets a minimum shear rate of 0.1 1/s for the same reason. Trust the model only inside the process shear-rate window.

The next pitfall is to "treat the power-law index n as a single global value." $n$ is a local slope: change the shear-rate window and $n$ shifts. A polymer solution may show $n = 0.5$ between 1 and 10 1/s and $n = 0.7$ between 100 and 1000 1/s. Always fit Ostwald-de Waele inside the shear-rate range of the actual process; using a catalogue $n$ as if it applied everywhere causes large prediction errors between the gate region and the runner of an injection mould. Processes spanning multiple decades demand a Carreau or Cross fit.

Finally, do not "transition at $Re = 2100$ using the apparent Re." The transition Reynolds number of a power-law fluid depends on $n$: in Metzner-Reed terms it is about 2400 at $n = 0.5$ and 1800 at $n = 1.5$. The apparent Re displayed here (using the local $\mu_{\text{app}}$) is educational; use Metzner-Reed or the Ryan-Johnson correction for real transition prediction. Sweeping $n$ with the Play button shows the apparent Re moving by orders of magnitude even when $K$, $\dot{\gamma}$ and $\rho$ are unchanged.

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