Viscosity

A viscous material is gooey.  Runny materials have low viscosity.  There are various ways of measuring viscosity, the most common of which is to drop a ball-bearing into the liquid, and measuring its terminal speed.  Remember that at terminal speed, the upwards forces of upthrust and drag and the downwards force of the weight are balanced. 

The upthrust is the same as the weight of fluid displaced by Archimedes' principle.  Therefore, if the weight is greater than the upthrust, the object will accelerate downwards until the drag balances the difference between the weight and the upthrust.  This is true of all fluids, for example air, or water, or chocolate.

We can calculate the drag force by Stokes' Law: 

The terms of the equation are:

• F - the viscous drag (N);

• r - the radius (m);
• h - the coefficient of viscosity  (N s m^-2) [The strange looking symbol h is 'eta' a Greek letter long 'ē'];
• v - the terminal velocity (m/s);

We can rearrange the equation to give:

Note that temperature of the fluid is important; at higher temperatures fluids are less viscous.  If the fluid is a gas, like air, the pressure is important.

• Water - h = 1.000 × 10^-3 N s m^-2 at 27 ^oC;
• Air - h = 18.325 × 10^-6 N s m^-2 at a pressure of 1 atmosphere.

To determine the viscosity, we need to know:

• Density of the fluid;
• Density of the ball-bearing;

The mass of fluid displaced = volume of the ball bearing × density of the fluid.

The mass of the ball bearing = volume of the ball bearing × density of the steel.

To get the weight, we need to multiply these by g (= 9.81 m/s²).

Since:

Weight = upthrust + viscous drag

we can write the expression in Physics code:

Another useful expression for the pressure at a certain depth is:

where:

• P - pressure (N/m²);

• h - depth (m);

• r - density of the fluid (kg/m³);

• g - the acceleration due to gravity (9.81 m/s²).

Stokes' law applies for either an object falling through a fluid, or a fluid passing around an object.

Laminar and Turbulent Flow

Stokes' Law depends on the fluid not being turbulent.

• Laminar flow means that the flow of fluid does not make an abrupt change in direction or speed.  We can imagine the fluid as layers, with very little mixing, except on a molecular level.

• Turbulent flow occurs when there is large scale mixing and eddies.

The first picture shows laminar flow:

While this picture shows turbulent flow behind an aeroplane:

Thixotropy

Some fluids a thixotropic, which means their viscosity changes.  When a shear force is applied, they become much less viscous.  Such fluids are positively thixotropic.  Margarine is an example.

On the other hand, it is easy to move a spoon slowly through custard.  When you increase the speed of the spoon, the viscosity increases rapidly.  Custard shows negative thixotropy.

The change is temporary.

In general, thixotropy is a property of non-Newtonian pseudoplastic fluids.  You didn't know that?  You learn something new every day.

Measuring Flow Rates

There are all sorts of different flow meters.  One of the simplest is a piece of card attached to a spring.  The card would be deflected by a flow of air, and the speed of the air would be calibrated on a scale.  Early aeroplanes had such an arrangement as a speed indicator.

Other types of flow meter are:

• Blades of a small turbine turning a small generator.  The output voltage represents the flow.

• Blades cut the beam from a photodiode.  The pulses are processed electronically.

• Diaphragm flow meter, for example a gas meter.

• Thermal flow meter in which a heater warms up the fluid between two temperature sensors.  The temperature difference represents the mass per unit time.

Let's look at this closely:

Suppose the fluid had a specific heat capacity c (J kg^-1 K^-1).  We know that:

DQ = mcDq

Let the temperature at Probe 1 be q1 and the temperature at Probe 2 be q2.  So we can write:

DQ = mc(q2 - q1)

We also that the heater gives out a power P and:

P = DQ

     Dt

So we can combine these two relationships to give:

P = mc(q2 - q1)

Dt

And rearranging:

                                                                                           m =            P       

                                                                                                                Dt         c(q2 - q1)

The m/Dt term is the mass per unit time (i.e. how much material is flowing every second).

Viscous properties can also be affected by electric fields.  Chocolate can even go solid in an electric field.  The study of this is called electro-rheology.