Compare Newtonian, power-law, Bingham plastic, and Herschel-Bulkley fluid models on shear stress vs shear rate diagrams. Calculate temperature effects and pipe flow applications.
The core of rheology is the relationship between shear stress (τ), which is the force per area causing flow, and shear rate (˙γ), which is the velocity gradient within the fluid. The simplest model is for Newtonian fluids:
$$ \tau = \mu \dot\gamma $$Here, τ is shear stress (Pa), ˙γ is shear rate (1/s), and μ is the dynamic viscosity (Pa·s), a constant. This is the straight line you control with the "Dynamic Viscosity μ" slider.
Many real-world fluids are more complex. The Herschel-Bulkley model combines a yield stress with a power-law dependence, making it very versatile:
$$ \tau = \tau_0 + K \dot\gamma^n $$In this equation, τ₀ is the yield stress (Pa) – the stress needed to start flow. K is the consistency index (Pa·sⁿ), a measure of thickness. n is the flow index (dimensionless): n < 1 means shear-thinning, n > 1 means shear-thickening, and n = 1 recovers the Bingham model. Play with all three sliders to see how the curve shape changes.
Food & Consumer Products: Ketchup (Bingham plastic) must stay on the burger but flow when you shake the bottle. Toothpaste needs a yield stress to stay on the brush. Mayonnaise and yogurt are also carefully designed non-Newtonian fluids where rheology controls texture and stability.
Paints & Coatings: Paint is strongly shear-thinning (Power-Law with n < 1). It's thick so it doesn't drip off the roller, but under the high shear of brushing or rolling, its viscosity drops dramatically for smooth, even application.
Oil & Gas Drilling ("Drilling Mud"): Drilling mud is a classic Bingham plastic. Its yield stress allows it to suspend rock cuttings when circulation stops, preventing them from settling and blocking the wellbore. The plastic viscosity controls flow resistance during pumping.
Polymer Processing & 3D Printing: Molten plastics and printing filaments are often shear-thinning. This allows them to be pumped easily through extruders and nozzles under high shear but hold their shape after deposition. The temperature effect, modeled by the Arrhenius equation μ(T) = A e^(Ea/RT), is critical here as viscosity is highly temperature-sensitive.
First, understand that you cannot directly compare "viscosity μ" and "consistency coefficient K" because their units are different. The unit for Newtonian fluid viscosity μ is [Pa·s], but the unit for the power-law K is [Pa·sn]. The dimension changes if the flow index n is not 1. For example, is a fluid with K=10 Pa·s0.5 "more viscous" than a Newtonian fluid with a viscosity of 10? You can't say that definitively. They are equal at a shear rate of 1 1/s, but the relationship can reverse at different speeds. Get into the habit of using the simulator to compare both models side-by-side across the entire range.
Next, note that the "plastic viscosity μp" of a Bingham plastic is not its "apparent viscosity" after yielding. While it is the parameter representing resistance to flow after yielding begins, the actual apparent viscosity η is given by η = τ/γ̇ = τ0/γ̇ + μp. This means the apparent viscosity becomes extremely high at low shear rates due to the τ0/γ̇ term. This is why the resistance you feel is completely different when squeezing a tube slowly versus squeezing it rapidly. When fitting real data, you must properly capture data in the low shear rate region to correctly estimate μp and τ0.
Finally, remember that the "power-law index n" is not a universal parameter. Even for shear-thinning fluids (n<1), extremely high shear rates can cause molecular chains to break, often leading to a return to Newtonian behavior. Think of the power law as an "approximate model" that describes the intermediate shear rate region well. In practice, you must always check if the shear rates in your target process (e.g., during coating or filling) fall within the model's valid range.
The rheology models handled by this tool are essential knowledge in the field of polymer processing. For instance, when injecting molten plastic into a mold, the resin passes through the nozzle at high shear rates (where shear-thinning lowers viscosity, aiding filling). Subsequently, the temperature dependence of viscosity during cooling in the mold directly affects shrinkage and warpage. Practicing with the simulator to combine the "power law" and "temperature dependence" is a first step in understanding this process.
They are also widely applied in food engineering. Beyond the toothpaste (Bingham plastic) mentioned earlier, mayonnaise, ketchup, yogurt, etc., all have a yield stress. This property enables the functionality of being "easy to dispense from a container, yet retaining shape on bread or a plate." Furthermore, chocolate tempering is essentially the precise control of the relationship between viscosity and temperature (Arrhenius equation).
Moreover, in the cosmetics and healthcare fields, the design of "application feel" depends on rheology. Products are designed to meet complex requirements: shear-thinning for easy spreading on skin with low force (n<1), yet flowing smoothly from the bottle (low τ0), while maintaining a dome shape on the hand (moderate τ0). Understanding how each parameter affects the graph shape in the simulator allows you to touch upon the core of such product design.
As a next step, make the concept of "apparent viscosity" completely your own. The apparent viscosity η for all models is defined as η = τ / γ̇. Try substituting each constitutive equation into this formula. For the power law, for example, it becomes η = K γ̇n-1. If n<1, you can clearly see from the equation that η decreases as γ̇ increases (shear-thinning). Being able to visualize this "apparent viscosity curve" in your mind will significantly enhance your ability to interpret real data.
From a mathematical background, it's also interesting to adopt the perspective of "Taylor expansion". Many non-Newtonian fluid models can be viewed as one form of attempt to represent more complex phenomena using a series with shear rate γ̇ as the variable. For example, taking the logarithm of the power law gives log τ = log K + n log γ̇, a straight line. If a log-log plot of measured data is linear, it means the power law can describe it well. Thinking about "how to plot data to make it appear linear" is fundamental to engineering research.
Finally, to build upon what you've learned with this tool, turn your attention to "time-dependent" behavior. The models covered so far only consider the instantaneous stress for a given shear rate (steady flow). However, phenomena like thixotropy—where stirred yogurt returns to its original firmness over time—involve stress and viscosity changing with "time." This directly relates to issues like paint "sagging" or "brush marks." Once your understanding of steady flow is solid, I encourage you to step into the world of time-dependent behavior.