Upper-Convected Derivative

To understand and predict the behavior of complex materials like polymer melts or biological fluids, we must use the language of continuum mechanics. A central challenge in this field is describing how material properties change in a system that is constantly flowing, stretching, and tumbling. Standard time derivatives are often insufficient, as they can be "fooled" by simple rotations, violating the fundamental physical principle that laws of nature should not depend on the observer's frame of reference. This article addresses this knowledge gap by introducing the concept of an objective time rate. Across the following sections, you will discover the core principles behind objective derivatives and see how they are built. The article culminates by exploring how one of the most important of these, the upper-convected derivative, serves as a master key for modeling and understanding the real-world behavior of complex fluids.

To truly understand the strange and wonderful world of materials like polymer melts, bread dough, or even the Earth's mantle, we must learn to speak their language. This is the language of continuum mechanics, a way of thinking about how matter flows, stretches, and deforms. Our central challenge is to measure change in a world that is constantly in motion. How do we create a physical law when our very laboratory—the material itself—is flowing and tumbling through space?

Imagine you are a scientist on a small raft, floating down a river. Your job is to measure the properties of the water, perhaps some internal stress tensor, which we'll call $\boldsymbol{T}$. The most natural thing to do is to dip your probe in the water right next to your raft and see how the reading on your meter changes over time. You are measuring the change as you follow the water. This rate of change, following a material particle, is what mathematicians call the ​​material derivative​​, denoted $\frac{D\boldsymbol{T}}{Dt}$. It's the sum of the change happening at a fixed location ($\frac{\partial \boldsymbol{T}}{\partial t}$) and the change due to you being carried to a new location with a different value of $\boldsymbol{T}$ (the convective term, $\boldsymbol{u}\cdot\nabla\boldsymbol{T}$).

Now, suppose your friend is observing you from a giant, rotating merry-go-round on the riverbank. They also have a probe to measure the stress tensor in the same water particle you are tracking. Should their measurement of the rate of change agree with yours? According to a fundamental tenet of physics, the ​​Principle of Material Frame Indifference​​ (or ​​objectivity​​), the answer must be yes. A physical law cannot depend on the arbitrary spinning or movement of the observer.

Here lies the problem: the simple material derivative is not objective. If a block of Jell-O is simply rotating rigidly like a spinning top, its internal state isn't changing in any meaningful physical way. Yet, an observer fixed in the lab would see the orientation of the stress tensor changing, and the material derivative $\frac{D\boldsymbol{T}}{Dt}$ would be non-zero. The material derivative is a "liar"; it confuses a simple rotation of the material with a genuine physical change within it. To write meaningful constitutive laws—the rules that govern a material's behavior—we need a "time derivative" that is honest, an ​​objective time rate​​. We need a clock that isn't fooled by rotation.

To build an honest clock, we must first understand the nature of motion itself. Any complex fluid motion, when you look at an infinitesimally small neighborhood, is a combination of translation, rotation, and deformation. The key to describing this local motion is the ​​velocity gradient tensor​​, $\boldsymbol{L} = \nabla\boldsymbol{u}$. This tensor is a compact dictionary that tells us how the velocity changes from one point to a neighboring point.

The true beauty here is that we can split any matrix, including $\boldsymbol{L}$, into a symmetric part and a skew-symmetric part. For the velocity gradient, this decomposition reveals the two fundamental characters of fluid motion:

$\boldsymbol{L} = \boldsymbol{D} + \boldsymbol{W}$

Here, $\boldsymbol{D} = \frac{1}{2}(\boldsymbol{L} + \boldsymbol{L}^{\mathrm{T}})$ is the symmetric part, called the ​​rate-of-deformation tensor​​. It describes how a small fluid element is being stretched, squashed, or sheared. It's the part of the motion that changes the element's shape.

The other part, $\boldsymbol{W} = \frac{1}{2}(\boldsymbol{L} - \boldsymbol{L}^{\mathrm{T}})$, is the skew-symmetric part, known as the ​​spin​​ or ​​vorticity tensor​​. It describes how the fluid element is undergoing a rigid-body rotation, like a spinning top, without any change in shape.

This simple mathematical split is profound. It tells us that any local fluid motion can be thought of as a superposition of pure stretching/shearing and pure spinning. With this insight, we now have the tools to correct our lying clock.

