Lower-Convected Derivative

In the study of deforming materials like fluids and solids, accurately describing how physical properties change is a fundamental yet complex challenge. The laws of physics are expected to hold true regardless of the observer's frame of reference—a concept known as the principle of material frame-indifference, or objectivity. However, the most intuitive way to measure change, the material time derivative, fails to meet this requirement, as its value depends on the observer's rotational motion. This discrepancy creates a significant gap in our ability to formulate universal physical laws for continuum mechanics.

This article addresses this problem by introducing the family of objective time derivatives, which are mathematical tools designed to provide a frame-independent measure of change. Over the next sections, you will gain a comprehensive understanding of these crucial concepts. The "Principles and Mechanisms" section will dissect the problem of observer-dependence and introduce the lower-convected derivative, explaining its mathematical structure and profound geometric connection to deforming surfaces. Subsequently, the "Applications and Interdisciplinary Connections" section will demonstrate how these theoretical constructs are applied to model the behavior of real-world materials in fields ranging from polymer rheology to geophysics, illustrating their indispensable role in modern science and engineering.

To understand the world, we must measure how it changes. But what seems like a simple task is surprisingly subtle. Imagine drawing a picture on a sheet of rubber and then stretching and rotating the sheet. If you ask, "How is the picture changing?", the answer depends entirely on your point of view. Are you a tiny observer standing on the rubber sheet, being stretched and rotated along with it? Or are you a stationary observer watching from afar? The laws of physics, we believe, should not depend on the arbitrary choice of the observer. This fundamental idea, known as the ​​principle of material frame-indifference​​, or more simply ​​objectivity​​, is the starting point of our journey.

In the mechanics of continuous materials like fluids and solids, we often track physical properties represented by mathematical objects called ​​tensors​​. You can think of a simple vector, like velocity, as a first-order tensor. But many properties are more complex. For instance, the stress within a material—the internal forces that particles exert on each other—is a second-order tensor, because at any point, you have forces acting on surfaces with different orientations. Another example is a tensor describing the average orientation and stretch of polymer molecules in a fluid.

The most straightforward way to measure the rate of change of such a tensor, $\boldsymbol{A}$, is to follow a tiny piece of the material and see how $\boldsymbol{A}$ changes with time. This is called the ​​material time derivative​​, denoted as $\dot{\boldsymbol{A}}$. It tells you the rate of change you'd see if you were surfing on a particle as it moves through space.

Here’s the catch: this simple derivative is not objective. To see why, consider a fluid undergoing a pure rigid-body rotation, like coffee being stirred in a cup. Imagine a tensor $\boldsymbol{A}$ that represents some fixed microstructural feature, like the alignment of tiny, frozen-in fibers. To an observer rotating with the fluid, this feature is constant; nothing is changing. Yet, to a stationary observer watching the cup, the fibers are clearly rotating. The material derivative $\dot{\boldsymbol{A}}$ will capture this rotation and be non-zero. It turns out that for a pure rotation described by a spin tensor $\boldsymbol{W}$, the material derivative is $\dot{\boldsymbol{A}} = \boldsymbol{W}\boldsymbol{A} - \boldsymbol{A}\boldsymbol{W}$. This rate of change is not "real" in a physical sense; it's an artifact of our stationary viewpoint. It tells us about the rotation of the object, not about any intrinsic change within it. Since the laws of physics should describe intrinsic changes, the material derivative $\dot{\boldsymbol{A}}$ is not a suitable ingredient for building physical laws.

To formulate laws of nature that are independent of the observer's motion, we need to invent new kinds of derivatives—​​objective time derivatives​​. These are mathematical tools ingeniously designed to subtract the "spurious" changes due to rigid rotation, leaving only the changes that correspond to actual stretching and deformation of the material. They are like special camera filters that remove the motion of the camera itself, letting us see only what's truly happening in the scene.

There isn't just one way to do this, which gives rise to a "family" of objective rates, each with its own perspective on what constitutes a "true" change. The most important members of this family are the Jaumann rate, the upper-convected derivative, and the lower-convected derivative.

