Newton's Law of Viscosity

Equation

\[\boxed{\,\tau = \mu\,\frac{du}{dy}\,}\]

Physical meaning

Shear stress between fluid layers is proportional to how fast the velocity changes across them. It is a constitutive law — a statement about a material, not a conservation law — defining what "Newtonian fluid" means and giving viscosity its operational definition: \(\mu\) is the slope of stress vs shear rate.

Variables

Symbol	Meaning	SI unit
\(\tau\)	shear stress	Pa
\(\mu\)	dynamic viscosity	Pa·s
\(du/dy\)	velocity gradient (shear rate)	s⁻¹

Typical \(\mu\) values: air \(1.8\times10^{-5}\), water \(1.0\times10^{-3}\), honey ~\(10\) Pa·s.

Assumptions

Newtonian fluid: \(\mu\) independent of shear rate (water, air, most oils — but not paint, blood, or cornstarch slurry)
Simple shear geometry (the tensor generalization \(T_{ij}^{visc} = \mu(\partial_i v_j + \partial_j v_i)\) feeds the Navier–Stokes equation)

Interpretation

Momentum diffuses down its gradient, exactly like heat down a temperature gradient (Fourier) or species down a concentration gradient (Fick). The diffusivity is \(\nu = \mu/\rho\): kinematic viscosity, units m²/s — compare it directly with thermal diffusivity to get the Prandtl number.

Applications

Measuring viscosity (rotational viscometers implement Couette flow)
Wall shear stress and skin-friction drag
Lubrication films, blood-vessel wall stress, syrup coating

Limitations

Non-Newtonian fluids need \(\mu(\dot\gamma)\) or full viscoelastic models; rarefied gases break the continuum premise.

Navier–Stokes equation — where it enters as the stress model
TKE transport equation — its dissipative fingerprint in turbulence

Quiz

Q1 (computational). A 0.1 mm oil film (\(\mu = 0.1\) Pa·s) separates a 0.1 m² sliding block from the floor. Force to slide at 0.5 m/s?

Answer

\(F = \tau A = \mu \frac{U}{h}A = 0.1\times\frac{0.5}{10^{-4}}\times0.1 = 50\) N.

Q2 (conceptual). Ketchup refuses to flow, then gushes. Which viscosity class, and what does the \(\tau\)–\(\dot\gamma\) curve look like?