11.4 Rheology of non-Newtonian fluids

Non-Newtonian fluid properties

Non-Newtonian fluids don't follow the simple linear relationship between shear stress and shear rate that defines Newtonian fluids. Instead, their viscosity changes depending on the applied shear rate or shear stress. This behavior shows up everywhere, from ketchup and blood to drilling muds and polymer melts, and understanding it is critical for designing processes that handle these fluids correctly.

Shear rate vs. shear stress

For a Newtonian fluid, plotting shear stress ($\tau$) against shear rate ($\dot{\gamma}$) gives a straight line through the origin, and the slope is the viscosity. Non-Newtonian fluids break this pattern. Their curves can be concave (shear thinning), convex (shear thickening), or offset from the origin (yield stress fluids).

The apparent viscosity at any point on the curve is the local slope $\tau / \dot{\gamma}$. Because the curve isn't linear, apparent viscosity changes with shear rate. This is the single most important distinction from Newtonian behavior.

Viscosity dependence on shear rate

• Shear thinning (pseudoplastic): Viscosity decreases with increasing shear rate. Molecular chains or suspended particles align in the flow direction, reducing internal resistance. Examples: paints, ketchup, polymer solutions.
• Shear thickening (dilatant): Viscosity increases with increasing shear rate. Particle clusters form or particle interactions intensify under faster deformation. Examples: concentrated cornstarch suspensions, some colloidal systems.

Most non-Newtonian fluids encountered in engineering are shear thinning. Shear thickening is less common but can cause serious problems if unexpected (e.g., jamming in high-speed mixing).

Yield stress fluids

Some fluids require a minimum stress, called the yield stress ($\tau_0$), before they begin to flow at all. Below $\tau_0$, the material behaves like a solid and resists deformation. Once $\tau_0$ is exceeded, the material flows.

Toothpaste is a classic example: it holds its shape on your brush (below yield stress) but flows when you squeeze the tube (above yield stress). Other examples include mayonnaise, drilling muds, and fresh concrete.

Shear thinning vs. shear thickening

Types of non-Newtonian fluids

Non-Newtonian fluids are classified by how their shear stress relates to shear rate. Each type has one or more constitutive models that describe its behavior mathematically. Choosing the right model matters because it determines how accurately you can predict pressure drops, flow profiles, and process performance.

Pseudoplastic fluids

Pseudoplastic fluids are shear thinning: their viscosity drops as shear rate increases. The power law model describes them with a flow behavior index $n < 1$. A smaller $n$ means more pronounced thinning.

Examples include polymer solutions, blood, and latex paints. Paint, for instance, thins when you brush it on (high shear) but thickens when it sits on the wall (low shear), preventing drips.

Dilatant fluids

Dilatant fluids are shear thickening: their viscosity rises with increasing shear rate. The power law model applies here too, but with $n > 1$.

The most familiar example is a concentrated cornstarch-water suspension. At low shear rates it flows easily, but if you punch it or run across a pool of it, the sudden high shear rate causes it to resist deformation and behave almost like a solid.

Bingham plastic model

The Bingham plastic model describes fluids that have a yield stress and then flow with a constant (Newtonian) viscosity once that yield stress is exceeded:

$\tau = \tau_0 + \mu_p \dot{\gamma} \quad \text{for } \tau > \tau_0$

• $\tau_0$ = yield stress
• $\mu_p$ = plastic viscosity (constant slope above yield)

This model works well for toothpaste, some drilling muds, and concentrated suspensions where the stress-shear rate relationship is roughly linear above $\tau_0$.

Herschel-Bulkley model

The Herschel-Bulkley model generalizes the Bingham plastic by allowing non-linear (shear thinning or thickening) behavior above the yield stress:

$\tau = \tau_0 + K \dot{\gamma}^n \quad \text{for } \tau > \tau_0$

• $\tau_0$ = yield stress
• $K$ = consistency index
• $n$ = flow behavior index

When $n = 1$, this reduces to the Bingham plastic. When $\tau_0 = 0$, it reduces to the power law. Many food products (yogurt, mayonnaise) and drilling fluids are well described by this three-parameter model.

Casson model

The Casson model also describes yield stress fluids but uses a different mathematical form:

$\sqrt{\tau} = \sqrt{\tau_0} + \sqrt{\mu_c \dot{\gamma}}$

• $\tau_0$ = Casson yield stress
• $\mu_c$ = Casson viscosity

This model is particularly useful for blood and certain food products like chocolate and tomato sauce. It tends to fit experimental data for these fluids better than the Bingham model, especially at low shear rates.

Constitutive equations

Constitutive equations are the mathematical relationships that link shear stress to shear rate (and sometimes to time or deformation history) for a given fluid. Picking the right equation determines how well your predictions match reality. Simpler models (power law) are easy to use but limited in range; more complex models (Carreau, Cross) capture behavior over wider shear rate ranges at the cost of more parameters to fit.