A first attempt at an "honest" derivative might be to simply subtract the apparent change caused by the fluid's spinning. This leads to the ​​Jaumann derivative​​ (or co-rotational derivative):

$\overset{\circ}{\boldsymbol{T}} = \frac{D\boldsymbol{T}}{Dt} - (\boldsymbol{W}\boldsymbol{T} - \boldsymbol{T}\boldsymbol{W})$

The term we subtract, $\boldsymbol{W}\boldsymbol{T} - \boldsymbol{T}\boldsymbol{W}$, is precisely the rate of change a tensor would have if it were just passively rotating with the fluid at a spin rate given by $\boldsymbol{W}$. So, the Jaumann derivative measures the rate of change of $\boldsymbol{T}$ from the perspective of an observer who is spinning along with the fluid element. It successfully ignores rigid rotation and is, in fact, an objective rate. It's a much more honest clock than the material derivative. But is it the whole story? What about the stretching described by $\boldsymbol{D}$?

For many physical quantities, especially those tied to the material's fabric itself, even the change due to pure stretching is a form of "trivial" convection that we want to ignore. We want to find the rate of change that is truly intrinsic to the material's response, stripped of all effects from the bulk flow. This brings us to the hero of our story: the ​​upper-convected derivative​​.

At first glance, its definition looks a bit intimidating:

$\overset{\triangledown}{\boldsymbol{T}} = \frac{D\boldsymbol{T}}{Dt} - \boldsymbol{L}\boldsymbol{T} - \boldsymbol{T}\boldsymbol{L}^{\mathrm{T}}$

But let's unpack its meaning. First, we can use our decomposition $\boldsymbol{L} = \boldsymbol{D} + \boldsymbol{W}$. A little bit of algebra shows something remarkable:

$\overset{\triangledown}{\boldsymbol{T}} = \left( \frac{D\boldsymbol{T}}{Dt} - (\boldsymbol{W}\boldsymbol{T} - \boldsymbol{T}\boldsymbol{W}) \right) - (\boldsymbol{D}\boldsymbol{T} + \boldsymbol{T}\boldsymbol{D})$

Look closely! The first part in the parentheses is just the Jaumann derivative. The upper-convected derivative does what the Jaumann derivative does—it subtracts rotation—but then it does more. It also subtracts the term $\boldsymbol{D}\boldsymbol{T} + \boldsymbol{T}\boldsymbol{D}$, which represents the rate of change that happens simply because the material lines in which the tensor is defined are being stretched and sheared by the flow. It is an objective rate, proven to be so for any motion, compressible or incompressible. It answers the question: "What is the rate of change of $\boldsymbol{T}$ that is not due to the material element being passively spun and stretched by the flow?"

There is another, perhaps more intuitive, way to see this. Imagine a net drawn on the fluid at time $t$. A short time later, at $t + \Delta t$, the fluid has flowed, and our net is now distorted and in a new position. The upper-convected derivative is the result of a conceptual experiment: take the tensor $\boldsymbol{T}$ measured on the distorted net at the later time, mathematically "pull it back" along the flow paths and "un-stretch" it so the net has the same shape as it did at time $t$, and then calculate how much the tensor has changed. This "pull-back" operation is precisely what the terms $-\boldsymbol{L}\boldsymbol{T} - \boldsymbol{T}\boldsymbol{L}^{\mathrm{T}}$ accomplish. It is the derivative in a coordinate system that is convected and deformed with the material.

The true power and elegance of the upper-convected derivative are revealed when we use it to measure the change of a very special tensor: the ​​Left Cauchy-Green strain tensor​​, $\boldsymbol{B} = \boldsymbol{F}\boldsymbol{F}^{\mathrm{T}}$. This tensor, built from the deformation gradient $\boldsymbol{F}$, is a fundamental measure of the total finite deformation a material has experienced from some reference state. It essentially carries the history of all the stretching that has occurred.

Now, let's ask a simple kinematic question: how does $\boldsymbol{B}$ change in time? Through the rules of calculus and kinematics, one can derive a beautiful and exact identity:

$\frac{D\boldsymbol{B}}{Dt} = \boldsymbol{L}\boldsymbol{B} + \boldsymbol{B}\boldsymbol{L}^{\mathrm{T}}$

This equation tells us how the material derivative of the strain history is related to the current velocity gradient. Now for the magic. Let's compute the upper-convected derivative of $\boldsymbol{B}$:

$\overset{\triangledown}{\boldsymbol{B}} = \frac{D\boldsymbol{B}}{Dt} - \boldsymbol{L}\boldsymbol{B} - \boldsymbol{B}\boldsymbol{L}^{\mathrm{T}}$

Substituting our identity into this definition, we find:

$\overset{\triangledown}{\boldsymbol{B}} = (\boldsymbol{L}\boldsymbol{B} + \boldsymbol{B}\boldsymbol{L}^{\mathrm{T}}) - (\boldsymbol{L}\boldsymbol{B} + \boldsymbol{B}\boldsymbol{L}^{\mathrm{T}}) = \boldsymbol{0}$

The result is zero. Always. This is a moment of mechanical zen. The upper-convected derivative of the Left Cauchy-Green strain tensor is identically zero. This tells us that from the special viewpoint of the upper-convected derivative, the finite strain tensor $\boldsymbol{B}$ does not change at all. It is purely convected by the flow in a way that the derivative is designed to ignore. This confirms our intuition: the upper-convected derivative is the natural rate of change for quantities that are intrinsically woven into the deforming fabric of the material itself.

The upper-convected derivative is a key member of a whole family of objective rates. Its "dual" is the ​​lower-convected derivative​​, given by $\overset{\triangle}{\boldsymbol{A}} = \frac{D\boldsymbol{A}}{Dt} + \boldsymbol{L}^{\mathrm{T}}\boldsymbol{A} + \boldsymbol{A}\boldsymbol{L}$. This rate is natural for describing the evolution of covariant tensors (which transform differently than contravariant ones like $\boldsymbol{B}$). Other rates, like the ​​Truesdell rate​​, are also objective and are elegantly related to the upper-convected rate via different stress measures like the Kirchhoff stress.

This family of objective rates provides a complete toolkit for describing the physics of deforming media. The upper-convected derivative, however, holds a special place. By accounting for both rotation and stretching, it provides a perfect kinematic reference, allowing us to isolate the truly interesting parts of a material's response—the physics of relaxation, entanglement, and flow—from the trivial background of the material simply being carried along for the ride. It is the honest clock we set out to build, a masterpiece of kinematic reasoning.

Having grappled with the mathematical machinery of the upper-convected derivative, you might be wondering: what is this all for? Is it merely an exercise in tensor gymnastics? The answer, I hope you will find, is a resounding no. This concept is not a mathematical abstraction; it is the key that unlocks the door to understanding and predicting the strange and wonderful behavior of a vast class of materials we encounter every day—from the molten plastic in a factory and the dough in a bakery to the synovial fluid in our joints. It is a cornerstone of rheology, the science of flow, and its branches reach deep into materials science, chemical engineering, geophysics, and even biology.

The journey from abstract principle to tangible application begins with a question of fundamental physics: how can we write down laws of nature that are true for everyone?

Imagine you are observing a stirring vat of polymer soup from the side of the vat. Now, imagine your friend is on a spinning merry-go-round, also observing the same vat. You both see the same physical reality, but your descriptions of velocities and rotations will be wildly different. A physical law, if it is to be of any use, must not depend on whether the physicist is standing still or spinning. This principle is called ​​material objectivity​​ or ​​frame indifference​​.

A simple time derivative, like $\frac{d\boldsymbol{A}}{dt}$, fails this test spectacularly. It cannot distinguish between changes in a material property and the trivial rotation of the object as a whole. The upper-convected derivative is the physicist's ingenious solution to this problem. It is constructed precisely to be "objective"—it gives the same result no matter how the observer is moving or spinning. It isolates the true, intrinsic deformation of the material from any rigid-body motion. By proposing a constitutive law using this derivative, we are, by construction, proposing a law that respects a fundamental symmetry of nature. It is the proper way to ask a material how it is changing, in a coordinate system that deforms and tumbles along with it.

Once we have an objective language, we can begin to explore. One of the first things a physicist does with a new equation is to make it dimensionless, to strip away the peculiarities of units and expose the universal truths within. When we non-dimensionalize the constitutive equations built upon the upper-convected derivative, a crucial number naturally emerges: the Weissenberg number, $Wi$.

Here, $\lambda$ is the material's characteristic relaxation time (think of it as its "memory"), while $L/U$ is the characteristic time scale of the flow process. The Weissenberg number is a simple ratio, but its meaning is profound: it is the ratio of the material's memory time to the process time.

When $Wi \ll 1$, the process is so slow that the material has ample time to relax and "forget" its previous state. It behaves like a simple, viscous liquid. When $Wi \gg 1$, the flow is so rapid that the material has no time to relax. Its elastic nature, its memory, dominates. The Weissenberg number is our compass in the world of viscoelasticity, telling us whether we should expect liquid-like or solid-like behavior.

Let's use our compass to explore the simplest of flows: steady simple shear, like a deck of cards being pushed from the top. For a simple Newtonian fluid like water, the only stress generated is the shear stress resisting the sliding motion. But for a viscoelastic fluid, something much more interesting happens.

