Viscosity and Shear Stress

Viscosity is a fluid's resistance to shear deformation — its internal "stickiness". Honey is highly viscous; water is only mildly so; air has very low viscosity.

Newton's Law of Viscosity

For a Newtonian fluid, shear stress $\tau$ is proportional to the velocity gradient perpendicular to the flow:

$\tau = \mu \frac{du}{dy}$

where:

• $\tau$ — shear stress (Pa)
• $\mu$ — dynamic viscosity (Pa·s), also written mPa·s or cP
• $du/dy$ — velocity gradient (s⁻¹); how fast velocity changes across the fluid layer

The shear force on an area $A$ is simply:

$F = \tau A = \mu \frac{du}{dy} A$

Kinematic Viscosity

Kinematic viscosity $\nu$ normalises dynamic viscosity by density:

$\nu = \frac{\mu}{\rho}$

Units: m²/s (also written as cSt, where 1 cSt = 10⁻⁶ m²/s). It appears naturally in the Reynolds number $Re = vL/\nu$.

Temperature Dependence

Liquids become less viscous as temperature rises (thermal agitation breaks intermolecular bonds). Gases become more viscous as temperature rises (more molecular collisions transfer momentum).

Your Task

Implement:

• shear_stress(mu, du_dy) — returns shear stress $\tau = \mu \cdot (du/dy)$ in Pa
• kinematic_viscosity(mu, rho) — returns $\nu = \mu / \rho$ in m²/s
• shear_force(mu, du_dy, A) — returns shear force $F = \mu \cdot (du/dy) \cdot A$ in N