Buckingham-Reiner Equation

Some materials, like water, flow with the slightest push. Others, like toothpaste or wet concrete, stubbornly refuse to move until a significant force is applied. These materials, known as Bingham plastics, defy the simple laws governing common fluids and are ubiquitous in industry and nature. The central challenge they present is predicting their behavior: how do you describe the flow of something that is part-solid, part-liquid? This article bridges that knowledge gap by exploring the fundamental physics of these complex fluids.

This article will guide you through the core principles that govern Bingham plastics. In the first section, "Principles and Mechanisms," we will deconstruct the concepts of yield stress and the fascinating phenomenon of plug flow, building up to the elegant Buckingham-Reiner equation that ties it all together. Subsequently, in "Applications and Interdisciplinary Connections," we will see this equation in action, discovering its profound impact across diverse fields such as civil engineering, food science, and the design of life-saving biomedical devices.

Imagine squeezing a tube of toothpaste. Nothing happens, nothing happens... and then, with a little more force, it suddenly yields and begins to flow. Now, contrast this with turning on a water tap. The slightest pressure sends water flowing, and more pressure simply means more flow. This everyday observation holds the key to a fascinating class of materials that straddle the line between solid and liquid. Water is a ​​Newtonian fluid​​; its motion is governed by a simple, elegant relationship between the force you apply and how fast it flows. Toothpaste, ketchup, wet concrete, and even some advanced thermal pastes for cooling electronics are different. They are stubborn. They are ​​Bingham plastics​​.

What makes these materials so stubborn? They possess a property called ​​yield stress​​, which we can denote by the symbol $\tau_y$. Think of it as an internal resistance, a threshold that must be overcome. If the internal "rubbing" forces between layers of the material—what physicists call ​​shear stress​​, $\tau$—are less than this yield stress, the material simply refuses to deform. It acts like a solid. It might move as a whole block, but its layers won't slide past one another.

Once you push hard enough, and the local shear stress $|\tau|$ exceeds the yield stress $\tau_y$, the material gives way and begins to flow like a liquid. But it doesn't forget its stubborn nature. The force needed to keep it flowing is still higher than for a simple fluid like water. The relationship for a Bingham plastic is beautifully simple: the total stress you feel is the yield stress you had to overcome, plus an extra bit that depends on how fast you want it to flow. Mathematically, we write this as:

$|\tau| = \tau_y + \mu_p |\dot{\gamma}|$

Here, $|\dot{\gamma}|$ is the ​​shear rate​​, which measures how fast adjacent layers of the fluid are sliding past each other, and $\mu_p$ is the ​​plastic viscosity​​, a property analogous to the normal viscosity of water. This simple equation is the constitution, the fundamental law, governing these peculiar fluids.

Now for the really interesting part. What happens when we pump a Bingham plastic through a simple circular pipe? Let's think about the forces. The fluid is pushed forward by a pressure difference, $\Delta P$, across the length of the pipe, $L$. This push is resisted by the shear stress at the pipe wall. A simple balance of forces tells us that the shear stress is zero right at the center of the pipe ($r=0$) and grows linearly until it reaches its maximum value at the wall ($r=R$). The shear stress at any radius $r$ is given by:

$\tau(r) = \frac{\Delta P}{2L} r$

Do you see the spectacular consequence? Since the stress is zero at the center and grows outwards, there must be a region in the middle of the pipe where the stress is less than the yield stress, $\tau_y$. And what does a Bingham plastic do when the stress is below the yield stress? It acts like a solid!

This leads to a remarkable phenomenon: a central core of the fluid moves down the pipe as a solid, undeformed cylinder. This is known as the ​​plug flow​​ region. It's as if a solid rod is being lubricated by a surrounding layer of flowing fluid. The concept is general, appearing not just in pipes but also in flow between flat plates. The radius of this plug, let's call it $r_p$, is precisely the point where the stress equals the yield stress: $\tau(r_p) = \tau_y$. Any closer to the center, the stress is too low to cause flow. Any further out, the fluid is yielded and sheared.

Of course, if the pressure push is too weak, the stress even at the wall might not be enough to overcome $\tau_y$. In that case, nothing flows at all. There is a ​​minimum pressure drop​​, required just to get things started:

$\Delta p_{min} = \frac{2L\tau_y}{R}$

This is the price of admission for flowing a Bingham plastic.

Now we can build a complete picture of the flow. The total amount of fluid moving through the pipe per second—the ​​volumetric flow rate​​, $Q$—is the sum of the flow from the central plug and the flow from the sheared annulus around it.

1. ​​The Plug:​​ This is a solid cylinder of radius $r_p$ moving at a constant velocity, $u_p$. Its contribution to the flow is easy: $Q_{plug} = (\pi r_p^2) u_p$.
2. ​​The Annulus:​​ In the region from $r_p$ to the wall $R$, the fluid is flowing according to the Bingham plastic rule. We can integrate the velocity across this annular region to find its contribution, $Q_{annulus}$.

Putting these two pieces together, after a bit of calculus that involves finding the velocity profile by integrating the constitutive law, we arrive at a magnificent result known as the ​​Buckingham-Reiner equation​​:

$Q = \frac{\pi R^{4} \Delta P}{8 \mu_p L} \left[ 1 - \frac{4}{3} \left( \frac{\tau_y}{\tau_w} \right) + \frac{1}{3} \left( \frac{\tau_y}{\tau_w} \right)^{4} \right]$

Here, $\tau_w = \frac{R \Delta P}{2L}$ is the shear stress at the pipe wall. This single equation beautifully captures the entire behavior. The first part, $\frac{\pi R^4 \Delta P}{8 \mu_p L}$, is just the famous ​​Hagen-Poiseuille equation​​ for a Newtonian fluid, with the plastic viscosity $\mu_p$ taking the place of regular viscosity. The part in the brackets is a dimensionless correction factor that contains all the physics of the yield stress.

This equation is more than just a formula; it's a story.

First, let's do a sanity check. What if we have a fluid with no yield stress, $\tau_y = 0$? The material is no longer "stubborn". In this case, the entire bracketed term becomes simply 1, and the Buckingham-Reiner equation magically transforms into the Hagen-Poiseuille equation for a Newtonian fluid! This is a beautiful example of the unity of physics. The more complex theory correctly contains the simpler one as a special case.

Second, the presence of the plug fundamentally changes the shape of the flow. For a Newtonian fluid, the velocity profile is a perfect parabola, fastest at the center and smoothly going to zero at the wall. For a Bingham plastic, the velocity is constant across the entire plug region and then drops off in the annulus. This makes the profile blunter. We can quantify this by comparing the maximum velocity (the plug speed) to the average velocity. This ratio depends only on the relative size of the plug, $\phi = r_p/R$, and tells us just how dominant the "solid-like" behavior is.

The Buckingham-Reiner model provides a powerful framework, but the real world is always richer. What happens if we relax our assumptions?

• ​​Slippery Walls:​​ We assumed the fluid sticks to the pipe wall (the "no-slip" condition). But what if the wall has a special coating that makes it slippery when the stress gets high enough? We can add this effect to our model. The total flow rate simply becomes the flow we calculated before plus an extra term from the entire plug of fluid sliding along with a certain slip velocity at the wall. This shows how physical models can be built up in a modular way to capture more complex effects.

• ​​The Price of Flow: Viscous Heating:​​ Where does the energy from the pump go? It works against the viscous forces and the yield stress, and this work is dissipated as heat. This is ​​viscous heating​​. We can calculate the total heat generated per unit length of the pipe. Amazingly, the resulting equation has a structure that is a dead ringer for the Buckingham-Reiner equation itself. This is no coincidence; it reveals a deep and beautiful connection between the transport of momentum (the flow rate) and the dissipation of energy (the heat generation). Pushing a stubborn fluid through a pipe costs energy, and that cost is paid in the form of heat.

• ​​A Family of Models:​​ The Bingham model, with its sharp transition from solid to liquid, is an idealization. Other materials might have a more gradual transition. The ​​Casson fluid​​ model, for example, uses a square-root relationship and often describes things like blood or melted chocolate better. We can apply the very same principles—a linear stress profile in a pipe leading to a central plug—to derive a flow rate equation for a Casson fluid. The math is a bit different, but the physical reasoning, the soul of the model, is identical.

From the simple act of squeezing toothpaste, we've journeyed through the concepts of yield stress, plug flow, and energy dissipation, arriving at a powerful equation that unifies the behavior of solids and liquids. The Buckingham-Reiner equation is not just a tool for engineers; it's a window into the rich and complex personality of the materials that make up our world.

Now that we have grappled with the principles behind Bingham plastics and the Buckingham-Reiner equation, we might be tempted to file this knowledge away as a neat piece of physics, a specific solution to a specific problem. But to do so would be to miss the forest for the trees! The real magic of a fundamental physical law is not in its elegance on a blackboard, but in its surprising and widespread power to describe the world around us. The behavior of a fluid that refuses to flow until it's pushed hard enough is not an esoteric curiosity; it is a recurring theme played out in our kitchens, in massive industrial plants, in our own bodies, and at the frontiers of technology.

Let us begin our journey of discovery in a familiar place: the kitchen. Imagine you are decorating a cake. You have a piping bag full of frosting, and you squeeze. The frosting flows smoothly through the nozzle, creating a beautiful rosette. You stop squeezing, and the rosette sits perfectly on the cake, holding its delicate shape against the pull of gravity. Have you ever wondered about the peculiar genius of frosting? It must be willing to flow like a liquid when you squeeze it, yet act like a solid when you don't. This dual personality is the very essence of a Bingham plastic. The pressure you apply with your hands generates shear stresses in the nozzle that exceed the frosting's yield stress, $\tau_y$, allowing it to flow. Once on the cake, the only force is the frosting's own weight, which creates stresses that are less than $\tau_y$. The frosting resolutely refuses to flow, and the decoration remains intact. The Buckingham-Reiner equation allows a food scientist to take a measurement of the flow rate for a given pressure and work backwards to calculate the all-important yield stress, ensuring the product is neither too stiff to pipe nor too runny to hold its shape.

This same principle, this contest between an applied stress and an intrinsic yield stress, is scaled up to colossal proportions in civil and chemical engineering. Consider the challenge of pumping wet concrete, a quintessential Bingham plastic. A simple pump designed for water would be useless. The engineer must ensure the pressure is great enough to overcome the concrete's substantial yield stress, not just at one point, but along the entire length of the pipe. The Buckingham-Reiner equation, or its cousin for different geometries like flow between parallel plates, becomes an indispensable design tool. It tells us the minimum pressure required to even start the flow—a critical value that has no counterpart in the world of simple Newtonian fluids like water. Below this threshold pressure, the concrete remains a stubborn, unmoving plug. This same physics governs the transport of countless other industrial materials: drilling muds in the oil and gas industry, thick slurries in mining operations, and even the flow of sand or grain from a hopper. The frustrating phenomenon of a silo "jamming" is nothing more than a case where the stress generated by the material's own weight in the narrow exit is insufficient to overcome its yield stress. The material simply refuses to flow.

Perhaps the most profound applications are not in the machines we build, but in the one we inhabit: the human body. For over a century, we have modeled blood as a simple Newtonian fluid, a fruitful approximation that has taught us much. But it is not the whole story. At very low flow rates, red blood cells tend to clump together into stacks called "rouleaux," giving the blood a microscopic structure that must be broken up before it can flow. In other words, blood exhibits a yield stress.

This single fact has life-or-death consequences. In large, fast-flowing arteries, the shear stresses are so high that the yield stress is negligible. But consider a coronary artery that has been propped open with a stent. A stent is a complex mesh, and its very structure creates tiny regions near the artery wall where the blood flow can become sluggish, and the shear stress can plummet. If the local shear stress drops below blood's yield stress, the blood in that region stops moving altogether. It stagnates. These zones of stagnation are the perfect breeding ground for thrombosis—the formation of a life-threatening blood clot. Understanding the principles of Bingham flow is therefore critical for biomedical engineers designing safer stents, helping them to minimize regions of low shear and keep the blood moving.

Nature's use of non-Newtonian fluids is not limited to animals. The pressure-flow hypothesis, which describes how sugars are transported through the phloem of plants, is typically modeled using simple Newtonian physics. But phloem sap is a complex cocktail of proteins and polysaccharides. It is entirely plausible that it behaves as a shear-thinning fluid, flowing more easily the harder it is pushed. Or perhaps, in response to an injury, the proteins within the sap rapidly assemble to form a gel with a high yield stress, acting as a natural, self-sealing plug to prevent the plant from bleeding out its precious nutrients. By applying more sophisticated rheological models, like the Bingham model, botanists can paint a much richer and more accurate picture of the inner life of plants.

Having seen how this principle operates in our engineered world and in the natural world, we can look to the future, where we are learning not just to deal with a material's yield stress, but to control it. This is the domain of so-called "smart fluids." A magneto-rheological (MR) fluid, for instance, is a suspension of tiny iron particles in an oil. In the absence of a magnetic field, it flows easily. But apply a magnetic field, and the iron particles instantly align into chains, giving the fluid a substantial and controllable yield stress. It can transform from a liquid to a thick, semi-solid paste in milliseconds.

Imagine this fluid inside a car's shock absorber. As the car goes over a bump, a sensor detects the motion and an electromagnet adjusts the fluid's yield stress in real time. The damping force of the shock absorber, which is directly related to the pressure needed to push the Bingham fluid through small channels in the piston, can be tuned on the fly. The result is a suspension system that can be soft and comfortable one moment, and firm and responsive the next. A similar principle is at work in advanced 4D printing, where the "ink" might be a resin whose yield stress changes as it is cured by ambient light. To ensure a consistent and perfect print, the printer's control system must intelligently adjust the extrusion pressure, actively compensating for the material's changing personality.

From frosting to concrete, from blood clots to intelligent cars, the same fundamental idea reappears. A substance resists flow until the force is great enough. The Buckingham-Reiner equation is more than a formula; it is a lens through which we can see a hidden unity in the world. It reveals the shared physics that governs the slump of a sand pile and the function of a life-saving medical device, reminding us that a deep understanding of one small corner of nature can illuminate the workings of a great many others.