The upper-convected derivative in our models (like the simple Oldroyd-B model) predicts that as the fluid is sheared, the long-chain molecules within it not only resist the shear but are also stretched along the direction of flow. This stretching creates a tension along the streamlines, much like the tension in a stretched rubber band. This tension manifests as a normal stress difference—the stress in the flow direction becomes greater than the stress in the gradient direction. This is called the ​​first normal stress difference​​, $N_1$, and it is responsible for a host of bizarre phenomena, such as the famous ​​Weissenberg effect​​, where a viscoelastic fluid will climb up a rotating rod instead of being flung outwards by centrifugal force. The calculation of the microscopic conformation tensor from first principles reveals exactly how this molecular stretching leads to these macroscopic stresses.

But reality is subtler still. Experiments show a ​​second normal stress difference​​, $N_2$, which describes the stress anisotropy in the plane perpendicular to flow. The simplest models, like the Oldroyd-B, predict $N_2=0$. This is because they model polymers as simple, non-interacting dumbbells. To capture the fact that $N_2$ is typically non-zero and negative in reality, we need more sophisticated models. The Giesekus model, for instance, adds a nonlinear term to the constitutive equation—still within the objective framework of the upper-convected derivative—that represents an "anisotropic drag." This term accounts for the fact that a polymer chain finds it harder to move through its aligned neighbors than to slide along with them. This subtle addition breaks the symmetry and allows the model to correctly predict a non-zero $N_2$. This is a beautiful example of how the upper-convected framework serves as a robust scaffold upon which we can build increasingly realistic physical models, such as the Phan-Thien-Tanner (PTT) model which successfully predicts other key phenomena like shear-thinning viscosity.

The behavior of viscoelastic fluids in shear is strange, but their behavior in extensional flow—when they are stretched—is nothing short of dramatic. Think of pulling a piece of taffy.

When we apply the same constitutive models to a uniaxial extensional flow, the upper-convected derivative predicts a phenomenon called ​​strain hardening​​. Unlike in shear, where polymers can tumble and relax, in a strong extensional flow they are relentlessly aligned and stretched. This causes a massive resistance to further stretching, and the extensional viscosity can become many times larger than the shear viscosity. This ratio is known as the ​​Trouton ratio​​, and for viscoelastic fluids, it can grow to enormous values as the strain rate increases.

This effect has profound practical consequences. It is what makes polymer melts strong and stable during fiber spinning or film blowing processes. It is why a stream of shampoo can form a surprisingly long, stable filament before breaking.

The simplest model, the upper-convected Maxwell (UCM) model, even predicts that at a critical strain rate, the extensional viscosity will become infinite!. This is, of course, unphysical. A real polymer chain cannot be stretched indefinitely. But this "failure" of the model is incredibly instructive. It tells us that our physical picture of the polymer as a simple Hookean spring is too naive. It drove the development of more refined models that incorporate finite extensibility, while retaining the essential objective framework of the upper-convected derivative.

So far, we have spoken of steady flows. But one of the most powerful ways to characterize a material is to "listen" to its response to vibrations. In rheology, this is done through ​​Small Amplitude Oscillatory Shear (SAOS)​​. The material is placed between two plates, and one is gently oscillated back and forth at a specific frequency, $\omega$.

By analyzing the UCM model under these conditions, we can make a direct connection to experimental measurement. In the limit of very small oscillations, the nonlinear terms in the upper-convected derivative become negligible. The remaining linear equation elegantly predicts the material's response in terms of a ​​complex viscosity​​, $\eta^*(\omega)$. The real part of this viscosity relates to the energy dissipated as heat (the viscous, liquid-like response), while the imaginary part relates to the energy stored and recovered in each cycle (the elastic, solid-like response). By sweeping through different frequencies, rheologists can obtain a mechanical "spectrum" of a material, providing a detailed fingerprint of its internal structure and dynamics. This technique is the workhorse of modern rheology, used to design everything from paints that don't drip to foods with the perfect mouthfeel.

The journey of the upper-convected derivative takes us from the abstract symmetries of physics to the factory floor and the research lab. It is a unifying language that allows us to describe the complex dance of molecules in flowing materials. It ensures our physical laws are objective, gives rise to the critical parameters that govern behavior, and provides a framework for building models that predict the remarkable properties of the complex fluids that shape our world. Far from being a mere mathematical curiosity, it is an essential tool for any scientist or engineer who wishes to understand, predict, and control the flow of matter.