14.3 Flow Behavior of Non-Newtonian Fluids

Flow Behavior of Non-Newtonian Fluids

Non-Newtonian fluids have a viscosity that changes depending on how much stress or deformation you apply to them. Unlike Newtonian fluids (where viscosity is constant at a given temperature), these fluids can thin out, thicken up, or refuse to flow until a threshold force is reached. Understanding their flow behavior is essential for designing piping systems, pumps, and industrial processes that handle everything from polymer melts to food products.

Flow of Non-Newtonian Fluids

Non-Newtonian fluids fall into several categories based on how their viscosity responds to shear rate (the rate at which fluid layers slide past each other):

• Shear-thinning (pseudoplastic) fluids have a viscosity that decreases as shear rate increases. Ketchup is a classic example: it resists flowing in the bottle, but once you shake or squeeze it, the viscosity drops and it pours easily. Paint behaves similarly so it spreads smoothly under a brush but doesn't drip off the wall.
• Shear-thickening (dilatant) fluids have a viscosity that increases with shear rate. A cornstarch-water suspension ("oobleck") flows freely when stirred slowly but becomes nearly solid if you punch it or stir it fast.
• Bingham plastics don't flow at all until the applied stress exceeds a threshold called the yield stress ($\tau_0$). Once that threshold is exceeded, they flow roughly like a Newtonian fluid. Toothpaste and mayonnaise are good examples: they hold their shape on a surface but flow when you squeeze or spread them.

Pipe flow differences from Newtonian fluids:

In a Newtonian fluid, fully developed laminar pipe flow produces a parabolic velocity profile. Non-Newtonian fluids deviate from this:

• Shear-thinning fluids develop a flatter velocity profile in the core of the pipe. Because viscosity drops near the wall (where shear rate is highest), the fluid moves more uniformly across the cross-section.
• Shear-thickening fluids show steeper velocity gradients near the wall, since viscosity increases in the high-shear region close to the pipe surface.
• Bingham plastics develop a plug flow region in the center of the pipe where the shear stress is below $\tau_0$. In this central plug, the fluid moves as a solid block, with shearing only occurring in the annular region closer to the wall.

Flow in rectangular channels follows similar trends, but the velocity distribution depends on the channel geometry and the specific rheological model of the fluid.

Calculations for Non-Newtonian Fluids

Rheological Models

These models relate shear stress ($\tau$) to shear rate ($\dot{\gamma}$) and are the starting point for any flow calculation.

Power-law (Ostwald-de Waele) model:

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

• $K$ = consistency index (units depend on $n$; higher $K$ means the fluid is "thicker" overall)
• $n$ = flow behavior index:
  • $n < 1$: shear-thinning
  • $n > 1$: shear-thickening
  • $n = 1$: Newtonian (and $K$ reduces to dynamic viscosity $\mu$)

This model is simple and widely used, but it has no yield stress term, so it can't describe Bingham-type behavior. It also breaks down at very low and very high shear rates where most real fluids plateau to a constant viscosity.

Bingham plastic model:

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

• $\tau_0$ = yield stress (the minimum stress to initiate flow)
• $\mu_p$ = plastic viscosity (the slope of the stress-vs.-shear-rate curve once flow begins)

If $\tau < \tau_0$, the material behaves as a rigid solid and $\dot{\gamma} = 0$.

Pressure Drop in Pipe Flow

For power-law fluids, the modified Hagen-Poiseuille equation gives the pressure drop over a pipe of length $L$:

$\Delta P = \frac{2KL}{R^{1+\frac{1}{n}}}\left(\frac{3n+1}{4n}\right)^n\left(\frac{8V}{D}\right)^n$

where:

• $R$ = pipe radius, $D$ = pipe diameter
• $V$ = mean (bulk average) velocity
• $L$ = pipe length

Notice that when $n = 1$, the correction factor $\left(\frac{3n+1}{4n}\right)$ equals 1 and this reduces to the standard Hagen-Poiseuille result.

For Bingham plastic fluids, the Buckingham-Reiner equation applies:

$\frac{\Delta P}{L} = \frac{4\tau_0}{3R}\left(1-\frac{4}{3}\lambda+\frac{1}{3}\lambda^4\right) + \frac{8\mu_pV}{R^2}\left(1-\frac{\lambda}{2}\right)$

where the dimensionless yield stress parameter is:

$\lambda = \frac{R\tau_0}{2\mu_pV}$

The parameter $\lambda$ compares the yield stress to the viscous stress. When $\lambda \to 0$ (negligible yield stress), the equation approaches the Newtonian Hagen-Poiseuille result.

