14.1 Classification of Non-Newtonian Fluids

Classification of Fluids

Newtonian fluids follow a simple rule: shear stress is directly proportional to shear rate, with viscosity as the constant linking them. Non-Newtonian fluids break that rule. Their viscosity changes depending on how fast you shear them, how long you shear them, or both. Since most real-world fluids (blood, food products, polymers, drilling muds) are non-Newtonian, classifying their behavior is essential for choosing the right models in engineering analysis.

Newtonian vs Non-Newtonian Fluids

A Newtonian fluid obeys a linear constitutive equation:

$\tau = \mu \frac{du}{dy}$

where $\tau$ is shear stress, $\mu$ is the dynamic viscosity (a constant for a given temperature and pressure), and $\frac{du}{dy}$ is the shear rate (velocity gradient). On a plot of $\tau$ vs. $\frac{du}{dy}$, a Newtonian fluid gives a straight line through the origin. Water, air, and glycerin are common examples. Honey is also Newtonian, though its high viscosity sometimes makes people assume otherwise.

A non-Newtonian fluid has a nonlinear or offset relationship between shear stress and shear rate. Its apparent viscosity (defined as $\eta = \frac{\tau}{\dot{\gamma}}$) changes with shear rate, time, or both. Blood, ketchup, and toothpaste are everyday examples. Because the behavior varies so widely, non-Newtonian fluids are subdivided into three major categories.

Categories of Non-Newtonian Fluids

Time-independent (purely viscous) fluids have shear stress that depends only on the current shear rate, not on how long the fluid has been sheared. The viscosity at any moment is fully determined by the instantaneous shear rate. This category includes:

• Shear-thinning (pseudoplastic) fluids, where viscosity decreases as shear rate increases (paint, ketchup, blood)
• Shear-thickening (dilatant) fluids, where viscosity increases as shear rate increases (concentrated cornstarch-water suspensions, certain dense colloids)
• Viscoplastic fluids, which require a minimum yield stress before they flow at all (toothpaste, mayonnaise)

Time-dependent fluids have shear stress that depends on both the current shear rate and the duration of shearing. Their internal structure gradually builds up or breaks down over time:

• Thixotropic fluids thin over time at a constant shear rate. Their microstructure breaks down with sustained shearing and rebuilds at rest. Yogurt and many paints are thixotropic: they become easier to spread the longer you work them.
• Rheopectic fluids thicken over time at a constant shear rate. This is rarer. Some printer inks and gypsum pastes show rheopectic behavior.

Viscoelastic fluids exhibit both viscous (liquid-like) and elastic (solid-like) responses. When deformed, they partially flow and partially spring back. Polymer solutions, polymer melts, and silly putty all fall into this category. Viscoelastic behavior shows up as stress relaxation (stress decays when strain is held constant) and creep (strain increases slowly under constant stress).

Non-Newtonian Fluid Characteristics

Shear-Dependent Fluids and the Power-Law Model

Both shear-thinning and shear-thickening fluids can be described by the power-law (Ostwald-de Waele) model:

$\tau = K\left(\frac{du}{dy}\right)^n$

Here $K$ is the consistency index (units depend on $n$; higher $K$ means a "thicker" fluid overall) and $n$ is the flow behavior index, which controls the shape of the curve:

• $n < 1$: shear-thinning. Viscosity drops as shear rate rises. Blood ($n \approx 0.7$) and many polymer solutions fall here. Ketchup is a familiar example: shaking or squeezing lowers its resistance to flow.
• $n = 1$: the model reduces to the Newtonian case ($K$ becomes $\mu$).
• $n > 1$: shear-thickening. Viscosity increases with shear rate. A concentrated cornstarch-water mixture stiffens when you punch it because the suspended particles jam together under rapid deformation.

The power-law model is simple and widely used, but it has limitations. It predicts zero viscosity at infinite shear rate for shear-thinning fluids and infinite viscosity at zero shear rate, neither of which is physical. For more accurate behavior at extreme shear rates, models like the Carreau or Cross model are used.

Yield Stress in Viscoplastic Fluids

Some fluids won't flow at all until the applied shear stress exceeds a critical value called the yield stress $\tau_y$. Below $\tau_y$, the material behaves like a solid (it deforms elastically or not at all). Above $\tau_y$, it flows. Toothpaste sits on your brush without dripping because the stress from gravity is below its yield stress; squeezing the tube pushes the stress above $\tau_y$ and it flows.

Two common constitutive models describe viscoplastic fluids:

1. Bingham plastic model: $\tau = \tau_y + \mu_p \frac{du}{dy}$

   Once the yield stress is exceeded, the fluid flows with a constant plastic viscosity $\mu_p$. The relationship between stress and shear rate above $\tau_y$ is linear. Drilling muds and some slurries approximate Bingham behavior.

2. Herschel-Bulkley model: $\tau = \tau_y + K\left(\frac{du}{dy}\right)^n$

   This combines a yield stress with power-law behavior above the yield point. It's more general than the Bingham model because it captures shear-thinning ($n < 1$) or shear-thickening ($n > 1$) after yielding. Many real viscoplastic fluids (food pastes, cement, waxy crude oil) fit a Herschel-Bulkley model better than a simple Bingham model.