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Helium-4, when cooled below approximately -271 °C (the lambda point of 2.17 K), becomes a superfluid liquid with zero viscosity in its superfluid component, allowing it to climb container walls and flow through tiny pores or capillaries that viscous liquids cannot penetrate.

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A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, that is, it has variable viscosity dependent on stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid.
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The fascinating concept of Non-Newtonian fluids, which transition from a liquid state to a solid-like state when pressure is applied, has a rich history that spans several centuries. The study and understanding of these peculiar fluids have evolved over time, leading to a wide range of practical applications and scientific insights. One of the earliest references to Non-Newtonian behavior in fluids dates back to the 17th century when Sir Isaac Newton formulated the basic principles of fluid mechanics. Newton's laws of fluid motion primarily applied to Newtonian fluids, which exhibit constant viscosity and flow behavior regardless of the applied force or pressure. However, it soon became apparent that not all fluids behaved in this predictable manner. In the mid-19th century, a scientist named Thomas Andrews made significant contributions to the understanding of Non-Newtonian fluids. Andrews conducted groundbreaking experiments with carbon dioxide, revealing that under high pressure, this gas could transform into a liquid. This observation marked one of the earliest instances of pressure-induced phase changes in fluids. The term "Non-Newtonian" itself was coined in the 20th century to describe fluids that did not adhere to Newton's classical laws of fluid dynamics. These fluids exhibited a variety of behaviors, but one of the most intriguing was their ability to solidify or increase in viscosity when subjected to stress or pressure. One of the most famous examples of such behavior is cornstarch mixed with water, which forms a substance known as "oobleck" that becomes more solid when pressure is applied. In the modern era, Non-Newtonian fluids have found applications in various fields, including food science, engineering, and material science. They are used in products like quicksand, body armor, and even in the development of impact-resistant materials. One of the key insights that emerged from the study of Non-Newtonian fluids is the importance of understanding the relationship between stress and strain, as well as the influence of time-dependent properties on their behavior. This knowledge has led to advancements in rheology, the study of flow and deformation in materials, and has practical implications in areas such as industrial processing, medicine, and the design of everyday products.

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d’Alembert’s Paradox: ν → 0 Is Not ν = 0 In 1752, Jean-le-Rond d’Alembert proved a result that still trips people up: An inviscid, incompressible, steady flow exerts zero drag on a body. No wake. No resistance. The equations let the fluid slip past as if the object weren’t there. The first animation shows exactly that world. The flow is ideal Euler flow. Streamlines bend around the body, accelerate, slow down, then recombine perfectly downstream. Nothing is left behind. The motion you see comes from pathlines moving through a steady velocity field, not from any evolving structure in the flow itself. The setup is the ideal fluid model: Euler (ν = 0) ρ(∂u/∂t + (u·∇)u) = −∇p ∇·u = 0 Assume steady flow and zero viscosity and the picture locks in. If the flow is also irrotational, ∇×u = 0, you can write u = ∇φ and the problem collapses to potential flow: ∇²φ = 0 u = ∇φ The force on the body comes entirely from pressure: F = −∮ p n dS D = F·eₓ Under these assumptions the pressure field is perfectly front–back symmetric, so the integral gives D = 0 That’s the paradox. Not a small correction. Zero. Now look at the second animation. This is the same geometry and the same inflow, but with viscosity turned on, even if it’s only a small amount. Navier–Stokes (ν > 0) ρ(∂u/∂t + (u·∇)u) = −∇p + μ∇²u ∇·u = 0 That extra term changes everything. A thin boundary layer forms near the surface. Separation becomes possible. Vorticity is generated and shed. A wake appears. Drag is no longer optional. The contrast is the point. Letting viscosity go to zero is not the same thing as setting it to zero. The inviscid limit deletes the mechanism that breaks time-reversal symmetry and allows energy dissipation. Once that mechanism is gone, wakes can’t exist, and drag vanishes by construction. #FluidDynamics #NavierStokes #EulerEquations #DAlembertParadox #BoundaryLayer #Physics
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d’Alembert’s Paradox: ν → 0 Is Not ν = 0 In 1752, Jean-le-Rond d’Alembert proved a result that still trips people up: An inviscid, incompressible, steady flow exerts zero drag on a body. No wake. No resistance. The equations let the fluid slip past as if the object weren’t there. The first animation shows exactly that world. The flow is ideal Euler flow. Streamlines bend around the body, accelerate, slow down, then recombine perfectly downstream. Nothing is left behind. The motion you see comes from pathlines moving through a steady velocity field, not from any evolving structure in the flow itself. The setup is the ideal fluid model: Euler (ν = 0) ρ(∂u/∂t + (u·∇)u) = −∇p ∇·u = 0 Assume steady flow and zero viscosity and the picture locks in. If the flow is also irrotational, ∇×u = 0, you can write u = ∇φ and the problem collapses to potential flow: ∇²φ = 0 u = ∇φ The force on the body comes entirely from pressure: F = −∮ p n dS D = F·eₓ Under these assumptions the pressure field is perfectly front–back symmetric, so the integral gives D = 0 That’s the paradox. Not a small correction. Zero. Now look at the second animation. This is the same geometry and the same inflow, but with viscosity turned on, even if it’s only a small amount. Navier–Stokes (ν > 0) ρ(∂u/∂t + (u·∇)u) = −∇p + μ∇²u ∇·u = 0 That extra term changes everything. A thin boundary layer forms near the surface. Separation becomes possible. Vorticity is generated and shed. A wake appears. Drag is no longer optional. The contrast is the point. Letting viscosity go to zero is not the same thing as setting it to zero. The inviscid limit deletes the mechanism that breaks time-reversal symmetry and allows energy dissipation. Once that mechanism is gone, wakes can’t exist, and drag vanishes by construction. #FluidDynamics #NavierStokes #EulerEquations #DAlembertParadox #BoundaryLayer #Physics

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