Thanks to CuriosityStream for sponsoring this video. Everyone has an intuitive understanding of what the **viscosity** of a fluid is. Honey is more viscous than oil, and oil is more viscous than water. Viscous fluids feel thicker, and don’t flow as easily. But since it’s such a fundamental parameter that defines how fluids behave, it’s worth developing a more in depth-understanding of **viscosity.** To do this, let’s start by looking at fluid flowing over a flat surface. It’s sometimes useful to think of fluid as flowing in layers, with each layer moving at a different velocity. When two layers are moving relative to one another because they’re flowing at different velocities, a shear stress develops between them. This is similar to how a frictional force develops between two solid objects sliding relative to one another. The effect of this shear stress is most obvious in flow close to a wall. The wall imparts a large shear stress onto the fluid particles that are in contact with it, causing them to have zero velocity. This is called the no-sIip condition.

The large shear stress is transmitted through to adjacent layers of flow, slowing down flow close to the wall and resulting in a velocity profile that looks like this. The magnitude of the shear stress acting between the layers of the fluid is closely linked to the slope of the velocity profile, du/dy, where u is the fluid velocity and y is the distance from the wall. The slope is small in the free stream. The fluid layers are moving at almost exactly the same velocity, and so the shear stress will also be very small. But close to the wall the velocity changes very suddenly. In this region the slope is large, and so the shear stress is also large. For most fluids, the relationship between the shear stress and the slope of the velocity profile is linear. And the constant of proportionality is what we call fluid **viscosity.** In engineering it’s usually denoted using the Greek letter Mu. We can think of viscosity as the internal friction of a fluid in motion.

It has the effect of smoothing out differences in velocity, by increasing the shear stresses in proportion with the velocity gradient. We can express the linear relationship using this equation. This is called Newton’s Law of **Viscosity,** and fluids that obey it are said to be Newtonian. It describes how easily a fluid will flow. The **viscosity** Mu is sometimes called the dynamic viscosity, as this differentiates it from the kinematic viscosity, which is the dynamic viscosity divided by the fluid density. Let’s look at how a small fluid element like this one will deform as it flows. If du/dy is positive, the upper surface will move faster than the lower surface, and so over a time period Delta-T the upper surface will travel further by a distance equal to Delta-U times Delta-T.

This generates a shear strain, that we can calculate using trigonometry and the small angle approximation. By re-arranging we can see that the slope of the velocity profile is equivalent to the rate at which the shear strain is applied. So now we can write Newton’s law of Viscosity in a different form, like this. This looks a lot like Hooke’s law for shear that we see in solids, except instead of being a function of the strain, the shear stress is a function of the rate at which the strain is applied. This makes sense if we consider how solids and fluids respond to applied shear forces. Solids respond to a constantly applied shear force with a finite amount of deformation, whereas fluids respond by deforming continuously – they flow for as long as the shear force is applied.

This is why for fluids the shear stress is a function of the rate at which strain changes over time, instead of being a function of the shear strain itself, like it is for solids. If we do some quick dimensional analysis on Newton’s law of viscosity, we can see that the dynamic viscosity has units of Pascal-seconds. And the kinematic viscosity has units of meters squared per second. The Poise and Stokes are other units for the dynamic and kinematic viscosities, although they are often expressed as centipoise and centistokes. The centipoise unit is particularly convenient because at room temperature the viscosity of water is equal to one centipoise. Typical engine oil at room temperature has a viscosity of around 500 centipoise, so is 500 times more viscous than water. The viscosity of honey is ten thousand centipoise.

And if we keep going we get to fluids that are so viscous they appear to be solid. Pitch is a thick tar-like fluid that has extremely high viscosity, as is famously demonstrated by the pitch drop experiment. In 1927 at the University of Queensland, heated pitch was added to a funnel and once it cooled it was allowed to drop into a beaker placed below it. Because of its extremely high viscosity, only nine drops have fallen into the beaker since then. It is the world’s longest running continuous lab experiment. On the other end of the spectrum we have gases, which generally have much lower viscosity than liquids. Air has a viscosity of 0.018 centipoise at room temperature, for example. One very interesting substance is Helium-4, the common isotope of helium.

