Viscosity

Viscosity can be interpreted as the inner friction of liquids. In order to make quantitative statements possible, we can imagine the liquid to be subdivided into different layers in the $y$-direction. If a force $F$ acts on the liquid along the $x$-axis, the liquid flows in the same direction, and as a result shear forces act on the different layers. In order to increase the speed of the liquid, $F$ has to be increased. Experimentally it can be found that the speed is proportional to $F$: $$F\propto v$$ In addition, the required force increases proportionally to the area $A$ of each layer: $$F\propto A$$ Finally, the force is also proportional to the inverse thickness of the distance between the layers. In total, this can be written as $$F\propto A\frac{v}{d}$$ Making the divisions infinitesimally small ($d\rightarrow \mathrm{d}y$) and introducing viscosity $\eta$ as a proportional constant, we obtain the following formula for the force. From many observations in our daily life, we know that water has a quite small viscosity, whereas the one of honey for instance is very huge. The following table shows some important examples. The viscosity of a liquid can be measured for instance in a viscometer which consists of a small sphere falling into a tube that is filled with the liquid. Inserting the area of a circle $$A = \pi r^2$$ and replacing the layer thickness $d$ with the radius of the sphere, the resulting force can be written in the following form which is called Stoke's Law. The additional factor 6 follows from a very detailed and complex derivation. In the viscosimeter, the sphere is accelerated until the sum of three forces friction, gravitation, and buoyancy gets $0$: $$F_\mathrm{S} + F_\mathrm{G} + F_\mathrm{B} = 0$$ Inserting all formulas leads to $$6\pi\eta r v + \frac{4}{3} \pi r^3 \varrho_\mathrm{Sp} g + \frac{4}{3} \pi r^3 \varrho_\mathrm{M} g = 0$$ This equation can then be solved for $\eta$.