Non-Newtonian Fluid Flow in Pipes

Understanding how fluids like ketchup, concrete, or blood move through pipes is critical for designing efficient industrial processes, medical devices, and food production lines. Unlike water, these fluids do not obey simple rules; their resistance to flow changes depending on how fast you try to move them. This breaks down the engineering analysis for two major classes of these complex fluids—power-law fluids and Bingham plastics—providing you with the modified tools to predict pressure drops and flow behavior in real-world systems.

Beyond Constant Viscosity: Defining Non-Newtonian Behavior

A Newtonian fluid, like water or simple oils, has a constant viscosity (its internal resistance to flow) regardless of the applied shear rate—the speed gradient within the fluid. In pipe flow, this leads to the classic parabolic velocity profile described by the Hagen-Poiseuille equation. The key departure for non-Newtonian fluids is that their apparent viscosity is not a constant property; it is a function of the shear rate itself. This means the fluid's "thickness" changes as it flows faster or slower. This behavior is categorized using rheological models, the most common for pipe flow analysis being the power-law and Bingham plastic models. You encounter these fluids daily: shampoos, polymer melts, paint, blood, and food products like yogurt or soup.

Analyzing Power-Law Fluid Flow

Power-law fluids are characterized by a viscosity that varies with shear rate according to a simple mathematical relationship. Their flow behavior is defined by two parameters: the consistency index (K), which indicates the general "thickness" of the fluid, and the flow behavior index (n), which determines how shear-thinning or shear-thickening the fluid is.

• If $n<1$, the fluid is shear-thinning (pseudoplastic): its apparent viscosity decreases with increasing shear rate. Think of paint—it flows easily when brushed (high shear) but doesn't drip off the brush at rest (low shear). Most polymers and biological fluids are shear-thinning.
• If $n>1$, the fluid is shear-thickening (dilatant): its apparent viscosity increases with shear rate. A mixture of cornstarch and water is a classic example—it acts like a solid under sudden force but flows like a liquid when handled gently.
• If $n=1$, the model reduces to a Newtonian fluid with $K$ representing the constant viscosity.

To analyze steady, laminar flow of a power-law fluid in a circular pipe, you must modify the Newtonian Hagen-Poiseuille approach. The derivation starts with the power-law constitutive equation and integrates the force balance across the pipe radius. The resulting key equations for volumetric flow rate ($Q$) and pressure drop ($\Delta P$) over pipe length ($L$) are:

$$\Delta P=\frac{2KL}{R}{\left(\frac{(3n+1)Q}{n\pi R^3}\right)}^n$$

and the more common form:

$$Q=\frac{n\pi R^3}{3n+1}{\left(\frac{R\Delta P}{2KL}\right)}^{1/n}$$

Notice how the Newtonian result ($Q\propto \Delta P$) is recovered when $n=1$. For shear-thinning fluids ($n<1$), a given pressure drop yields a higher flow rate than a Newtonian fluid with the same zero-shear viscosity would, due to the viscosity dropping in the high-shear region near the pipe wall.

Analyzing Bingham Plastic Flow

Bingham plastics require a minimum stress to initiate flow, known as the yield stress (${\tau }_0$). Below this stress, the material behaves like a rigid solid. Once the applied shear stress exceeds ${\tau }_0$, it flows like a Newtonian fluid with a constant plastic viscosity (${\mu }_p$). This is modeled as $\tau ={\tau }_0+{\mu }_p\dot{\gamma }$ for $\tau >{\tau }_0$.

This yield stress has a profound effect on the velocity profile in a pipe. In the central region of the pipe, where the shear stress (which increases linearly from zero at the centerline to a maximum at the wall) is less than ${\tau }_0$, the fluid does not yield. This creates a plug flow core region—a solid-like plug moving with uniform velocity. Flow only occurs in an annular region near the pipe wall where the local shear stress exceeds the yield stress.

The key parameter for analysis is the yield stress radius ($r_0$), the radial position where the shear stress equals ${\tau }_0$. It is defined by $r_0=(2{\tau }_{0}L)/(\Delta P)$. The flow rate equation for a Bingham plastic, known as the Buckingham-Reiner equation, is more complex:

$$Q=\frac{\pi R^4\Delta P}{8{\mu }_{p}L}\left[1-\frac{4}{3}\left(\frac{{\tau }_0}{{\tau }_w}\right)+\frac{1}{3}{\left(\frac{{\tau }_0}{{\tau }_w}\right)}^4\right]$$

where ${\tau }_w=(R\Delta P)/(2L)$ is the wall shear stress. For flow to occur at all, you must ensure ${\tau }_w>{\tau }_0$. This model accurately describes materials like toothpaste, drilling mud, certain food pastes, and fresh concrete.

Practical Applications and Model Selection

The power-law and Bingham plastic models are indispensable for designing systems involving polymers (in extrusion and molding), slurries (in mining and wastewater transport), blood (in cardiovascular device design), and food processing fluids (like sauces and purees). Selecting the correct model is the first critical step.

Use the power-law model for fluids without a true yield stress but whose viscosity changes continuously with shear rate. It is excellent for polymer solutions and melts. Use the Bingham plastic model for fluids that demonstrably sit still under low stress, like ketchup in a bottle or a paste on a trowel. Some materials, like certain clays or greases, may require more complex models (e.g., Herschel-Bulkley, which combines a yield stress with power-law behavior), but the two covered here form the essential foundation.

Common Pitfalls

1. Assuming Laminar Flow Without Verification: The transition to turbulence for non-Newtonian fluids is not defined by a simple Reynolds number of 2100. You must use a generalized Reynolds number. For power-law fluids, it is defined as $R{e}_{gen}=(\rho {\bar{V}}^{2-n}{D}^n)/(K{8}^{n-1})$, where criteria for transition depend on the flow behavior index $n$. Neglecting this can lead to severe pressure drop miscalculations.

1. Misapplying the Shear Stress Profile: A fundamental truth in steady, fully-developed pipe flow is that the shear stress varies linearly from zero at the centerline to a maximum at the wall: $\tau (r)=(r\Delta P)/(2L)$. This holds regardless of whether the fluid is Newtonian or non-Newtonian. The mistake is to assume the shear rate or velocity profile is linear. The fluid's rheological model (power-law, Bingham) dictates how shear rate and velocity relate to this linear stress distribution.

1. Forgetting the Yield Stress in System Startup: When designing a pumping system for a Bingham plastic, calculating the operating pressure drop is not enough. You must also calculate the pressure required to initiate flow from a static condition, which must overcome the yield stress throughout the entire pipe volume. This startup pressure can be significantly higher than the maintenance flow pressure.

1. Extrapolating Model Parameters Beyond Measured Data: The power-law model is an empirical fit over a specific range of shear rates. Using your calculated $K$ and $n$ values to predict behavior at shear rates far outside the measured range can lead to wildly inaccurate results. Always note the valid shear rate range for your rheological data.

Summary

• Non-Newtonian fluids, such as polymers, slurries, and food products, do not have a constant viscosity; it changes with the applied shear rate.
• Power-law fluids are described by a consistency index ($K$) and a flow behavior index ($n$). Analysis requires modifying the Hagen-Poiseuille equation, resulting in a nonlinear relationship between flow rate and pressure drop.
• Bingham plastics possess a yield stress (${\tau }_0$) that must be exceeded before flow begins, leading to a characteristic plug flow core region in pipe flow.
• Correct system design requires using the appropriate rheological model, verifying laminar flow conditions with a generalized Reynolds number, and accounting for startup stresses in yield-stress fluids.