Velocity Profiles

Power-law fluid in a pipe:

$v(r) = \left(\frac{n}{n+1}\right)\left(\frac{\Delta P R}{2KL}\right)^{\frac{1}{n}}\left[R^{\frac{n+1}{n}}-r^{\frac{n+1}{n}}\right]$

• $v(r)$ = velocity at radial position $r$ from the centerline
• As $n$ decreases below 1, the profile flattens toward plug-like flow
• At $n = 1$, this reduces to the standard parabolic profile

Bingham plastic in a pipe:

$v(r) = \frac{\Delta P R^2}{4\mu_p L}\left(1-\frac{r^2}{R^2}\right) - \frac{\tau_0 R}{2\mu_p}\left(1-\frac{r}{R}\right)$

This expression applies in the sheared annular region ($r \geq r_{\text{plug}}$). Inside the plug radius, the velocity is constant and equal to the value at $r = r_{\text{plug}}$.

Challenges with Non-Newtonian Fluids

Pumping and transport present several difficulties:

• Pressure drops are generally higher than for Newtonian fluids at the same flow rate, which means more pumping power is needed.
• Flow instabilities can develop, especially in shear-thickening fluids or near the yield-stress transition in Bingham plastics. These instabilities can cause non-uniform flow and processing defects.
• Pump selection matters: positive-displacement pumps (like progressive cavity pumps) are often preferred over centrifugal pumps for high-viscosity or yield-stress fluids.

Heat transfer is affected because viscosity influences convective transport:

• Standard Newtonian heat transfer correlations (like Dittus-Boelter) don't apply directly. Modified correlations that account for the shear-rate-dependent viscosity are required.
• High-viscosity regions (e.g., the plug zone in a Bingham plastic) reduce convective mixing, which can lower heat transfer efficiency and create hot spots.

Mixing and agitation require special attention:

• Higher viscosities and yield stresses demand more impeller power. Dead zones can form in regions where the local stress doesn't exceed the yield stress, leading to poor mixing and inconsistent product quality.
• Impeller choice is critical: anchor, helical ribbon, or planetary mixers are common for high-viscosity non-Newtonian fluids, whereas turbine impellers work well for low-viscosity Newtonian systems.

Rheological characterization underpins all of the above:

• Accurate measurement of $K$, $n$, $\tau_0$, and $\mu_p$ requires testing across the relevant shear rate range for your application.
• Temperature and time-dependent effects (thixotropy, rheopexy) can shift the rheological parameters, so testing conditions should match process conditions as closely as possible.

Applications of Non-Newtonian Fluids

Polymer processing is one of the largest application areas:

• Extrusion: Molten polymers are strongly shear-thinning, which is actually beneficial because viscosity drops as the material is forced through the die, reducing the required pressure. Products include plastic sheets, pipes, and films.
• Injection molding: The flow behavior index $n$ directly affects how the melt fills the mold cavity. Poor rheological understanding leads to short shots, weld lines, and warped parts.
• Fiber spinning: Elongational (extensional) viscosity, not just shear viscosity, controls whether a fiber can be drawn stably without breaking. This is critical for producing synthetic fibers and textiles.

Food engineering relies heavily on non-Newtonian fluid mechanics:

• Pumping products like ketchup, mayonnaise, and yogurt requires accounting for yield stress and shear-thinning behavior to size pumps and pipelines correctly.
• Extrusion of pasta and snack foods involves starch-based materials that exhibit complex rheology depending on moisture, temperature, and shear history.
• Dough mixing is a high-viscosity, yield-stress problem where dead zones lead to inconsistent product.

Other industrial applications:

• Drilling fluids (oil and gas) are designed to be shear-thinning with a yield stress: they flow easily during pumping but gel when circulation stops, suspending rock cuttings in the wellbore.
• Blood flow (biomedical engineering): Blood is shear-thinning due to red blood cell aggregation at low shear rates and deformation at high shear rates. This matters for designing artificial organs, stents, and dialysis equipment.
• Slurry transport (mining, construction): Concentrated particle suspensions can be shear-thickening or exhibit yield stresses, affecting pipeline design and energy costs.
• Printing inks and coatings: These formulations are tuned to be shear-thinning so they flow through nozzles or onto rollers but set quickly on the substrate.