Newtonian Constitutive Relation

How do we mathematically describe the familiar "stickiness" of fluids like water or honey? While solids resist being deformed, fluids resist the rate at which they are deformed. The Newtonian constitutive relation provides a beautifully simple yet profoundly powerful answer, forming a cornerstone of modern fluid dynamics. It establishes a linear connection between the forces within a fluid (stress) and the way it moves and changes shape (strain rate). This article addresses the fundamental question of how this internal friction, or viscosity, is modeled in a rigorous and general way. The reader will gain a deep understanding of the physical principles behind this law and its surprisingly broad impact. We will begin by exploring the core "Principles and Mechanisms" that define the relationship, deconstructing stress, strain, and the physical reasoning behind the model. Subsequently, the article will demonstrate the law's immense utility by examining its "Applications and Interdisciplinary Connections," revealing how this single equation helps describe the world, from the flow of magma deep within the Earth to the development of living organisms.

Let us begin with a simple observation. Take a rubber band, stretch it, and hold it. It pulls back, storing the energy you put into it. It has a memory of its original shape and will snap back if you let go. Now, dip your finger in a jar of honey and stir. The honey resists your motion; it feels thick and "viscous." But the moment you stop stirring, the honey stops flowing. It makes no attempt to undo the swirling you created. It has forgotten its original, placid state completely.

This simple comparison reveals the fundamental difference between an ideal solid and an ideal fluid. A solid resists a sustained deformation (a strain), storing energy like a spring. A fluid, on the other hand, resists the rate of deformation (a strain rate).

Imagine a material trapped between two large, flat plates. If we slide the top plate a small distance and hold it there, what happens? If the material is a solid, like a block of rubber, it will deform and exert a continuous, steady force back on the top plate. It sustains a stress. But if the material is a fluid, like water or air, it will flow and rearrange itself. Once the top plate stops moving, the fluid also comes to rest, and the force required to hold the plate in its new position drops to zero. A fluid at rest cannot sustain a shear stress. Its defining characteristic is its ability to flow and forget its shape. The resistance it offers exists only when it is in motion. This resistance is the essence of viscosity.

If a fluid only resists motion, how can we describe this resistance? Let's return to our two plates, but this time, we keep the top plate moving at a constant, slow velocity. To do this, we must continually apply a force to it, counteracting the drag from the fluid. It is as if the layers of fluid are sticking to each other, creating a kind of internal friction. This "stickiness" is what we call ​​viscosity​​.

For a great many common fluids—air, water, oil, honey—Isaac Newton found a wonderfully simple law. He proposed that the force per unit area needed to slide one layer of fluid past another (the ​​shear stress​​, denoted by $\tau$) is directly proportional to how fast it is being sheared (the ​​shear rate​​). In our plate example, the shear rate is simply the speed of the top plate, $U$, divided by the distance between the plates, $h$. So, we have:

$\tau = \mu \frac{U}{h}$

The constant of proportionality, $\mu$, is the fluid's ​​dynamic viscosity​​. It is a measure of the fluid's intrinsic resistance to this shearing motion. Honey has a high $\mu$; air has a very low one. This simple linear relationship is the defining feature of a ​​Newtonian fluid​​.

What happens to the energy we expend to keep the plate moving? It doesn't get stored as it would in a spring. Instead, it is continuously converted into heat, warming the fluid. The power required to overcome the viscous drag is a direct measure of this energy dissipation. Viscosity, then, is fundamentally a dissipative phenomenon; it's a process that turns organized motion into the disorganized, random motion of molecules we call heat.

Of course, fluid flows in the real world are rarely as simple as the motion between two plates. A fluid element in a river or around an airplane wing is being stretched, squeezed, and sheared in all directions at once. To describe this complex state of affairs, we need a more sophisticated tool than a single force value: the ​​Cauchy stress tensor​​, $\boldsymbol{\sigma}$.

Think of an infinitesimally small cube of fluid. The state of "stress" at that point is the complete description of all the forces acting on the faces of that tiny cube. This object, which must describe not only the magnitude and direction of the forces but also the orientation of the surfaces they act upon, is a tensor.