A gas under normal conditions, it becomes a liquid when cooled to below 4 Kelvin, and at these low temperatures researchers have found that it can behave like it has zero viscosity. Substances that behave in this way are called superfluids. So we know how to define viscosity, but what actually causes it on the molecular level? In liquids, viscosity is caused by inter-molecular cohesive forces, which bond layers of the fluid together. And in gases, it’s caused by inter-molecular collisions, which create interaction between adjacent layers of the fluid. Viscosity is highly dependent on temperature. In liquids it decreases as temperature increases. This is because at the molecular level an increased temperature allows molecules to more easily escape the attractive forces of the adjacent molecules. The viscosity of water for example reduces from 1 centipoise at 20 degrees Celsius to 0.5 centipoise at 50 degrees Celsius.

But temperature has the opposite effect in gases. Viscosity increases with increasing temperature, because the higher temperature means that gas molecules have more random motion, which results in more inter-molecular collisions. The temperature dependence of viscosity can be modelled using simple empirical correlations. We can use Andrade’s equation for liquids and Sutherland’s equation for gases, where the various constants are determined for each fluid by experimentation.

Viscosity also varies with pressure, although to a much lesser extent than temperature, and so pressure dependence is often neglected. Because it fundamentally affects how fluids behave, viscosity is a very important parameter in Fluid Mechanics. It appears in the equation for Reynolds number, for example, and so affects whether flow is likely to be laminar or turbulent. Flow of high viscosity fluids is more likely to be laminar, because any small turbulent disturbances are more easily dampened out by the larger shear stresses. Viscosity also causes the pressure drop along a pipe. Without viscosity there would be no shear stresses imparted by the pipe wall and the velocity profile would look like this. The existence of viscosity can make modelling flow quite complicated, and so engineers often try to neglect viscous forces where possible. The assumption that viscous forces are negligible is typically only applied to certain regions within a larger flow system. It’s never valid for flow close to a boundary, for example, where viscous forces are always significant. But outside of the boundary layer, at high Reynolds numbers, viscous forces can often be neglected.

Flow where viscous forces can be neglected is said to be inviscid. Assuming that flow is inviscid doesn’t mean we’re assuming the fluid has no viscosity. It just means that we’re neglecting viscous forces because they’re small compared to other forces. Doing this makes analysis of the flow much easier. The Navier-Stokes equations define the behaviour of fluids, but are very difficult to solve. Assuming that flow is inviscid means that we can neglect the viscous terms, which contain the higher order derivatives.

The resulting equations for inviscid flow are called the Euler equations, and they’re much easier to solve. Inviscid flow is also a key assumption in the derivation of Bernoulli’s equation. We said earlier that fluids that obey Newton’s law of viscosity are said to be Newtonian. The shear stress increases linearly with the strain rate. But for some fluids the relationship between shear stress and strain rate is non-linear. These are called non-Newtonian fluids, and there are two main types – shear thickening fluids, and shear thinning fluids. For shear thickening fluids the apparent viscosity increases as the strain rate increases. And for shear thinning fluids the apparent viscosity reduces as the strain rate increases. Paint is an example of a shear thinning fluid. It’s easily applied with a brush but doesn’t drip from the wall once it has been applied. Viscosity really is a fundamental parameter in the study of fluids.

Engine oil is one example of an application where viscosity and the effect of temperature play crucial roles. I’ve included a bit more discussion about engine oil viscosity in the extended version of this video over on Nebula. Nebula is a streaming platform built by independent educational creators. It’s a place where you can find loads of amazing original content, like Mustard’s The Origins of Stealth, that explores the history of the F-117 Nighthawk. But it also has extended versions of videos from your favourite creators, with what we’re calling Nebula+. The Nebula+ version of this video includes a few extra minutes on engine oil viscosity, and that content replaces this sponsor segment, since Nebula is completely ad free. To make Nebula even better we’ve teamed up with CuriosityStream. CuriosityStream is the best place to go for high quality documentaries.

It has thousands of titles, like Autonomy, a film that explores the implications of us giving up control to self-driving cars, or the series A World Without NASA, that examines how technologies developed by the space agency affect out everyday lives. So to get access to all of the great documentaries on CuriosityStream and my bonus content on Nebula, head over to curiositystream.com/efficientengineer, or click on the link in the description. You’ll get a 26% discount on the CuriosityStream annual plan, AND you’ll get Nebula for free. All for less than 15 dollars a year. It’s an amazing deal, and signing up is a great way to support this channel! And that’s it for this look at fluid viscosity! Thanks for watching!