Power law model

The power law (Ostwald-de Waele) model is the simplest and most widely used:

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

• $K$ = consistency index (units depend on $n$)
• $n$ = flow behavior index

Interpretation of $n$:

• $n < 1$: shear thinning
• $n = 1$: Newtonian ($K$ becomes the viscosity)
• $n > 1$: shear thickening

Limitation: The power law predicts viscosity going to infinity as $\dot{\gamma} \to 0$ and to zero as $\dot{\gamma} \to \infty$. Real fluids have finite Newtonian plateaus at both extremes. This means the power law is only valid over a limited range of shear rates.

Carreau model

The Carreau model fixes the power law's plateau problem by including Newtonian behavior at both low and high shear rates:

$\eta = \eta_\infty + (\eta_0 - \eta_\infty)[1 + (\lambda \dot{\gamma})^2]^{(n-1)/2}$

• $\eta_0$ = zero-shear-rate viscosity (low-shear plateau)
• $\eta_\infty$ = infinite-shear-rate viscosity (high-shear plateau)
• $\lambda$ = relaxation time constant (controls where thinning begins)
• $n$ = power law index

This four-parameter model is excellent for polymer solutions and melts where you need predictions across many decades of shear rate.

Cross model

The Cross model serves a similar purpose to the Carreau model but uses a different functional form:

$\eta = \eta_\infty + \frac{\eta_0 - \eta_\infty}{1 + (K \dot{\gamma})^m}$

• $\eta_0$, $\eta_\infty$ = zero-shear and infinite-shear viscosities
• $K$ = time constant
• $m$ = dimensionless exponent controlling the shear thinning transition

The Cross model is especially popular for polymer solutions and melts. In practice, both the Carreau and Cross models fit similar data well; the choice often comes down to convention in a particular field.

Ellis model

The Ellis model is a three-parameter model that works well when you mainly care about the transition from the low-shear Newtonian plateau into the shear thinning region:

$\frac{\eta}{\eta_0} = \frac{1}{1 + (\tau / \tau_{1/2})^{\alpha - 1}}$

• $\eta_0$ = zero-shear-rate viscosity
• $\tau_{1/2}$ = shear stress at which viscosity drops to half of $\eta_0$
• $\alpha$ = parameter controlling the steepness of thinning

One advantage of the Ellis model is that it's written in terms of shear stress rather than shear rate, which can simplify certain pipe flow calculations.

Experimental characterization

Measuring the rheological properties of non-Newtonian fluids requires specialized instruments called rheometers. The choice of rheometer and measurement geometry depends on the fluid's viscosity range, the shear rates of interest, and whether you need steady-shear or oscillatory data. Accurate experimental data is what allows you to select the right constitutive model and fit its parameters.

Rheometer types

Three main categories cover most applications:

1. Rotational rheometers apply a controlled rotation (or controlled torque) to the sample and measure the resulting torque (or rotation). Common geometries include cone-and-plate, parallel plate, and concentric cylinder. These are versatile and cover low to moderate shear rates.

2. Capillary rheometers force fluid through a narrow tube and measure the pressure drop vs. flow rate. They reach high shear rates relevant to extrusion and injection molding.

3. Oscillatory rheometers apply small-amplitude sinusoidal deformations to probe viscoelastic properties without disrupting the fluid's microstructure. These are typically rotational instruments operated in oscillatory mode.

Cone-and-plate geometry

A cone-and-plate setup consists of a flat plate and a shallow cone (typically 1–4° angle). The key advantage is that the shear rate is nearly uniform across the entire sample gap, which gives clean viscosity measurements without needing corrections.

This geometry works well for polymer solutions, suspensions, and food products at low to moderate shear rates. It's not ideal for fluids with large particles (the narrow gap at the cone tip can cause jamming) or for very high shear rates.

Couette flow

Couette flow (concentric cylinder or bob-and-cup geometry) places the sample in the annular gap between two cylinders. Typically the inner cylinder (bob) rotates while the outer cylinder (cup) stays fixed.

This geometry is well suited for low-viscosity fluids that would flow off a cone-and-plate setup. The shear rate isn't perfectly uniform across the gap (it varies with radial position), so corrections may be needed for strongly non-Newtonian fluids unless the gap is kept narrow.

Capillary rheometer

A capillary rheometer pushes fluid through a tube of known diameter and length while measuring the pressure drop and volumetric flow rate. From these, you can extract the wall shear stress and apparent shear rate.

