Rheology is the branch of physics that studies the deformation and flow of matter. It applies to materials such as liquids, gases, and solids, and it plays an essential role in fields such as engineering, geology, and materials science. Rheological properties include viscosity, elasticity, plasticity, and viscoelasticity, which determine how a material responds to different stresses and strains.
The winners of the 2017 Ig Nobel Prize, the science award is given to “achievements that first make people laugh then make them think”. The Ig Nobel’s 27th year saw the announcement of ten winners in subjects ranging from physics to biology.
The Ig Nobel Prize for physics went to Marc-Antoine Fardin’s paper, On the Rheology of Cats, which looked into the fluid dynamics of cats in order to determine whether cats can flow like liquids. Fardin observed that: “The wetting and general tribology of cats has not progressed enough to give a definitive answer to the capillary dependence of the feline relaxation time”.
A liquid is traditionally defined as a material that adapts its shape to fit a container. Yet under certain conditions, cats seem to fit this definition. At the center of the definition of a liquid is an action: A material must be able to modify its form to fit within a container.
The action must also have a characteristic duration. In rheology, this is called “relaxation time”. Determining if something is liquid depends on whether it’s observed over a time period that’s shorter or longer than the relaxation time. If we take cats as our example, the fact is that they can adapt their shape to their container if we give them enough time. Cats are thus liquid if we give them the time to become liquid.
What cats show clearly is that determining the state of a material requires comparing two time periods: the relaxation time and the experimental time, which is the time elapsed since the onset of deformation initiated by the container. For instance, it may be the time elapsed since the cat stepped into a sink.
Conventionally, one divides the relaxation time by the experimental time, and if the result is more than 1, the material is relatively solid; if the result is lower than 1, the material is relatively liquid. This is referred to as the Deborah number[1], after the biblical priestess who remarked that on geological time scales (“before God”) even mountains flowed. On shorter time scales one can see glaciers progressively flowing down valleys.
For liquids, there is another dimensionless number that can be used to estimate whether the flow will be turbulent, with vortices, or whether it will calmly follow the outline of the container (we say that the flow is laminar[2]). If the flow speed is V and the container has a typical size of h perpendicular to the flow, then we can define the velocity gradient V/h. The inverse of this velocity gradient scales as time.
Comparing this duration and the relaxation time produces the Reynolds number[3] in the case of fluids dominated by inertia (like water), or the Weissenberg number[4] for those dominated by elasticity (like cake batter). If these dimensionless numbers are large in comparison to 1, then the flow is likely to be turbulent. If they’re small in comparison to 1 the flow is likely to be laminar.
In the absence of reliable extensional rheology data, we can only point to the fact that when cats are deformed along their principal axis, they tend to relax more easily, suggesting that the extensional time is smaller than the shear time. Transient strain-hardening[5] can nonetheless occur.
Second, because, the flows of cats are usually free surface flows, the surface tension between the cat and its surrounding medium can be important. Fardin ultimately determined that “cats are liquid if we give them time to become liquid,” with several influencing factors. Fardin proposes that the type of container, the degree of stress applied to the cat, and the age or relaxation time of the cat may affect feline liquefaction.
Footnotes
- The Deborah number is a dimensionless quantity used in rheology to describe the relative importance of elastic and viscous forces in a material. It is named after the biblical prophetess Deborah, who was known for her ability to lead with both strength and grace. The Deborah number (De) is defined as the ratio of the relaxation time of a material (τ) to the characteristic time scale of the deformation process (t), or De = τ/t. The relaxation time is a measure of the time it takes for a material to return to its original shape after being subjected to deformation, while the characteristic time scale of the deformation process depends on the frequency or rate at which the deformation is applied. A high Deborah number (De > 1) indicates that the material is more elastic than viscous, meaning that it can store and release energy without dissipating it as heat. In contrast, a low Deborah number (De < 1) indicates that the material is more viscous than elastic, meaning that it dissipates energy as heat and flows more easily. The Deborah number is commonly used in the study of complex fluids and soft matter, such as polymer solutions, gels, and biological tissues. It provides a useful tool for predicting the behavior of these materials under different conditions, such as shear flow or oscillatory deformation. [Back]
