The term viscosity describes the internal friction within a liquid or gas caused by the interaction between its particles.
Layer Model
To analyze this quantitatively, we use a model where the liquid is represented by a cuboid consisting of horizontal layers in the y-direction. This is known as the layer model of a liquid.
- The bottom layer is fixed (velocity $v = 0$).
- A force is applied to the top layer in the x-direction, causing it to move and exert shear forces on the layers below.
- Due to internal friction, the speed decreases from top to bottom.
To increase the layer velocity, the applied force must be increased. Experimentally, the following relationships are observed:
- $F \propto v$ (force proportional to velocity),
- $F \propto A$ (force proportional to area),
- $F \propto \frac{1}{d}$ (force increases when layer spacing $d$ decreases).
Combining these relationships gives:
$$
F \propto A \frac{v}{d}
$$
In the differential limit (infinitely small distances), this becomes:
$$
\boxed{F = \eta A \frac{\mathrm{d}v}{\mathrm{d}y}}
$$
Here, $\eta$ is the viscosity or dynamic viscosity of the fluid, with units of Pascal-seconds (Pa·s).
Measuring Viscosity
The viscosity of a fluid can be determined using a falling ball viscometer.
- A sphere falling through the fluid experiences a drag force $F_\mathrm{R}$ due to the viscosity.
- Assuming laminar flow, we apply the earlier shear force relation.
- Replacing the area with the sphere’s cross-sectional area $\pi r^2$, and layer distance $d$ with sphere radius $r$, gives the Stokes drag:
$$
\boxed{F_\mathrm{R} = 6\pi\eta r v}
$$
- $v$: fall velocity,
- $\eta$: viscosity,
- The factor 6 is derived from a detailed hydrodynamic analysis and is experimentally confirmed.
The sphere accelerates due to the difference between its gravitational force $F_\mathrm{G}$ and the buoyant force $F_\mathrm{A}$, until the drag force balances the sum of the other forces.
Using:
$$
F_\mathrm{G} = mg = \frac{4}{3} \pi r^3 \varrho_\mathrm{K} g
$$
Summing all forces at terminal velocity:
$$
F_\mathrm{R} + F_\mathrm{G} + F_\mathrm{A} = 6\pi\eta r v + \frac{4}{3} \pi r^3 \varrho_\mathrm{K} g + \frac{4}{3} \pi r^3 \varrho_\mathrm{Fl} g = 0
$$
Solving for viscosity $\eta$:
$$
\boxed{\eta = \frac{2(\varrho_\mathrm{K} - \varrho_\mathrm{Fl})}{9v}r^2 g}
$$
Note: The viscometer can also be tilted, causing the ball to roll instead of fall, reducing the velocity. In this case, the above formula no longer applies.