Steps for a capillary rheometry measurement:

1. Force the fluid through the capillary at a controlled flow rate.
2. Measure the pressure drop $\Delta P$ across the capillary.
3. Calculate the wall shear stress: $\tau_w = \frac{\Delta P \cdot R}{2L}$ (where $R$ is the capillary radius and $L$ is its length).
4. Calculate the apparent shear rate: $\dot{\gamma}_{app} = \frac{4Q}{\pi R^3}$ (where $Q$ is the volumetric flow rate).
5. Apply the Weissenberg-Rabinowitsch correction to get the true wall shear rate for non-Newtonian fluids.
6. Apply the Bagley correction to account for entrance and exit pressure losses.

Capillary rheometers are the standard tool for characterizing polymer melts at the high shear rates seen in extrusion and injection molding.

Oscillatory shear measurements

Oscillatory measurements apply a small sinusoidal strain $\gamma(t) = \gamma_0 \sin(\omega t)$ and measure the resulting stress. For a viscoelastic material, the stress response is out of phase with the strain, and this phase difference reveals the balance between elastic and viscous behavior.

The two key quantities extracted are:

• Storage modulus ($G'$): Measures the elastic (energy-storing) component. Higher $G'$ means more solid-like behavior.
• Loss modulus ($G''$): Measures the viscous (energy-dissipating) component. Higher $G''$ means more liquid-like behavior.

When $G' > G''$, the material is predominantly elastic (gel-like). When $G'' > G'$, it's predominantly viscous (liquid-like). The crossover frequency where $G' = G''$ gives a characteristic relaxation time of the material.

Viscoelastic behavior

Many non-Newtonian fluids are not purely viscous; they also store elastic energy during deformation. These viscoelastic fluids exhibit time-dependent responses: they can partially spring back after deformation, show stress relaxation under constant strain, and creep under constant stress. Polymer melts and solutions are the most common examples.

Elastic vs. viscous response

• Purely elastic (Hookean solid): Stress is proportional to strain. Deformation is instantaneous and fully recoverable. Think of a spring.
• Purely viscous (Newtonian fluid): Stress is proportional to strain rate. Deformation is irreversible. Think of a dashpot (piston in a cylinder of oil).
• Viscoelastic: The response depends on the timescale of deformation. Fast deformations emphasize the elastic response; slow deformations emphasize the viscous response.

The Deborah number ($De = \lambda / t_{obs}$) compares the material's relaxation time $\lambda$ to the observation time $t_{obs}$. High $De$ means the material behaves more elastically; low $De$ means it behaves more like a viscous fluid.

Maxwell model

The Maxwell model places a spring (elastic element) and dashpot (viscous element) in series. This arrangement captures stress relaxation: if you suddenly impose a constant strain, the stress decays exponentially over time.

$\tau(t) = \tau_0 \, e^{-t/\lambda}$

• $\lambda = \eta / G$ is the relaxation time (ratio of viscosity to elastic modulus)
• At times much shorter than $\lambda$, the material responds elastically
• At times much longer than $\lambda$, the material flows like a viscous fluid

The Maxwell model does not predict creep behavior well. Under a constant stress, it predicts unbounded linear deformation (like a pure fluid), which is unrealistic for many viscoelastic solids.

Kelvin-Voigt model

The Kelvin-Voigt model places a spring and dashpot in parallel. This arrangement captures creep behavior: under a constant applied stress, the strain increases gradually and asymptotically approaches a steady-state value.

$\gamma(t) = \frac{\tau_0}{G}\left(1 - e^{-t/\tau_r}\right)$

• $\tau_r = \eta / G$ is the retardation time
• The strain never exceeds $\tau_0 / G$ (the elastic limit)

The Kelvin-Voigt model does not predict stress relaxation. Under a sudden constant strain, it predicts a constant stress forever, which is unrealistic for fluids.

Generalized linear viscoelastic models

Real materials rarely have a single relaxation or retardation time. To capture more realistic behavior:

• Generalized Maxwell model: Multiple Maxwell elements arranged in parallel. Each element has its own relaxation time $\lambda_i$ and modulus $G_i$. The total stress relaxation modulus is $G(t) = \sum_i G_i \, e^{-t/\lambda_i}$. This gives a spectrum of relaxation times.
• Generalized Kelvin-Voigt model: Multiple Kelvin-Voigt elements arranged in series. Each element has its own retardation time. This gives a spectrum of retardation times for creep behavior.

These generalized models are fitted to experimental data (relaxation tests or creep tests) to extract the discrete spectrum of time constants.

Relaxation time vs. retardation time

These two quantities describe different experiments:

• Relaxation time ($\lambda$): How quickly stress decays when you hold strain constant. Measured in stress relaxation experiments. Associated with the Maxwell model.
• Retardation time ($\tau_r$): How quickly strain builds up when you hold stress constant. Measured in creep experiments. Associated with the Kelvin-Voigt model.

For a single Maxwell element, $\lambda = \eta / G$. For a single Kelvin-Voigt element, $\tau_r = \eta / G$. The numerical values are the same for these simple models, but they describe physically different processes. In generalized models with multiple elements, the relaxation and retardation spectra are related but not identical.