To understand their structure, we first need to dissect the motion itself. The local motion of a fluid is described by the ​​velocity gradient tensor​​, $\boldsymbol{L}$. This tensor contains everything about how the velocity changes from point to point. We can split $\boldsymbol{L}$ into two parts:

• The ​​rate-of-deformation tensor​​, $\boldsymbol{D} = \frac{1}{2}(\boldsymbol{L} + \boldsymbol{L}^{\mathrm{T}})$, which is symmetric and describes how the material is stretching and shearing.
• The ​​spin tensor​​ (or vorticity tensor), $\boldsymbol{W} = \frac{1}{2}(\boldsymbol{L} - \boldsymbol{L}^{\mathrm{T}})$, which is skew-symmetric and describes how the material is locally rotating.

With this decomposition, we can see how the different objective rates are constructed.

The ​​Jaumann rate​​ (or corotational rate) is the most intuitive fix. It takes the material derivative $\dot{\boldsymbol{A}}$ and simply subtracts the part due to pure rotation:

This rate measures change as seen by an observer who is spinning along with the local material element but is not stretching with it.

The ​​convected derivatives​​, however, are more profound. They describe the rate of change from the perspective of an observer who is not only spinning but also stretching and deforming with the material. They come in two flavors: upper and lower.

The ​​upper-convected derivative​​ is defined as:

And the ​​lower-convected derivative​​ is defined as:

These definitions might seem arbitrary at first, but they are precisely what is needed to ensure objectivity. The extra terms involving $\boldsymbol{L}$ and $\boldsymbol{L}^{\mathrm{T}}$ conspire to perfectly cancel all the non-objective terms that arise from a change in observer. If we use our decomposition of $\boldsymbol{L}$ into $\boldsymbol{D}$ and $\boldsymbol{W}$, the structure of these derivatives becomes crystal clear:

This beautiful result shows that all three rates share the same core correction for spin (the Jaumann rate). They differ only in how they treat the stretching, $\boldsymbol{D}$. The upper-convected rate subtracts the stretching effects, while the lower-convected rate adds them. The difference between the two is directly related to the deformation; for instance, for the strain-rate tensor $\boldsymbol{D}$ itself, their difference is $\stackrel{\triangledown}{\boldsymbol{D}} - \underset{\triangledown}{\boldsymbol{D}} = -2(\boldsymbol{D}\boldsymbol{D} + \boldsymbol{D}\boldsymbol{D}) = -4\boldsymbol{D}^2$.

So, what is the physical meaning behind the lower-convected derivative? Why would we want to add terms related to stretching? The answer lies in the geometric nature of the quantity we are measuring.

The lower-convected derivative is the natural way to measure the rate of change of ​​covariant tensors​​. A simple way to think of a covariant quantity is as something associated with a ​​surface element​​ that is embedded in the material. Imagine a tiny square painted on a deforming rubber sheet. As the sheet deforms, the square deforms with it. The lower-convected derivative measures how a property associated with this deforming square (like its area or a force acting per unit area) changes from the perspective of the square itself. Because this perspective is "attached" to the deforming material, the resulting rate of change is inherently objective.

This geometric interpretation is beautifully illuminated by a few key examples. In mechanics, the ​​identity tensor​​, $\boldsymbol{I}$, can be thought of as the metric of our space—it's the tool we use to measure lengths and angles. If we ask how this metric changes as the material flows, we must use an objective derivative. Calculating the lower-convected derivative of the identity tensor gives a stunningly simple result:

This equation is a profound statement: the objective rate of change of our spatial measuring stick is exactly twice the rate at which the material itself is stretching and deforming. The trace (sum of diagonal elements) of this equation tells us that the rate of change of the volume of the metric is twice the rate of change of the volume of the material element ($\nabla \cdot \boldsymbol{v}$), directly linking the geometry of space to the physics of compressibility.

The true power of the lower-convected derivative is revealed when it yields a result of zero. In continuum mechanics, the ​​Finger tensor​​, $\boldsymbol{B}^{-1}$, is a measure of deformation. It is a covariant tensor that essentially tracks how surface elements have been stretched and sheared from their original shape in an undeformed reference state. If we compute its lower-convected derivative, we find:

This result is far from trivial. It means that from the perspective of an observer embedded within and deforming with the material surfaces, the Finger tensor appears constant. It is the natural, unchanging measure of strain for this viewpoint. It’s like having a ruler made of perfectly elastic rubber; even as the ruler is stretched to twice its length, an ant living on the ruler would still measure its length as "one ruler-length." This consistency is what makes the combination of the Finger tensor and the lower-convected derivative so powerful in building constitutive models for materials like viscoelastic fluids and rubbery solids.

