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Viscosity

Intuition

Drag a spoon through water, then through honey. Honey resists — not because it's heavier (it barely is), but because its layers grip each other as they slide. Viscosity is internal friction between fluid layers in relative motion: fast layers drag slow ones forward, slow layers hold fast ones back, and mechanical energy quietly leaks into heat.

Formal definition

For simple shear (velocity \(u(y)\) varying across the flow), Newton's law of viscosity:

\[\boxed{\,\tau = \mu\,\frac{du}{dy}\,}\]
  • \(\tau\) — shear stress (Pa)
  • \(\mu\) — dynamic viscosity (Pa·s), a material property
  • \(du/dy\) — velocity gradient / shear rate (s⁻¹)

The kinematic viscosity \(\nu = \mu/\rho\) (m²/s) is momentum's diffusion coefficient — see the Navier–Stokes equation and Newton's law of viscosity.

Newtonian vs non-Newtonian

Class Behavior Examples
Newtonian \(\mu\) constant; \(\tau \propto du/dy\) water, air, most oils
Shear-thinning (pseudoplastic) \(\mu\) falls with shear rate paint, blood, ketchup
Shear-thickening (dilatant) \(\mu\) rises with shear rate cornstarch + water
Viscoelastic stores and dissipates polymers, biological fluids

This course stays Newtonian; the rich world beyond is rheology.

What viscosity does to a flow

  • No-slip condition. At a solid boundary the fluid velocity equals the wall velocity. This single boundary condition creates most of real-fluid physics.
  • Velocity gradients. Layers must shear to reconcile no-slip walls with a moving interior — hence profiles like Couette (linear) and Poiseuille (parabolic).
  • Boundary layers. At high Reynolds number viscosity retreats into thin layers near walls — but never disappears.
  • Dissipation. Kinetic energy → heat, at rate \(\sim \mu (du/dy)^2\) per unit volume; the endpoint of the turbulent energy cascade.

Physical interpretation

Microscopically, molecules wander between layers, carrying their momentum with them (gases) or drag each other via intermolecular forces (liquids). That's why gas viscosity rises with temperature (faster wanderers) while liquid viscosity falls (looser grip).

Common mistakes

  • "Viscous fluid = dense fluid." Mercury is dense but runny; pancake syrup is light but viscous. \(\mu\) and \(\rho\) are independent — that's why \(\nu = \mu/\rho\) exists.
  • Dropping no-slip for "small viscosity". However small \(\mu\), the wall velocity matches exactly; the adjustment just happens in a thinner layer.
  • Applying inviscid results in the boundary layer — d'Alembert's paradox (zero drag) is the famous punishment.

Knowledge graph position

Prerequisites: What is a fluid?, Eulerian description. Leads to: Navier–Stokes, Reynolds number, Turbulence.

Quiz

Q1 (computational). Two plates 2 mm apart; the top one slides at 1 m/s. For water (\(\mu = 10^{-3}\) Pa·s), what shear stress acts on each plate?

Answer

Linear (Couette) profile: \(\tau = \mu U/h = 10^{-3}\times 1/0.002 = 0.5\) Pa.

Q2 (conceptual). Why does stirring cornstarch-in-water slowly work fine, but punching it feels like hitting a solid?

Answer

It is shear-thickening: viscosity increases dramatically with shear rate. Slow stirring = low shear rate = low \(\mu\); a punch = huge shear rate = enormous \(\mu\).

Q3 (multiple choice). The no-slip condition states that at a stationary wall:

  • (a) shear stress vanishes (b) normal velocity vanishes only
  • (c) the full fluid velocity vanishes (d) pressure vanishes
Answer

(c). Both normal and tangential components match the wall (zero). Inviscid theory can only enforce (b) — that's why it misses drag.