The concept of viscosity

Newton saw that viscosity produced an effect just like solid friction in that mechanical energy was just lost into the liquid as it might be into a solid. His concept has obvious similarities with the way we think about solid friction as figures 6-3 and 6-4 show. In solid friction we put  and call m the coefficient of friction.  In the system shown in figure 6-4 a liquid is imagined to fill the space between the flat and level plane and the flat plate, of surface area , which is distance  above it. A horizontal force  is imagined to act on the block and to cause it to move at a steady speed . Newton then used this system to define a coefficient of viscosity for the liquid. He put:-

                                       where m is the coefficient of viscosity.

Clearly Newton has proposed a system, under which the fluid is made to move with continual shearing, that is both well defined and easily visualised. I interpret Newton’s system as behaving as I have shown in figures 6-5a, b and c where a straight line through the liquid layer remains straight as the layer is continually distorted and there is a uniform rate of .

 

We need to have some mental model of what goes on in that layer because this is where mechanical energy is ultimately converted to the random motion of internal energy and I want to look towards the molecular motion of water as an example that might also apply to other liquids.

 

However it is easiest for me to start with gases. The kinetic theory of gases gives a very good picture of the structure of a gas. It postulates that a gas is composed of separate molecules, (which may be single atoms). The molecules “fly” freely at high speed, colliding frequently with other molecules and with the walls of the container in which they are enclosed. The scale of the structure of a gas is indicated by the following figures. The common gases at room temperature and at a pressure of one atmosphere have about  molecules per cubic centimetre. Even with this concentration there is still space between the molecules for them to move freely at high speed (about 350 m/s) through a distance of about 7 molecular "diameters" between collisions and the number of collisions made by each molecule each second is about . These are large numbers that I can accept but cannot imagine.

 

The molecules of the gas have mass. They move with high linear speed, rotate, and where the molecules have two or more atoms they can vibrate in the inter-atomic bonding. Kinetic energy can be stored in the gas in these motions. (Measurements seem to suggest that little energy is stored in vibration in a gas, nevertheless it does have this degree of freedom.) Consequently the gas may be regarded as having a stock of kinetic energy that is stored in a random manner in its rectilinear motion and, in thermodynamics, this is called internal energy.

 

The temperature and the pressure of a gas are measurements of two different aspects of the concentration of kinetic energy in the structure of the gas. The pressure is the result of the very large number of collisions that occur between the molecules of the gas and the solid surfaces containing it[1]. Each collision involves a very small force acting for a very short time at some angle to the surface and the continuous uniform pressure is the aggregate of all these short-lived forces. Of course the pressure acts normally to the surface because all the tangential components of the impact forces cancel out. So, in some way, pressure depends on the mass of the molecule, the mean kinetic energy of the molecules, and the concentration of molecules in the space (which, of course, determines the frequency of collisions with the walls). If, for a given gas, the mean kinetic energy were to be kept constant, the pressure would simply depend on the concentration of molecules, that is, on the density.

 

Temperature is a concept that springs from our natural concept of hot and cold and this appears to be essential to human survival because we are more vulnerable to temperature changes than animals, birds and reptiles. In order to measure the temperature of a gas we bring some thermometric device into contact with it. The atoms of the material of the device are held together by atomic bonds, that appear to be perfectly elastic, and they vibrate in a random manner within the limits imposed by the bonding. The molecules of the gas, when they collide with the surface of the device, do not meet a rigid, flat, stationary surface but one that is irregular in shape, and at the molecular scale, is in violent motion. The molecules of the gas and the atoms of the surface continually exchange energy. We wait until equilibrium is established between the average rate at which energy is given to the surface of the device by the molecules of the gas and the average rate at which energy is given to the gas by the atoms in the surface of the device. Thermodynamicists call this thermal equilibrium and say that the gas and the solid are at the same temperature. Thus temperature is an independent variable, to which we can relate other properties, and which is a measure, in the case of a gas, of the mean total kinetic energy of each molecule, which physicists observe to be the same for most gases at the same temperature.

 

The thermometric device will have been chosen so that some easily observed feature of it changes during the process of reaching thermal equilibrium and reaches a steady value. The device is calibrated to read temperature in an arbitrary system of units. The important thing to note, is that the reading of the device, that is the temperature, is dependent on the mean kinetic energy of any single molecule, whereas pressure is a measure of the concentration in many molecules of random kinetic energy in the space occupied by the gas.