This also clarifies why the different objective rates are not interchangeable. For a pure rigid rotation where the material spins but does not stretch ($\boldsymbol{D} = \boldsymbol{0}$), all three objective rates—Jaumann, upper-, and lower-convected—become identical. As expected, they all correctly report a rate of zero for a tensor that is merely rotating with the body, whereas the non-objective material derivative reports a non-zero rate. The choice of which derivative to use in a physical law depends on the geometric character of the tensor in question: is it associated with lines (contravariant, use upper-convected), surfaces (covariant, use lower-convected), or is it something else entirely? This deep connection between physics and geometry reveals the underlying unity and elegance of continuum mechanics.

Having grappled with the principles and mechanisms of objective derivatives, you might be tempted to view them as a somewhat esoteric piece of mathematical machinery. But nothing could be further from the truth. These concepts are not just abstract formalism; they are the very language we use to describe how real materials—from the polymer chains in a shampoo bottle to the rock mantle of our planet—behave under the duress of flow, stretching, and rotation. This is where the physics truly comes alive, connecting our mathematical framework to tangible, measurable phenomena. It is a journey from the abstract world of tensors to the practical world of engineering, geology, and chemistry.

First, let's ask a fundamental question: what does it mean for a derivative to be "objective"? Imagine you are a tiny observer, a Maxwell's demon of sorts, riding along on a small parcel of fluid. This fluid parcel might be tumbling and spinning wildly as it moves. If a property of the fluid, say its internal stress, appears constant to you in your tumbling frame of reference, then any "objective" rate of change you measure must be zero. This is the essential test of objectivity.

In a pure rigid-body rotation, every part of the material spins together without any stretching or distortion. For any tensor property that is simply "carried along" by this rotation—meaning it is constant in the co-rotating reference frame—its material derivative is decidedly not zero. An observer in a fixed laboratory frame sees the tensor's components changing simply because the object is turning. The magic of objective derivatives, like the upper-convected, lower-convected, and Jaumann rates, is that they are specifically designed to subtract out this illusory change due to pure rotation. When applied to such a tensor in a rigid rotation, they all correctly yield a value of zero, passing the litmus test of a true, frame-independent physical measurement. They capture the rate of change that is intrinsic to the material's deformation, not the observer's perspective.

At first glance, the existence of multiple objective derivatives, like the upper- and lower-convected rates, might seem like an unnecessary complication. Why more than one? The answer reveals a beautiful and profound duality in the heart of continuum mechanics, a symmetry that reflects the different ways geometric objects can be transported by a deforming medium.

Imagine a line of dye injected into a clear, flowing syrup. As the syrup deforms, this line stretches and rotates. The vector connecting two points on this line is a contravariant vector, an object that is "pushed forward" and stretched by the flow. The evolution of structures built from such vectors, like the conformation of a polymer chain, is most naturally described by the ​​upper-convected derivative​​. In fact, if we model the long-chain molecules in a polymer solution as tiny dumbbells that are carried affinely by the flow, their average shape, described by a conformation tensor $\boldsymbol{A}$, evolves precisely in a way that its upper-convected derivative is zero (in the absence of thermal relaxation). This derivative is tailor-made for things that stretch with the material.

Now, consider a different object: a small surface element embedded in the same flowing syrup. Think of it as the face of a tiny cube. As the material deforms, this surface not only moves, but its orientation and area change. The normal vector to this surface is a covariant vector; it transforms differently from a stretching line element. Its evolution is described by a "pull-back" from the reference state. The ​​lower-convected derivative​​ is the natural language for describing the evolution of tensors built from these covariant objects. Indeed, if one constructs a tensor representing a convected material surface, its lower-convected derivative is found to be zero. This duality is not a mathematical quirk; it is a reflection of the fundamental geometric structure of deformation.

This distinction is not merely academic. The choice of objective derivative has dramatic and experimentally verifiable consequences in the field of rheology, the science of the flow and deformation of matter. Consider a simple "Maxwell" fluid, a basic model for materials like polymer melts or Boger fluids that exhibit both viscous (liquid-like) and elastic (solid-like) properties.