The beauty of this tensor is that we can decompose it into parts that have clear physical meaning. First, we can find the average of the pushing forces on each face. This part of the stress acts equally in all directions, trying to change the fluid element's volume but not its shape. We call this the ​​hydrostatic pressure​​, $p$. It's what you feel pressing on your eardrums as you dive into a pool. Mathematically, it is defined from the trace (the sum of the diagonal elements) of the stress tensor: $p = -\frac{1}{3}\operatorname{tr}(\boldsymbol{\sigma})$.

The part of the stress that is left over after we subtract out the pressure is called the ​​deviatoric stress tensor​​, $\boldsymbol{\tau}$. This is the shape-changing part of the stress. It represents the shearing and stretching forces that arise from the fluid's motion. Thus, the total stress at any point in a fluid can always be written as the sum of these two parts:

$\boldsymbol{\sigma} = -p\mathbf{I} + \boldsymbol{\tau}$

where $\mathbf{I}$ is the identity tensor. It is this deviatoric stress, $\boldsymbol{\tau}$, that is directly related to viscosity.

So, the viscous stress $\boldsymbol{\tau}$ is caused by the fluid's motion. But what specific aspect of the motion is responsible? The key is not the velocity itself, but how the velocity changes from one point to its neighbors. This information is captured by the velocity gradient tensor, $\nabla\mathbf{u}$.

A wonderful piece of mathematics reveals that any local fluid motion described by $\nabla\mathbf{u}$ can be broken down into two distinct components. One part is a pure rotation of the fluid element, as if it were a tiny, rigid spinning top. This is described by the skew-symmetric part of the velocity gradient, which is related to the ​​vorticity​​, $\boldsymbol{\omega}$. The other part is a pure deformation—a stretching and shearing motion that changes the element's shape. This is described by the symmetric part of the velocity gradient, known as the ​​rate-of-deformation tensor​​, $\mathbf{D}$.

A deep question then presents itself: does the viscous stress depend on the rotation, the deformation, or both? The answer, dictated by the fundamental principles of physics, is that it can only depend on the deformation, $\mathbf{D}$. There are two beautiful reasons for this.

First, consider energy. Viscosity causes dissipation. The rate at which mechanical energy is turned into heat is given by the viscous stress doing work. A bit of tensor algebra shows that a symmetric tensor (the viscous stress, which must be symmetric to ensure angular momentum is conserved does zero work on a skew-symmetric tensor (the rotation part of the motion). Therefore, pure rotation does not dissipate energy. Only the deforming part of the motion, $\mathbf{D}$, can interact with the viscous stress to generate heat.

Second, consider how things look to different observers. The laws of physics must be the same for everyone, whether they are standing still or spinning on a merry-go-round. This is the ​​principle of material frame-indifference​​. Now, think of coffee in a cup that is spun at a constant rate. The coffee rotates as a rigid body. It has plenty of vorticity, but its parts are not moving relative to one another. It is not deforming, and so it should not experience any viscous stress. The rate-of-deformation tensor $\mathbf{D}$ is zero for this rigid-body rotation, but the vorticity is not. If stress depended on vorticity, it would be non-zero, which makes no physical sense. Therefore, the viscous stress must depend only on $\mathbf{D}$, the quantity that correctly identifies that no deformation is occurring.

We can now assemble our masterpiece. We are seeking a ​​constitutive relation​​—an equation that connects the cause (the rate of deformation, $\mathbf{D}$) to the effect (the viscous stress, $\boldsymbol{\tau}$). The very definition of a Newtonian fluid is one for which this relationship is the simplest imaginable: a linear one.

