Notes on Stress, Shear, and Viscosity: Newtonian and Non-Newtonian Fluids

Foundational quantities in solid mechanics and fluids

• Pressure and normal stress
  • Pressure is the normal force per unit area: $P = \frac{F}{A}$
  • In solid mechanics, this is viewed as normal stress (often denoted by sigma, $\sigma$)
  • Stress is a tensor with multiple components; we also have shear stress (not just normal stress)
• Normal vs. shear stress
  • Normal stress is perpendicular to a surface
  • Shear stress is parallel to a surface: defined as the resistance to deformation due to shear force
  • Shear stress is denoted by $\tau$ (tau); common shorthand in class: tau for shear stress
  • Example notation: $\tau_{z}^{x}$ in an $x,y,z$ coordinate system indicates shear stress in the x-direction on a surface with normal in the z-direction; the first footnote indicates the velocity-gradient direction, the second notes the direction of the shear force
  • Units: shear stress has the same units as pressure (Pa in SI)
• Deformation under stress: solids vs fluids
  • For solids: applying shear leads to deformation (shape changes) while the body resists; the deformation can be visualized as a stack of layers sliding relative to each other
  • Strain (deformation measure) is percent deformation; for elongation, $\varepsilon = \frac{\Delta L}{L_0}$
  • In fluids: fluids offer little resistance to permanent deformation; they flow; deformation is continuous under shear
  • Viscosity is the mediator of shear-induced deformation in fluids; it links shear stress to the rate of deformation
• Shear stress vs deformation in fluids and the velocity gradient
  • Shear rate (gamma dot) is the rate of shear deformation, i.e., velocity gradient
  • Basic definition (simple shear in x-direction across y): $\dot{\gamma} = \frac{du_x}{dy}$ (sometimes written as $\frac{d u}{d y}$ depending on notation)
  • Conceptual picture: adjacent fluid layers move with different velocities; the interaction between layers (intermolecular forces) governs the rate of momentum transfer and hence viscosity
  • If the top layer is dragged faster than the layer below, momentum transfers downward and upward, with dissipation as heat due to intermolecular interactions
  • The velocity gradient quantifies how different layers move; a larger gradient means larger shear rate and typically more shear stress for a given fluid
• Viscosity and the Newtonian constitutive relation
  • Viscosity concept: a material property that describes how resistant a fluid is to shear and how it mediates the conversion of shear stress into deformation rate
  • Constitutive relation (Newtonian fluids): $\tau = \mu \dot{\gamma}$ where $\dot{\gamma}$ is the shear rate and $\mu$ is the dynamic viscosity
  • The proportionality constant (viscosity) is assumed constant for Newtonian fluids (independent of the shear rate); for many real fluids, viscosity can depend on the shear rate (non-Newtonian behavior)
  • Apparent viscosity: $\mu<em>{app} = \frac{\tau}{\dot{\gamma}}$; if $\mu</em>{app}$ is constant across different shear stresses and shear rates, the fluid is Newtonian; if not, it is non-Newtonian
  • In many liquids, especially simple liquids like water, the Newtonian linear relationship holds over a wide range; complex fluids (paints, starch suspensions, blood) often show non-Newtonian behavior
• Why the relationship is often first-order and linear (small-strain intuition)
  • Consider an infinitesimally thin layer of fluid; for a very small strain, the change in angle and velocity can be related linearly to the applied shear stress
  • For tiny deformations, $\tan(\Delta\theta) \approx \Delta\theta$, and the ratio $\frac{\Delta u}{\Delta t}$ relates to the velocity gradient across a small separation $\Delta y$
  • This leads to a proportionality between shear stress and velocity gradient in the infinitesimal limit, i.e., a linear constitutive relation with viscosity as the proportionality constant
  • In many texts, this linear, first-order relation underpins the Newtonian model; deviations at high shear rates or for complex fluids yield non-Newtonian behavior
• Why viscosity is a constitutive property and its units
  • Viscosity depends on the molecular structure and intermolecular interactions of the fluid; it is not a universal constant but a property of the material
  • SI unit of dynamic viscosity: $[\mu] = \text{Pa} \cdot \text{s}$ (or kg/(m·s))
  • Some alternate units include poise (P) and centipoise (cP): 1 Pa·s = 10 Poise; 1 cP = 0.001 Pa·s
• Typical viscosity values for common fluids (at around room temperature unless stated otherwise)
  • Water: $\mu \approx 1.0 \times 10^{-3} \text{ Pa}\cdot\text{s}$
  • Air: about two orders of magnitude smaller than water; commonly quoted around $\mu_{air} \sim 1.8 \times 10^{-5} \text{ Pa}\cdot\text{s}$ (roughly 10^(-5) to 10^(-5) range)
  • Plasma (component of blood): similar to water, slightly higher; roughly $\mu_{plasma} \approx 1.2 \text{ cP} = 1.2 \times 10^{-3} \text{ Pa}\cdot\text{s}$
  • Blood: viscosity depends on hematocrit (cell concentration); large-vessel viscosity around $\mu_{blood} \approx 4 \text{ cP} = 4 \times 10^{-3} \text{ Pa}\cdot\text{s}$ at 37°C ~ hematocrit ≈ 45%