If we build a constitutive model using the upper-convected derivative (the Upper-Convected Maxwell or UCM model) and compare it to one using the lower-convected derivative (the Lower-Convected Maxwell or LCM model), they make startlingly different predictions about the fluid's behavior in a simple shear flow, like the flow between two parallel plates. While both models predict the same shear stress and the same first normal stress difference ($N_1 = \tau_{xx} - \tau_{yy}$), which is responsible for phenomena like the Weissenberg effect where a fluid climbs a rotating rod, they disagree on the second normal stress difference ($N_2 = \tau_{yy} - \tau_{zz}$). The UCM model predicts $N_2 = 0$, while the LCM model predicts a non-zero, negative $N_2$. Experiments on polymer solutions show that $N_2$ is indeed non-zero and typically much smaller than $N_1$. This tells us that neither simple model is perfect, but their differences are real and measurable, stemming directly from the choice of objective rate.

The Oldroyd-B model, a cornerstone of polymer rheology, elegantly combines these ideas. It models a dilute polymer solution as a mixture of a simple Newtonian solvent (like water) and a polymeric component whose stress is governed by the upper-convected Maxwell model. This choice is physically motivated: the upper-convected derivative correctly captures the affine transport of the contravariant conformation tensor of the polymer chains.

The utility of these concepts extends far beyond polymer solutions. In geophysics, modeling the flow of Earth's mantle or the deformation of tectonic plates involves materials subjected to immense pressures and temperatures over geological timescales. Rocks, under these conditions, can flow and undergo enormous rotations. To formulate a "hypoelastic" law that relates the rate of stress to the rate of deformation in a rock, one must use an objective stress rate to properly handle the large rotations of tectonic blocks. Geophysicists employ a variety of rates, including the Jaumann rate, the Green-Naghdi rate (which is based on the true rotation of the material's internal coordinate system), and the Truesdell rate, each with its own nuances and domain of applicability. The choice is critical for accurately predicting stress accumulation and release in processes like earthquakes and mountain building.

In the modern era, these theoretical principles have a direct impact on the billion-dollar industry of computer-aided engineering. When writing software for computational fluid dynamics (CFD) or computational solid mechanics (CSM), ensuring material objectivity is not an option; it is a requirement for a valid simulation. A simulation of a stirred tank reactor or a car's tire that does not use an objective formulation for the material's stress will produce physically meaningless results that depend on the arbitrary orientation of the laboratory's coordinate system. The mathematical verification of objectivity is a standard procedure in the development and validation of advanced computational codes.

Finally, the quest for better constitutive models brings us to a fascinating computational and physical puzzle: the High Weissenberg Number Problem (HWNP). The Weissenberg number ($Wi$) is a dimensionless quantity that compares a material's characteristic relaxation time to the rate of the imposed flow. At high $Wi$, the flow is fast, and the material doesn't have time to relax, leading to highly elastic behavior.

The simple and elegant Oldroyd-B model, with its upper-convected derivative, makes a dire prediction: in a strong extensional flow (a stretching flow, like pulling taffy), once $Wi$ exceeds a critical value (0.5 for planar extension), the polymer stress is predicted to grow without bound! This "stress blow-up" is a feature of the model that reflects a real physical phenomenon called the coil-stretch transition, where polymer chains abruptly unravel. However, this mathematical singularity poses an immense challenge for numerical simulations, causing them to fail catastrophically.

This is not a failure of the concept of objective derivatives, but a sign that our physical model is too simple. Nature has a more clever way of handling this. The Giesekus model introduces a more realistic physical ingredient: anisotropic drag. It posits that the friction a polymer chain feels depends on its orientation relative to its neighbors. This seemingly small refinement has a profound mathematical consequence. It introduces a nonlinear damping term into the evolution equation for the stress. This term is quadratic and acts as a powerful "sink" that grows rapidly as the polymer stretches, preventing the stress from ever reaching infinity. It tames the catastrophic blow-up predicted by the simpler model, leading to stress saturation that is both more physically realistic and computationally manageable.

This beautiful interplay—where a deeper physical insight leads to a more robust mathematical model that resolves a crippling computational problem—is the hallmark of modern continuum physics. It shows that the family of objective derivatives is not a closed chapter in a textbook, but an active and essential toolkit for scientists and engineers pushing the boundaries of what we can understand and simulate.