17.17 Drag and viscosity

Domestic pig lounging in the mud.

Try pushing your finger slowly through a glass of water: now try to do the same in a glass of treacle or honey. You will find that the treacle or honey exerts a force (post 16.13), resisting the motion of your finger (rather like friction in post 16.19), that is much greater than the force exerted by water. The effect is like wading through mud.

The force that a fluid exerts to resist motion of an object in a fluid is called drag or the drag force.

The simple experiment, described in the first paragraph, shows that the drag force depends on some property of the fluid: this property is called viscosity. Our experiment shows us that treacle and honey are more viscous than water.

When we increase the speed, v, of an object moving through a fluid, the drag force increases: doubling the speed increases the magnitude, F, of the drag force. We can express this result as the equation

F = Mv

where M depends on the viscosity of the liquid and the shape and dimensions of the object. Since force is a vector (post 16.50), it would be better to define M using the vector equation F = – Mv, where v is the velocity of the object and the minus sign shows that F acts in the opposite direction to v.

Now let’s find out more about M. Experiments on spheres falling through liquids show that, for a sphere,

M = 6πηr

where r is the radius of the sphere and η is a constant for a given fluid. We can use this result to define the viscosity of a fluid as

η = M/(6πr).

According to our first equation, M, is a force divided by a speed. So it is measured in the units of force divided by the units of speed which are N/(m.s-1). Since 6π is simply a number (see post 17.11 for a definition of π), and so is not measured in any units, the units of η are the units used to measure M divided by the units used to measure r (m). The result is N/(m2.s-1) or N.s/m2. But N/m2 or N.m-2 is the unit used to measure pressure (Pa) (post 17.5). So, the SI unit (posts 16.13 and 17.14) of viscosity is Pa.s.

When an object falls through a viscous fluid, it has less energy than an object that falls through a fluid of negligible viscosity. In a fluid of negligible viscosity, the resultant downward force on the object is simply the gravitational force (post 16.16). But in a viscous fluid, the resultant downward force is reduced by the drag that opposes motion. As a result, the viscosity reduces the acceleration of the object (posts 16.13 and 17.15); so it has less speed (post 16.13) and, therefore, less kinetic energy (post 16.21) than an object that falls for the same time, with the same initial potential energy, in a fluid of negligible viscosity.

So the object falling in a viscous fluid loses mechanical energy (post 16.21). According to the first law of thermodynamics (post 16.21), this energy doesn’t simply disappear but is converted into different form, usually heat (posts 16.21 and 16.35). We say that the object falling in the viscous fluid dissipates mechanical energy. Often this idea is simply expressed as “dissipating energy”; it is understood that the lost energy doesn’t vanish but is converted into another form which cannot directly do useful work. The drag force generated when an object moves through a fluid is, therefore, similar to the frictional force generated when an object moves over a solid surface (post 16.19).


We mustn’t forget that shape affects the value of M. Designers of cars try to find a shape that decreases the drag when a car moves through air and so reduce the energy dissipated. Reducing energy dissipation means that a higher proportion of the energy generated by burning fuel is made to do useful work – to propel the car.

My definition of viscosity is not the official definition. However, my definition and the official one are equivalent. The advantage of my definition is that it is based on everyday experience and is easy to understand. The official definition is based on velocity gradients in fluids; we don’t have an intuitive understanding of these velocity gradients and so the official definition is difficult to relate to everyday experience. The advantage of the official definition is that we can then use Newton’s laws (post 16.2) or the Navier-Stokes equation (post 17.15) to derive M for a sphere, or any other simple shape. But neither derivation is easy! For more complicated shapes, we first divide them into small parts, in much the same way as we divided the hull of a ship into parts to calculate its volume (post 17.8). We then calculate the drag on each part in a computer and add the results to obtain the total drag force – this method is called computational fluid dynamics (CFD) and is used to solve most real problem involving drag.


Related posts

16.19 Friction


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