For an incompressible fluid (one whose volume cannot change), the viscous stress is simply directly proportional to the rate-of-deformation tensor. Combining this with our earlier decomposition of the total stress, we arrive at the complete constitutive relation for an incompressible Newtonian fluid:

$\boldsymbol{\sigma} = -p\mathbf{I} + 2\mu\mathbf{D}$

This wonderfully compact equation is the three-dimensional generalization of Newton's original discovery. The pressure, $p$, is the part of the stress that enforces the incompressibility, and the viscous term, $2\mu\mathbf{D}$, describes the stresses that arise from, and resist, the fluid's motion and deformation. This single equation, when combined with Newton's second law of motion ($F=ma$), yields the celebrated ​​Navier-Stokes equations​​, which form the bedrock of modern fluid dynamics.

Our story has so far assumed the fluid is incompressible, which is a very good approximation for liquids like water in many situations. But what about gases like air, which are easily squeezed? When a fluid's volume can change, we must account for a new type of deformation: a uniform expansion or compression, whose rate is given by $\nabla \cdot \mathbf{u}$.

This introduces the possibility of a second type of viscous resistance: a resistance to the rate of change of volume. This effect is governed by a second coefficient of viscosity, known as the ​​bulk viscosity​​, $\zeta$ (related to another coefficient, $\lambda$, by $\zeta = \lambda + \frac{2}{3}\mu$). It represents an internal friction that appears only when the fluid is being rapidly compressed or expanded, as in a sound wave or a shock wave.

For many applications, it is common to make the ​​Stokes hypothesis​​, which is the assumption that this bulk viscosity is zero ($\zeta = 0$). For a simple monatomic gas like helium, kinetic theory confirms this is an excellent approximation. The physical reason is that in such a gas, energy is only stored in the translational motion of the atoms, and the energy from compression is distributed among them almost instantaneously.

However, the Stokes hypothesis is just an approximation, and its failure teaches us about deeper physics. In polyatomic gases like air (nitrogen and oxygen), energy from compression must be partitioned into not just translational motion, but also the rotation and vibration of the molecules. This process takes time. If you compress the gas faster than this internal "relaxation time," a sort of thermodynamic friction occurs, which manifests as a non-zero bulk viscosity. This is why bulk viscosity is crucial for understanding the absorption of high-frequency sound in air and the structure of shock waves,. Likewise, for many liquids and complex fluids, the Stokes hypothesis fails, and the bulk viscosity can be quite large. This nuance shows the limits of our simple model and points toward the rich internal dynamics of matter.

We have treated viscosity as a macroscopic property, but its origins lie in the chaotic, microscopic world of molecules. Viscosity is not a property of a single water molecule; it is an emergent property of trillions of them acting in concert.

Picture a fluid flowing in layers, with each layer moving slightly faster than the one below it. The molecules within the fluid are not confined to their layers; they are in constant, random thermal motion, jiggling and colliding endlessly. A fast-moving molecule from a higher, faster layer will occasionally jiggle down into the slower layer below. In its subsequent collisions, it imparts some of its excess momentum to its new, slower neighbors, giving them a slight push forward. Symmetrically, a slower molecule from a lower layer might jiggle up into a faster layer, and through its collisions, it acts as a drag, slowing that layer down.

This continuous microscopic exchange of momentum between adjacent layers of fluid is the true physical origin of viscous shear stress. The viscosity coefficient, $\mu$, is nothing more than a measure of how efficiently this molecular momentum transfer occurs. Kinetic theory for gases shows that viscosity increases with temperature and is largely independent of pressure, a surprising result that arises from the interplay between molecular density and the average distance molecules travel between collisions. It is a stunning bridge between the chaotic dance of individual particles and the smooth, predictable flow of the continuum world, allowing us to understand everything from the flow of honey to the magnificent swirls of Jupiter's atmosphere.

After our exploration of the principles and mechanisms behind the Newtonian constitutive relation, you might be left with a feeling of neat, but perhaps sterile, mathematical elegance. Does this simple, linear relationship between stress and the rate of strain truly have much to say about the complex, messy world we live in? The answer is a resounding yes, and the story of its applications is a journey across disciplines that reveals a surprising unity in nature's design. The same fundamental law that governs the flow of water in a pipe also sculpts the hearts of living embryos, dictates the behavior of molten rock deep within the Earth, and even provides a clever trick for taming the chaos of turbulence.

Let's begin in a familiar setting: a simple pipe. Every time we pump a fluid, from the water in our homes to the broth in an industrial bioreactor, we are fighting against an invisible force of resistance. This force, this drag, has its origins in the very heart of the Newtonian constitutive relation. As the fluid flows, the layer in contact with the pipe wall is stationary—the famous no-slip condition. A little further from the wall, the fluid is moving slowly, and a little further still, it moves faster. This change in velocity with position, the velocity gradient $\frac{du}{dr}$, is the rate of shear strain.

According to our law, this strain rate gives rise to a shear stress, $\tau = \mu \frac{du}{dr}$. This is the fluid trying to drag the stationary wall along with it, and conversely, the wall trying to hold the fluid back. This microscopic stress, acting all along the inner surface of the pipe, is the source of all viscous drag. When you sum up this stress over the entire length of the pipe, you get the total drag force that the pump must overcome.

But there's a cost to this struggle, a consequence dictated by the laws of thermodynamics. Work must be done to push the fluid against the viscous friction, and that energy does not simply vanish. It is dissipated, converted into the random motion of molecules—in other words, heat. The very same term, the velocity gradient, that determines the stress also tells us how much energy is being lost. The rate of energy dissipation per unit volume is proportional to the viscosity and the square of the velocity gradient, $\Phi = \mu (\frac{du}{dr})^2$. So, the stickier the fluid or the faster you try to shear it, the more rapidly you heat it up. Every time you stir honey, you are converting the mechanical energy from your arm into heat, making the honey infinitesimally warmer, a direct consequence of its Newtonian nature.

One of the beautiful things about fundamental physical laws is their blissful ignorance of context. They apply just as well to molten rock as they do to water. A geophysicist modeling the flow of magma in a volcanic conduit uses the exact same constitutive relation as a chemical engineer designing a factory pipe. The magma, an incredibly viscous silicate liquid, creeps upwards, its velocity profile shaped by the same balance of pressure and viscous forces. The shear stress it exerts on the conduit walls, calculated from $\tau = \mu (du/dr)$, can be immense, capable of fracturing the surrounding rock and influencing the dynamics of an eruption. The only difference is the scale of the numbers—the viscosity $\mu$ is billions of times higher than that of water, but the principle is identical.

The story gets even more remarkable. Can a solid be a Newtonian fluid? On human timescales, we think of glass as a quintessential solid. But over geological time, or at the high temperatures used in manufacturing, it flows. The atoms in an amorphous solid lack a fixed crystal lattice and can slowly rearrange themselves under stress. This slow, creeping flow can often be described as a highly viscous Newtonian fluid.

Consider the process of sintering, where fine powders are heated to form a solid object. Two tiny amorphous spheres, just touching, are not in their lowest energy state. Their vast surface area represents an excess of energy, which we call surface tension. This surface tension pulls on the particles, creating a stress that tries to minimize the surface area by fusing them together. This stress drives a slow, viscous flow, causing a "neck" of material to grow between the particles. Astonishingly, we can model this by relating the driving stress from surface tension to the rate of strain (the change in the neck's geometry) via the material's viscosity. The Newtonian constitutive relation allows us to predict how quickly powders will fuse, a process fundamental to ceramics, metallurgy, and 3D printing of certain materials. The "fluid" is a solid, and the "pump" is surface tension itself.

Nowhere is the subtle power of the Newtonian relation more apparent than in the domain of life. The forces exerted by flowing fluids are not just obstacles to be overcome; they are critical signals that guide the development and function of living organisms.

Let's journey into the heart of a developing zebrafish embryo, a tiny creature whose transparent body allows us to watch life unfold. In its nascent circulatory system, blood—which at this scale behaves as a simple Newtonian fluid—flows through developing vessels. As it does, it exerts a shear stress on the layer of endothelial cells that form the vessel walls. The magnitude of this stress, perhaps a mere $0.6$ Pascals, seems tiny. But to the cells, it is a shout.

Specialized proteins on the cell surface act as mechanosensors, detecting this physical tug. The cell "feels" the flow. This physical signal is then converted—transduced—into a biochemical cascade inside the cell, ultimately altering which genes are turned on or off. For instance, the expression of a gene called Klf2a is highly sensitive to shear stress. This gene, in turn, regulates the Notch signaling pathway, a master controller of development. The result? The physical force of blood flow tells the cells whether to become an artery or a vein, and it instructs them on how to sculpt themselves into the delicate leaflets of the heart valves. If the flow is too weak (perhaps due to lower blood viscosity) or absent, the signal is lost, and the heart fails to form correctly. This is mechanobiology: the Newtonian constitutive relation, in a very real sense, helps write the book of life.

The same law can also describe battles on a microscopic scale. Consider a biofilm, a slimy colony of bacteria that can cause infections or clog pipes. This colony is held together by a matrix of extracellular polymers, which gives it a certain cohesive strength. As water flows over it, a shear stress is applied. For a while, the biofilm holds firm. But if the flow becomes strong enough, the applied shear stress will eventually exceed the internal strength of the biofilm, and chunks will be ripped away in a process called sloughing. Crucially, the real biofilm isn't a perfectly smooth surface. It has bumps and towers. These bits of roughness poke into the flow, causing the fluid to accelerate around them. This leads to local "hot spots" of much higher shear stress, a phenomenon of stress concentration that allows the flow to find a weak point and initiate failure, even when the average stress is low.

In the 21st century, many of our greatest scientific and engineering feats happen inside a computer. The Newtonian constitutive relation is an indispensable building block in the vast edifice of computational modeling.

Think of something as delicate as the "tears of wine" that form inside a glass. This phenomenon, known as the Marangoni effect, is driven by a gradient in surface tension. At the air-liquid interface, the alcohol evaporates faster, increasing the local water concentration and thus the surface tension. This gradient in surface tension pulls fluid up the side of the glass. The fluid that is pulled up must be balanced by the viscous shear stress from the fluid below it. The boundary condition for the simulation is a direct statement of this balance: the shear stress at the surface, $\mu (\partial u / \partial y)$, must equal the surface tension gradient, $\partial \sigma / \partial x$. Without the constitutive relation, we couldn't model this beautiful effect.

When we simulate even more complex scenarios, like the interaction of wind with a bridge or blood flow through a prosthetic heart valve, we are solving a fluid-structure interaction (FSI) problem. The computer needs to know how the fluid pushes on the solid. This is governed by the dynamic interface condition, which is a mathematical expression of Newton's third law: the force vector (traction) exerted by the fluid on the solid must be equal to the force vector exerted by the solid on the fluid. The fluid's stress tensor, $\boldsymbol{\sigma}_f = -p_f \boldsymbol{I} + 2\mu_f \mathbf{D}(\mathbf{u}_f)$, containing our Newtonian relation, is what allows us to calculate the force the fluid exerts, making these life-saving simulations possible.

Finally, what about flows that are decidedly not simple and Newtonian, like a turbulent jet of air or the wake behind a ship? Here, the constitutive relation inspires a brilliant act of intellectual modeling. The chaotic, swirling eddies of turbulence create stresses that are far more complex than in a laminar flow. A full simulation of this chaos is often computationally impossible. The Boussinesq eddy-viscosity hypothesis offers a clever workaround. It suggests we model the average effect of all the turbulent eddies as if it were an extra-large viscosity, an "eddy viscosity" $\mu_t$. We then use a Newtonian-like constitutive relation to model the mean flow, but with this new, much larger viscosity. It's an admission that the underlying flow is not Newtonian, but it's a testament to the power of the original concept that we can create a powerful predictive model by simply preserving its mathematical form and inventing a new, effective parameter.

From the mundane to the living to the virtual, the simple premise of a linear relationship between stress and strain rate has proven to be an idea of extraordinary power and reach. It is a golden thread connecting disparate worlds, a beautiful example of how a simple piece of physics can illuminate the workings of the universe on every scale.