Viscosity — dynamic μ, kinematic ν = μ - ρ; Newtonian vs non-Newtonian

WHY do we even need viscosity?

WHAT problem are we solving? In an ideal fluid, layers slip freely — push a paddle and it never slows. But real fluids drag. Stir tea and it eventually stops. Something is converting bulk motion into heat. That "something" is viscosity.

WHY does the drag exist? Molecules jump between adjacent layers carrying their momentum. A fast layer donates momentum to a slow layer (speeds it up); the slow layer steals momentum from the fast one (slows it down). Net effect: a shear stress that opposes the velocity difference between layers.

So viscosity is fundamentally about a velocity gradient, not velocity itself. Uniform flow (all layers same speed) has zero viscous stress, even at high speed.

HOW to build the formula from scratch

Step 1 — What could the stress depend on? Experiment (Newton's guess): the force per area $\tau$ needed to keep the top plate moving is

• proportional to plate speed $U$ (faster → more drag),
• inversely proportional to gap $h$ (thicker cushion → less drag).

So $\tau \propto U/h$. Why? Because $U/h$ is exactly the velocity gradient $du/dy$ for the straight-line profile. The proportionality is local: stress depends on the local gradient.

Step 2 — Name the constant. Define the proportionality constant to be the dynamic (absolute) viscosity $\mu$:

Units, derived not memorized: $[\mu] = \frac{[\tau]}{[du/dy]} = \frac{\text{Pa}}{\text{s}^{-1}} = \text{Pa·s} = \frac{\text{kg}}{\text{m·s}}$ (Old CGS unit: 1 poise $= 0.1$ Pa·s; water ≈ 1 centipoise $= 10^{-3}$ Pa·s.)

Kinematic viscosity ν — WHY divide by ρ?

Why m²/s is beautiful: those are the units of a diffusion coefficient. $\nu$ literally tells you how fast momentum diffuses through the fluid. In time $t$, momentum spreads a distance $\sim\sqrt{\nu t}$.

Newtonian vs Non-Newtonian

A handy power-law model: $\tau = K\dot\gamma^{\,n}$. $n=1$ Newtonian; $n<1$ shear-thinning; $n>1$ shear-thickening.

Worked Examples

Flashcards

Connections

• Reynolds Number — uses $\nu$: $Re=UL/\nu$, ratio of inertia to viscous forces.
• Boundary Layer — where velocity gradients (and viscous stress) concentrate near walls.
• Poiseuille Flow — μ sets the flow rate in a pipe under pressure.
• Stokes Drag — drag on a sphere $F=6\pi\mu r v$, direct use of μ.
• Momentum Diffusion — $\nu$ as a diffusion coefficient; analogy to heat/mass diffusion.
• Terminal Velocity — balance of viscous drag and gravity.

Concept Map

Hinglish (regional understanding)