- Laminar refers to a type of fluid flow where the fluid particles move in parallel layers or laminae, without significant mixing between them. In laminar flow, the fluid velocity at any given point remains constant over time, and the flow is generally smooth and predictable. Laminar flow occurs at low flow rates or in highly viscous fluids, where the forces between the fluid particles dominate over the forces due to fluid turbulence. The opposite of laminar flow is turbulent flow, where the fluid particles move in a chaotic and random manner, resulting in eddies, vortices, and mixing between the fluid layers. The distinction between laminar and turbulent flow is important in many engineering applications, such as fluid transport, heat transfer, and chemical reactions, as the type of flow can have a significant impact on the efficiency and performance of the system. For example, laminar flow is desirable in some situations, such as in microfluidic devices or in the transport of highly sensitive fluids, where turbulence could damage the system or alter the fluid properties. [Back]
- The Reynolds number is a dimensionless quantity used in fluid mechanics to describe the relative importance of inertial and viscous forces in a fluid flow. It is named after Osborne Reynolds, a British engineer, and mathematician who first described the phenomenon in the late 1800s. The Reynolds number (Re) is defined as the ratio of inertial forces to viscous forces in a fluid flow. Mathematically, it is expressed as: Re = ρvL/μ where ρ is the fluid density, v is the fluid velocity, L is a characteristic length scale of the system (such as the diameter of a pipe), and μ is the fluid viscosity. The Reynolds number is a useful tool for predicting the behavior of fluid flow, as it can determine whether the flow is laminar or turbulent. At low Reynolds numbers, the flow is typically laminar, characterized by smooth and ordered layers of fluid particles. At high Reynolds numbers, the flow becomes turbulent, characterized by chaotic and unpredictable fluid motions, eddies, and vortices. The Reynolds number is commonly used in the design and analysis of fluid systems, such as pipelines, heat exchangers, and aerodynamic structures. It provides a way to predict the onset of turbulence and to determine the optimal operating conditions for a given system. [Back]
- The Weissenberg number is a dimensionless quantity used in rheology to describe the relative importance of elastic and viscous forces in a viscoelastic fluid. It is named after Karl Weissenberg, a German physicist who made significant contributions to the field of polymer rheology in the mid-1900s. The Weissenberg number (Wi) is defined as the product of the fluid’s elastic modulus (G) and a characteristic time scale (τ) of the system, divided by the fluid’s viscosity (μ). Mathematically, it is expressed as: Wi = Gτ/μ where G is the elastic modulus, τ is a characteristic time scale (such as the relaxation time of the polymer chains), and μ is the fluid viscosity. The Weissenberg number is a useful tool for characterizing the behavior of viscoelastic fluids, such as polymer solutions, gels, and biological fluids. It provides a way to predict the response of the fluid to external forces, such as shear or extensional deformation. At low Weissenberg numbers, the fluid behaves like a Newtonian fluid, with the elastic properties having little effect on the overall flow behavior. At high Weissenberg numbers, the elastic forces dominate, and the fluid can exhibit significant non-Newtonian behavior, such as strain hardening, shear thickening, or viscoelastic instabilities. [Back]
- Transient strain-hardening is a type of non-Newtonian behavior exhibited by some viscoelastic fluids, in which the material becomes stiffer and more resistant to deformation over time, even at a constant applied stress or strain rate. This behavior is characterized by a transient period of strain-hardening, followed by a steady-state response. During the transient period, the fluid undergoes an initial phase of elastic deformation, in which the polymer chains within the fluid are stretched and aligned in the direction of the applied stress. This results in an increase in the fluid’s elastic modulus, causing it to become more resistant to deformation. As the polymer chains continue to deform and relax over time, the material gradually becomes less stiff, eventually reaching a steady-state response. Transient strain-hardening is commonly observed in polymer melts, solutions, and gels, as well as in biological fluids such as blood and synovial fluid. It is important to understand this behavior when designing and analyzing systems that involve the flow of viscoelastic fluids, such as extrusion, injection molding, and lubrication. [Back]
Further Reading
Sources
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Trib Net
The Conversation
Telus World of Science Edmonton
- Larson, R. G. (1999). The structure and rheology of complex fluids. Oxford University Press.
- Meissner, J., & Schröder, A. (2006). Rheology of soft matter. Wiley-VCH.
- Wang, M., & Chen, W. (2011). The Deborah number and its implications in the study of viscoelasticity. Journal of Non-Newtonian Fluid Mechanics, 166(12-13), 763-773.
- Wilhelm, M., & Frey, E. (2016). The Deborah number as a predictor of the elasticity of polymer networks. Journal of Rheology, 60(3), 497-503.
- White, F. M. (2016). Fluid mechanics (8th ed.). McGraw-Hill Education.
- Fox, R. W., & McDonald, A. T. (2011). Introduction to fluid mechanics (8th ed.). John Wiley & Sons.
- Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2007). Transport phenomena (2nd ed.). John Wiley & Sons.
- Larson, R. G. (1999). The structure and rheology of complex fluids (Vol. 7). Oxford University Press.
- McKinley, G. H., & Tripathi, A. (1999). Transient and steady-state extensional rheology of dilute polymer solutions. Journal of Rheology, 43(5), 1213-1232.
- Tadmor, Z., & Gogos, C. G. (2006). Principles of polymer processing (2nd ed.). John Wiley & Sons.
