How liquids flow

Look at picture 6-1. It was taken in the River Medway in Kent in the UK. The flow in the river was greater than normal and the water flowed past bridge piers that have these pointed ends[1]. The water breaks away from the sharply angled corner and a swirling wake forms between the main flow and the side of the pier. Those who seek to make diagrams of this flow will usually just draw a lot of squiggly lines[2] to indicate no orderly flow. But, if you have the opportunity to look long enough, you can find some order and, in my view it is important to look for that order. In this flow there is a cyclic change in the flow pattern and what you need to know is that the foot of the pier has a large apron stretching over, and standing proud of, the natural bed of the river to protect it from scour. This affects the flow at the foot and these two flow patterns get out of phase. If you look at the wake you cannot avoid seeing eddies that form in the wake and get swept downstream. There is one in figure 6-2. These eddies are everywhere in the natural flow of water and air.

 

Picture 6-3 is from Prandtl’s book where there are several pictures of eddies made using paint on glass and aluminium powder sprinkled on the surface of eddying water. Here he is showing us how a two-dimensional flow that has become detached can become re-attached as the speed of flow increases. I am interested in these eddies. Look at them, they are not accidental or random, they are an important part of the overall flow pattern. Note that the eddies all have the same direction of rotation and having three or more eddies in a line like these is quite common. Inevitably where two eddies exist side-by-side there is contra flow between them. Eddies are not necessarily circular nor are they the same shape instant-by-instant but over time they have a well-defined shape and appear to be an essential and predictable part of the flow pattern. Eddies do not rotate like wheels, instead the angular velocity increases with decreasing radius and, throughout the eddies, there is shearing and this shearing takes place continuously both within the eddies and between them. It seems to me that Nature has designed a flow pattern in which to lose mechanical energy to internal energy very effectively.

 

You can see from the horizontal stone course in picture 6-1 that the surface of the river Medway suffered a drop in level as it flowed past this bridge. This means that potential energy has been given up and it is now contained partially in an increased velocity and partly in the wakes from the several piers. Beyond the bridge the velocity will return to normal and the energy given up in the drop in level will be contained in the flow. If you try to think of ways to contain energy without an increase in level or average velocity you will find that the options are very limited and there is only one way for this energy to be contained and that is in eddies rotating with all sorts of different axes.[3] Go and look at your local river and see the way that it eddies after it has passed an obstruction. The well-defined eddies will have disappeared from the surface leaving a flow with much finer grained disturbance in it  and the kinetic energy will be dispersing into the water. The mechanism doing the dispersion is the shearing and it is reasonable to suppose that some liquids resist this shearing more than others. This means that, say, crude oil flowing in the natural bed of a river would present us with a new and exciting flow pattern to study. We, in ordinary parlance, talk of thin and thick liquids and we think of them as being runny or slow to pour. It is all about shearing. But ultimately it is not the shearing that we can see that matters, it is the shearing that goes on at molecular level that is the basic mechanism by which mechanical energy is dispersed into the fluid as random internal energy. That motion is going on all the time. It is the character of this motion that we quantify when we measure viscosity.

 

I have been looking at flow with a random element like this water in a river and perhaps in the wind but liquids can also flow in a totally orderly way. Look at picture 6-3. It is from chapter 9 of my book on model yachts that is on this website. It was produced on a home-built Hele-Shaw apparatus to try to visualise the flow round sails. (See details in book.)[4]  Here I am interested in the blow-up of part of the flow pattern. It is the total orderliness that I find so astonishing. The water was flowing between glass plates spaced about 1 millimetre apart and the lines are of ordinary fountain pen ink. The water was flowing round a brass shape representing a sail attached to a mast. Water in contact with the plates is stationary and the velocity distribution between the plates is parabolic. When there is a change in direction of the lines this parabolic velocity distribution takes on a transverse component as well and the line widens because there is the same pressure difference to produce the essential centripetal acceleration and because the water moves at different tangential speeds the centripetal acceleration to make them stay together is not the same for all the flow. When you actually see the apparatus running you can see the detail of the lines widening and narrowing and those “shaded” areas are actually sheets of ink having this parabolic shape across the flow. It is all too evident that water can flow without any discernable mixing even though it must be subjected to continual shearing at molecular level. I like these images that come from Hele-Shaw, I suspect that the apparatus is more versatile and instructive than is generally supposed but it needs experimental skill to get results from it.

 

My examples above show two very different aspects of this shearing and I need to attempt to find a mechanism that accounts for all the effects that are attributed to viscosity. I think that I must start with Newton, who, at a time (1687) when there was no concept of a molecular structure for solids, liquids and gases, could offer no insight to the mechanism that could create this internal friction that he called viscosity but could cut through the maze of different effects of viscosity and suggest a way of quantifying it.