Viscous fluids

All ordinary fluids resist motion because of their viscosity.Our common experience tells us that fluids like water and air are less viscous than oil and syrup, but we need a way to quantify that difference. Students at AIMS began by dropping steel ball bearings of different diameters into golden syrup contained in a measuring cylinder (see Figure 1), and measuring the distance travelled by each ball as a function of time. Their visual experience, confirmed by their data, was that each ball fell at a constant speed. Therefore the forces on the sphere must have been in equilibrium. What were those forces?

The ball falls because gravity acts on it. The gravitational force on the ball is its

where ρs is the density of the (steel) ball, V is its volume and g is the acceleration due to gravity.

However, we also know that bodies submerged in fluids experience a buoyancy force: for example, corks rise upwards in water and heliumfilled balloons rise upwards in air. That buoyancy force is also called the

where ρf is the density of the fluid. This relationship, known as Archimedes principle, says that the upthrust on a body submerged in a fluid is equal to the weight of the fluid displaced by the body.

We can derive this result as follows. If the body moves downwards a distance z, as shown in Figure 2, then the change in potential energy of the system is

The body loses potential energy, but the fluid that the body displaces gains potential energy. Recall that potential energy is the work done against the force of gravity, and is equal to the force F times the vertical distance moved upwards, in this case −z. The net downwards force on the body is therefore

If the densities of the body and the fluid are different then there is a net gravitational force (weight−upthrust) on the body, which would cause it to accelerate if there were no other forces acting. The additional force that allowed the ball bearings to fall at constant speed, rather than accelerating, is the viscous shear stress.

Normal stress

Surface stress τ, which I shall abbreviate to stress, is force per unit area. It is a vector quantity because it has direction as well as magnitude.