14.2 Rheological Models

Rheological Models for Non-Newtonian Fluids

Rheological models are mathematical equations that describe how a fluid's shear stress relates to its shear rate. For Newtonian fluids like water, that relationship is simply a straight line through the origin with a constant slope (the viscosity). Non-Newtonian fluids break that linearity: their effective viscosity can change with shear rate, or they may require a minimum stress before they flow at all. Rheological models capture these behaviors so engineers can predict pressure drops in pipes, size pumps correctly, and design processes like extrusion or mixing for complex fluids.

Purpose of Rheological Models

• Provide a mathematical relationship between shear stress ($\tau$) and shear rate ($\dot{\gamma}$) that characterizes a fluid's flow behavior.
• Allow engineers to predict how a fluid will respond under different flow conditions, rather than relying on trial and error.
• Guide equipment selection and process design: choosing pipe diameters, pump types, and operating speeds all depend on knowing how the fluid behaves under shear.
• Newtonian fluids follow $\tau = \mu \dot{\gamma}$ with a constant viscosity $\mu$. Non-Newtonian models extend this to capture variable viscosity, yield stress, or both.

Power-Law Model (Ostwald–de Waele)

The power-law model is the simplest and most widely used model for fluids whose viscosity changes with shear rate. It's written as:

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

• $\tau$ = shear stress
• $K$ = consistency index, which represents the fluid's effective viscosity at a shear rate of 1 $\text{s}^{-1}$. Higher $K$ means a "thicker" fluid overall.
• $\dot{\gamma}$ = shear rate
• $n$ = flow behavior index, which controls how the viscosity changes with shear rate

The value of $n$ tells you everything about the fluid's character:

• Shear-thinning (pseudoplastic): $n < 1$. Viscosity decreases as shear rate increases. Most polymer solutions, blood, and paint fall here. This is why paint spreads easily under a brush but doesn't drip off a wall.
• Shear-thickening (dilatant): $n > 1$. Viscosity increases as shear rate increases. Concentrated cornstarch-in-water suspensions are the classic example.
• Newtonian: $n = 1$. The model collapses to $\tau = K\dot{\gamma}$, and $K$ simply equals the Newtonian viscosity $\mu$.

Bingham Plastic Model

Some fluids behave as if they're solid until the applied stress exceeds a critical value called the yield stress ($\tau_0$). Below that stress, there's no flow. Above it, the fluid shears linearly like a Newtonian fluid. The Bingham plastic model captures this:

$\tau = \tau_0 + \mu_p \dot{\gamma}$

• $\tau_0$ = yield stress, the minimum shear stress needed to initiate flow
• $\mu_p$ = plastic viscosity, the constant slope of the $\tau$ vs. $\dot{\gamma}$ curve once flow begins
• $\dot{\gamma}$ = shear rate

Toothpaste is a good example: it sits in the tube without flowing (stress below $\tau_0$), but once you squeeze hard enough, it flows steadily. Mayonnaise, drilling muds, and some cement slurries also follow this model reasonably well.

Herschel–Bulkley Model

The Herschel–Bulkley model is essentially a hybrid of the power-law and Bingham models. It accounts for both a yield stress and a nonlinear viscosity response:

$\tau = \tau_0 + K \dot{\gamma}^n$

• $\tau_0$ = yield stress
• $K$ = consistency index
• $n$ = flow behavior index
• $\dot{\gamma}$ = shear rate

This makes it the most versatile of the three models. It reduces to the others as special cases:

Because it has three fitting parameters instead of two, the Herschel–Bulkley model fits a wider range of real fluids, particularly yield-stress fluids that also shear-thin ($n < 1$) or shear-thicken ($n > 1$) above the yield point. Examples include waxy crude oils, food pastes, and many biological fluids.

The trade-off is that fitting three parameters requires more experimental data, and the model can sometimes overfit noisy rheometer measurements. For fluids that are well-described by the simpler power-law or Bingham models, there's no advantage to using Herschel–Bulkley.