 

We now have a picture of the gas in which each molecule is involved in many millions of collisions every second. The image brought to mind is of a continuous exchange of kinetic energy, both between the molecules of the gas, and between the molecules of the gas and the atoms of the material of the container. Such a process tends to produce uniform values for the mean velocity etc of the molecules. If there were to be some way in which the kinetic energy of the molecules of the gas and the atoms of the isolated container could dissipate their kinetic energy there would be a steady fall in temperature and pressure. Neither of these things happens. We can only conclude that the energy of the molecules and atoms is not dissipated and therefore the molecular interactions are perfectly elastic.

 

 Now I can look at water as a liquid that behaves in a way that is typical of many other liquids. In chapter 2, I mentioned that the molecule of water has a T shape but one might prefer to see it as three atoms joined in a triangle. I have no mental image of an atom, except as a nucleus with electrons whizzing round it, but I know that atoms vary in physical size so that the atom of oxygen is larger than the atom of hydrogen and that the molecule of water will have an odd shape because it has three atoms. Water is an agglomeration of densely packed molecules of . I have said that, in the commonly used, simple image of a molecular “structure”, molecules are regarded as being perfectly elastic. This means that, for millions of millions of molecules all moving in a seething mass, collisions and other exchanges of kinetic energy do not lead to a change in the sum total of kinetic energy. Only an external effect can produce a change. I need some model of the system in which this kinetic energy is stored by the molecules. As I understand it (I may be totally wrong.) the molecules seem to experience an intermolecular attraction when they are close together and a repulsion when they are too close together. As both the attraction and the repulsion increase as the distance between molecules decreases I have supposed, in order to make progress, that it might be possible to represent these two forces by the graph 6-1. For the system of two molecules to have my claimed character these two plots must cross as I have shown them. At some point P the attraction will equal the repulsion and when the two molecules get closer the net force will act to push them apart and when they are further apart than P they will be attracted.

 

The repulsion will obviously go on rising rapidly as the separation decreases and this must mean that a large number of molecules forming a continuum will be effectively incompressible. The implication of having an attraction is that the same continuum will be capable of withstanding tension, which is, of course, the case. This tension has a significant value compared with the forces ordinarily impressed on water in engineering applications and the tensile forces created during shearing are easily resisted by water without rupture at the molecular level.

 

The molecules have mass and if they also have motion the molecules could oscillate between the two positions that I have shown as  and  and the combination of forces of attraction and repulsion produce a motion like that of a weight suspended from a non-linear spring. The amplitude of the oscillation will vary with the total kinetic energy of the two molecules. It looks to me as though the kinetic energy can have any value between zero and some value when the attraction is too weak to prevent the molecules flying apart if that is, in fact, physically possible.[2]

 

Molecules of water that are part of a continuum of molecules do not join to form pairs. They are very closely packed so it follows that any molecule will also be in engaging with several other molecules and have some complex system of oscillation with all of them simultaneously. Given that the molecules are not symmetrical this makes one wonder how many molecules can be in contact simultaneously. I looked at tennis balls and they do not fit snugly round a central ball. I suppose that 8 or 9 can be in loose contact at once. New Scientist says that water molecules form clusters of ever-changing numbers with a probable size of 5 molecules and that fits with my figure.

 

Somehow we have to form a mental picture of innumerable closely-packed molecules oscillating in complex ways in ever-changing groups to form an extremely active continuum of molecules. Clearly kinetic energy will be contained at some mean level in every molecule. The kinetic energy has no mean direction so it cannot give the mass centre of the water a direction and so there is no way that the sum of all this kinetic energy can be extracted and stored in the gravitational field. It is totally random energy.

 

There is a further piece of information that is common experience is that water is a poor conductor of heat. The sum total of the random kinetic energy can be increased by heating the water and this proves to be a slow process if it takes place by dispersion through the molecular structure. This is consistent with this model.

 

Now I need to look at this model of molecular behaviour in the layer of water envisaged by Newton. He applies a force to the layer of liquid. Presumably, if an external force tending to compress the system of molecules were to be applied, it would occur through the whole system instantly. A depression would also affect the system instantly. So it would be reasonable to expect a shearing force to be exerted instantly throughout the molecular structure and not depend on a slow dispersion process.

 

This seems to me to be a plausible model for the molecular structure and its behaviour and that it is adequate to continue this study of viscosity.