Viscosity is something that many of us possibly take for granted and, yet, it is implicated across a remarkably diverse set of physical situations: from trying to pour honey without making a mess to the distance of the Moon from the Earth!

So what is viscosity? 

In essence, it's a measure of the internal friction of a fluid, which can be anything from the air we breathe to the aforementioned sticky, sweet substance bees make and is so beloved of bears. It's why we can stir coffee in a cup, or row a boat over a lake. Ironically, it's also what makes rowing so exhausting.

The easiest way to understand viscosity is to picture a fluid as formed of many layers sliding over one another. If we imagine layer X slipping over layer Y (resulting from the shear stress, due to a force along the boundary layer), as the molecules in layer X approach those in layer Y, the molecules of X are increasingly attracted to the molecules of Y, momentarily bonding as they pass by, only to feel an opposing 'tug' as they break free. This dynamic creates what we call viscous drag, and is responsible for kinetic energy being lost as heat. The viscosity of a fluid is temperature dependent, decreasing for liquids as their temperature rises. However, perhaps counterintuitively, viscosity in a gas increases with rising temperature, as the number of collisions increases along with the kinetic energy of the molecules.

Let's now turn to a viscous fluid filling a gap of cross-sectional area \(A\) and depth \(z\)between two plates: one fixed and the other moving with velocity \(v\). At the boundary layer between the fluid and the surface of each plate, the fluid can be considered to be at rest with respect to the plates. However, as we move away from the fixed plate, there is an increase in the fluid’s velocity as a function of depth consistent with the driving force \(F\) on the moving plate opposing the viscous drag. Where there is a linear relationship between the shear stress, \(τ = {F\over A} \), and the velocity gradient \(Δv\over Δz\), as in the case of a so-called Newtonian fluid, the constant of proportionality is known as the coefficient of viscosity \(η\), where:

\(η = {τ\over {Δv\over Δz}}\)

All of the above also applies to viscous flow in a tube or pipe. Here the boundary layer will be where the stream of fluid is in contact with the inner surface, the velocity increasing radially from zero at the wall of the pipe, reaching its maximum at the centre. In this case, the driving force opposing the viscous drag comes from the pressure difference \(Δp\) over a particular length. For a length of pipe \(L\), the flow rate \(Q\) is proportional to the fourth power of the pipe’s radius \(R\). According to Poiseuille's Law:

\(Q = {πΔpR^4\over 8ηL} \)

Rearranging this equation to make \(Δp\) the subject demonstrates that even a small reduction in the radius of a pipe carrying a viscous fluid will result in a large increase in the pressure difference required to maintain its flow rate. This has important ramifications for the amount of work required by the heart in a patient who is suffering from a narrowing of the arteries.

Fluid flow can broadly be described as being turbulent, like a stream in a rocky gorge, or laminar like golden syrup being poured from a bottle, and a useful measure of this characteristic, in the study of fluid dynamics, is the dimensionless Reynolds number, \(Re\), which describes the ratio of inertial to viscous influences on the behaviour of a fluid:

\(Re = {ρvL\over μ}\)

Here, \(μ\) is the dynamic viscosity of a fluid of characteristic length \(L\) and density \(ρ\). We can see that high viscosity, in conjunction with a low flow speed, gives a low Reynolds number, which is consistent with laminar flow; whereas a fast-flowing, low-viscosity fluid will be characteristic of a high Reynolds number and hence turbulent flow. In slow viscous fluids, the effects of irregularities in the boundary layer tend to be damped out whereas, in the example of a lively stream, the low viscosity and high velocity tend to exaggerate the effects of irregularities, causing chaotic flow lines for the constituent molecules.

Reynolds numbers are of particular use in aerodynamics and wind tunnels. An interesting experiment is to inject a blob of dye into a layer of highly viscous fluid between two glass cylinders. Rotating the inner cylinder a few times will cause the blob to smear out; however, twisting the inner cylinder back by the same number of rotations will (if the fluid exhibits a suitably low Reynolds number), cause the blob to practically regain its original form!

Let's finish with a look at the diversity of situations in which viscosity plays an important role. It's the viscosity of air that allows footballers, tennis players and golfers to do seemingly miraculous things with spin. Accretion discs surrounding black holes rely on the viscosity of hot plasma to redistribute angular momentum radially outward, thus allowing material to “fall in”! So-called bottom friction, hopefully not as painful as it sounds, between the Earth and its oceans, drags the tidal bulge caused by the Moon ahead of the Moon in its orbit, speeding it up and consequently slowly, over time, pushing it into an ever higher orbit. And the nature of volcanic eruptions is highly dependent on the viscosity of magma and the behaviour of subsequent lava flows. 

Even we humans, when in suitably large crowds, behave like fluids to such a degree that we can be forced to collectively move like a laminar fluid, simply by being encouraged to walk in files past obstacles, rather than pushing and shoving in a 'turbulent' fashion, whilst entering and leaving football stadiums and rock concerts!

Images

• Working with water photo by Buddha Elemental 3D on Unsplash
• Planar Couette flow by Duk at the English-language Wikipedia, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4168566
• Transition from laminar to turbulent by Joseasorrentino, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons