Quantifying viscosity

I think that it is worth examining Newton’s expression. Newton, like everyone else, must have been aware that the there are plenty of liquids that flow more easily when they are heated and have expected that his coefficient of viscosity would decrease as the temperature rises. However I do not remember whether he said anything about this or about the fact that the coefficient of viscosity might be dependent of the velocity gradient  as well.

Anyone thinking of using this system to actually quantify the coefficient of friction for the liquid in the space finds out very quickly that it is not easy because any mechanical system that is devised causes effects other than those due to viscosity. No doubt many people tried to devise ways of realising Newton’s idea in practical hardware but the hardware that was needed did not appear until Hagen, who was an engineer and ignored by physicists, gave a basis for it in 1839 only for it to be rediscovered by Poiseuille in 1840. It is interesting to contemplate what went on in the years between 1687 and 1840. We know that Robert Hooke, who was contemporary with Newton and often entertained the fellows of the new Royal Society with demonstrations much as the Royal Society entertains the public now, appears to have had reasonably well-developed engineering skill at his disposal. This must mean that various devices must have been invented that were of a mechanical nature e.g. time of discharge to measure liquid mobility, rotating discs to measure shear, falling balls and so on. This suggests that the failure to exploit Newton’s proposal to produce a method that could give an accurate measurement of viscosity did not lie with the technology of the time. The one system that did work was dependent on the use of calculus. Newton “invented” calculus but he wrote his Principia in Latin and, even though it was translated into English in 1729 it is still very difficult to read and understand. This must have delayed the dissemination of his ideas and, as the methods used by both Poiseuille and Hagen depended on being able to integrate, the delay may have been inevitable. I suppose that both Hagen and Poiseuille showed that when a liquid flows in a tube of small diameter the pressure drop is proportional to the flow. Perhaps they thought that this was an unlikely result because it is not true for larger pipes but they both pursued it. They found a method of quantifying viscosity that has proved to be useful well beyond the direct result. It made possible the non-dimensional group that we call Reynolds’ number and the use of Reynolds’ number opened the way to storage and retrieval systems for immense amounts of valuable experimental data. It was a big step on the way to constructing an empirical science.

 

Hagen and Poiseuille knew that, for a small diameter pipe, the pressure drop was proportional to the flow and they must also have known that the actual pressure drop varied with temperature. They required something that took account of both temperature and the liquid that was flowing in the pipe. Newton’s definition of the coefficient of viscosity was there to use and we can reconstruct their method. It is implicit in Newton’s work that the liquid in contact with a fixed surface behaves as if it is stationary and this means that in flow through a pipe the liquid in contact with the pipe has zero velocity. Obviously there must be velocity elsewhere in the pipe and the most reasonable supposition is that across any section the velocity at a given radius is uniform and that this velocity changes from zero to at the outer radius to some maximum on the axis. Poiseuille imagined the flow to take place as if a series of coaxial cylindrical layers moving in an orderly way at different speeds. This clearly reflects Newton’s method of defining a coefficient of viscosity but does not provide uniform velocity in the layers nor a constant area.[1]

 

This is how the rest of the argument goes. Now we can look at figure 6-4. It shows, at some instant, a section of the flow of a liquid in a small tube of radius [2] between two plane surfaces  apart. The pressure on the upstream section is  and on the downstream section is . In figure 6-4 I have drawn a section of a cylindrical surface of radius  that is coaxial with the tube. There is a net force acting on this element equal to:-

 

                                                  = 

If the flow pattern is taken to be symmetrical about the axis, we can replace the  of Newton’s definition by , use Newton’s relationship, and put the force exerted on the liquid in the small tube equal to:-

                                                               

Then :- , where the minus sign allows for the fact that  will be negative.

Then we can rearrange and put:-

                                                       which can be integrated to give :-

                                                       where  is a constant.

Using Newton’s simplifying decision that the fluid behaves as if the solid boundary is the same as a stationary layer of liquid we get  where  and then   and :-

                                                        , which is the equation to a parabola.

This is not yet in the form of a rational expression because c is not a measurable quantity. We need an expression in terms of the mean velocity   and:-

                                                        

                                                =

                                                =  or, changing to diameter, = 

If this is now integrated along the pipe we get:-

                                                         and this has been named after Poiseuille and sometimes also called the Hagen-Poiseuille expression.

 

It is not in the form that is best for engineering. We need:-

                                                                

This is an expression that contains only measurable quantities and it is a rational expression on which the coefficient has been based for experiment. When it was first produced no one knew whether liquid could flow with this parabolic velocity distribution. Its success would depend on whether it turned out to be useful in practice. It was remarkably successful for liquids like water and oil. Look in any science data book and you will find values of  quoted at 20°C for a string of liquids. Presumably these liquids share the fact that they are all agglomerations of one type of molecule and not of curious mixtures of liquids, solids and semi-solids and, as a result, the coefficient does not depend on the rate of shearing. We do not find paint, milk or confectionary cream on the list.