Flow behavior of non-Newtonian fluids

When non-Newtonian fluids flow through pipes, dies, and channels, several phenomena arise that don't occur (or are negligible) with Newtonian fluids. These effects stem from shear-dependent viscosity and elastic memory, and they have direct consequences for equipment design and product quality.

Entrance effects in pipes

When fluid enters a pipe, the velocity profile transitions from a roughly uniform (plug-like) profile at the entrance to a fully developed profile downstream. The distance over which this transition occurs is the entrance length.

For non-Newtonian fluids, entrance lengths can differ significantly from the Newtonian case:

• Shear thinning fluids develop flatter velocity profiles (closer to plug flow) than Newtonian fluids, which can alter the entrance length.
• Viscoelastic fluids store elastic energy during the developing flow, which can create additional pressure losses and vortex formation near the entrance.

Failing to account for entrance effects leads to errors in pressure drop calculations, especially in short pipes or dies.

Drag reduction

Adding small concentrations of high molecular weight polymers (parts per million) to a turbulent flow can reduce friction drag by up to 80%. This is called the Toms effect, and it's one of the most striking non-Newtonian phenomena.

The mechanism involves polymer molecules stretching in the turbulent flow and suppressing the small-scale eddies that contribute to turbulent energy dissipation. Applications include:

• Long-distance oil pipelines (the Trans-Alaska Pipeline uses drag-reducing agents)
• Fire hoses (longer throw distance for the same pump pressure)
• Marine vessel hull coatings

Shear thinning alone can also reduce drag in turbulent flow, since the effective viscosity drops in the high-shear near-wall region.

Extrudate swell

When a viscoelastic fluid exits a die or nozzle, its cross-section expands. This is extrudate swell (or die swell). The fluid "remembers" the elastic deformation it experienced inside the die and partially recovers that deformation once the constraining walls are gone.

Typical swell ratios (exit diameter / die diameter) range from about 1.1 to 3.0 or more, depending on:

• The fluid's elasticity (higher normal stress differences produce more swell)
• The length-to-diameter ratio of the die (longer dies allow more relaxation, reducing swell)
• The flow rate and temperature

Die swell in extrusion

Die swell is the specific manifestation of extrudate swell in extrusion processes, where polymer melts are forced through shaped dies to produce pipes, sheets, films, and profiles.

Controlling die swell is essential because the final product dimensions differ from the die dimensions. Engineers compensate by:

1. Designing the die opening smaller than the desired product cross-section.
2. Using longer die land lengths to allow partial stress relaxation before exit.
3. Adjusting extrusion speed and melt temperature.
4. Applying post-die calibration (cooling fixtures that constrain the extrudate to the desired shape).

Melt fracture

At high extrusion rates, the surface of the extrudate can become distorted. This instability is called melt fracture and appears in several forms of increasing severity:

• Sharkskin: Fine surface roughness caused by tensile failure of the melt at the die exit.
• Stick-slip: Alternating smooth and rough sections, caused by periodic transitions between slip and no-slip at the die wall.
• Gross melt fracture: Severe, irregular distortions of the entire extrudate cross-section.

Melt fracture sets an upper limit on production rate. Mitigation strategies include using dies with gradual tapers (avoiding sharp corners), adding processing aids (e.g., fluoropolymer coatings on die walls), and adjusting melt temperature.

Applications of non-Newtonian fluids

Non-Newtonian behavior shows up across a wide range of industries. In each case, the shear-dependent viscosity or viscoelastic properties are either exploited for a functional purpose or must be accounted for to avoid process failures.

Polymer processing

Polymer melts and solutions are almost always non-Newtonian. Their shear thinning behavior is actually beneficial during processing: it means the melt flows more easily through narrow die channels and mold cavities at the high shear rates encountered during extrusion and injection molding, but maintains structural integrity at rest.

Key rheological concerns in polymer processing include:

• Selecting the right constitutive model (Carreau or Cross for viscosity; viscoelastic models for die swell and melt fracture predictions)
• Controlling die swell to achieve target product dimensions
• Avoiding melt fracture by staying below critical wall shear stress values
• Accounting for temperature-dependent viscosity changes during cooling

Blood flow in arteries

Blood is a shear thinning fluid. At low shear rates (in veins and small vessels), red blood cells aggregate into stacks called rouleaux, increasing viscosity. At higher shear rates (in large arteries), these aggregates break apart and cells deform and align, reducing viscosity.

The Casson model is commonly used to describe blood rheology, with a yield stress of roughly 0.005–0.01 Pa and a Casson viscosity that captures the shear thinning behavior. Understanding blood rheology is critical for designing artificial heart valves, stents, and dialysis equipment, and for predicting flow patterns in diseased arteries where geometry changes (stenosis) create regions of abnormal shear.