• Kinematic viscosity and momentum diffusivity
  • Kinematic viscosity: $\nu = \frac{\mu}{\rho}$ where $\rho$ is density
  • Units: $\nu$ has units of $\text{m}^2/\text{s}$; describes diffusivity of momentum instead of momentum per unit area per unit time
  • Relationship: for a given fluid, higher density lowers the kinematic viscosity for the same dynamic viscosity
• Practical demos and contexts to connect concepts
  • Torque converters and lubricants: viscous fluids (oil) enable momentum transfer and heat removal in transmissions; oil’s viscosity is central to coupling momentum and cooling
  • The role of viscosity in everyday devices and engineering systems: viscosity affects how momentum is transmitted and dissipated as heat or used for coupling
• Special case: viscosity at very low temperatures (helium experiments and two-fluid model)
  • Helium II (superfluid phase below lambda point): demonstrates a paradox where capillary flow appears to show zero viscosity, while a rotating cylinder reveals a finite (normal) component with viscosity
  • Two-fluid model for He II: normal component with viscosity that drags with macroscopic flow, and superfluid component with essentially zero viscosity that can flow through very narrow channels with no resistance
  • Capillary flow experiments show normal component controls resistance in narrow capillaries; rotating-cylinder experiments reveal the normal component’s viscosity; the superfluid component does not participate in the slow, viscous transport in the capillaries
  • This duality leads to the concept that He II behaves as two interpenetrating fluids: a normal viscous component and a zero-viscosity superfluid component
• Temperature dependence and limits of viscosity
  • As temperature approaches absolute zero, molecular motion drops; classical viscosity tends toward zero but a residual viscosity can persist due to quantum and other effects in real systems
  • In classical fluids, viscosity is generally nonzero at finite temperatures because molecules continually interact
• Non-Newtonian fluids: definitions and examples
  • Apparent viscosity concept: $\mu_{app} = \frac{\tau}{\dot{\gamma}}$
  • Newtonian fluid: $\mu_{app}$ is constant regardless of the applied shear stress or shear rate
  • Non-Newtonian fluid: $\mu_{app}$ changes with shear rate or shear stress; viscosity is not constant
  • Everyday examples
  • Paints: polymer particles in water; mixing paint initially hard; as shear rate increases, viscosity decreases (shear-thinning) or sometimes complicates behavior depending on formulation
  • Starch suspensions: nonlinear and can be shear-thickening (viscosity increases with shear rate) in some systems
  • Blood: viscosity depends on hematocrit (cell concentration) due to interactions and collisions among cells; higher hematocrit -> higher viscosity
• Non-slip boundary conditions (mention for later)
  • Boundary conditions in fluid mechanics; non-slip condition states that fluid velocity at a solid boundary equals the boundary velocity (often zero for stationary walls)
  • Will be discussed further in subsequent topics
• Connections to broader themes and practical implications
  • Viscosity connects to energy dissipation, heat generation, and momentum transport in fluids
  • Viscosity is central to fluid design in pipelines, lubrication, microfluidics, and biological systems (blood flow, capillary transport)
  • The concept of Newtonian vs non-Newtonian fluids helps explain why some real-world fluids behave counterintuitively under different flow conditions (e.g., paints that become easier to stir at high speeds, starch suspensions that stiffen with rapid stirring)
• Quick glossary of key notations and concepts used in this material
  • Pressure: $P = \frac{F}{A}$
  • Normal stress: $\sigma$
  • Shear stress: $\tau$
  • Shear rate: $\dot{\gamma}$; often $\dot{\gamma} = \dfrac{du}{dy}$ in simple shear
  • Dynamic viscosity: $\mu$; unit $\text{Pa}\cdot\text{s}$
  • Kinematic viscosity: $\nu = \dfrac{\mu}{\rho}$; unit $\text{m}^2/\text{s}$
  • Relationship between shear stress and rate: $\tau = \mu \dot{\gamma}$ (Newtonian)
  • Apparent viscosity: $\mu_{app} = \dfrac{\tau}{\dot{\gamma}}$
  • Strain (solid deformation): $\varepsilon = \frac{\Delta L}{L_0}$
  • Capillary flow in helium: He I (normal viscous), He II (two-fluid: normal component with viscosity + superfluid with zero viscosity), capillary vs rotating-cylinder experiments
• Summary takeaway
  • Stress, strain, and viscosity are core concepts for connecting forces to deformations in solids and fluids
  • Viscosity acts as the mediator between shear stress and velocity gradients in fluids; it is a material property that can be constant (Newtonian) or depend on shear rate (non-Newtonian)
  • Momentum diffusion in fluids is characterized by viscosity and, in the case of momentum diffusion in fluids, by the related kinematic viscosity; real fluids exhibit a rich set of behaviors from simple Newtonian to complex non-Newtonian and quantum fluids (e.g., helium II)