VOL. 6. No. 6
RAEOLOGY. I. TRE NATURBOR FLUIDFLOW
1113
RHEOLOGY. I. THE NATURE OF FLUID FLOW EUGENE C. BINGHAM, LAFAYETTE COLLEGE, EASTON, PENNSYLVANIA The science of the flow of matter very early got off to a brave start, when the Greek philosopher Heraclitus announced that "everything flows." The flow of water in pipes, the plasticity of clay and bronze, even the flow of the sand in the "hour glass" offered a challenge to investigators, but progress has not kept pace with the demands of science and industry. Surely no one would think of consulting a work on hydraulics in designing a new alloy, artificial textile or lacquer. What has been the matter? Certainly cats' backs and thunderstorms could hardly have presented so promising a subject for the investigator a t the beginning, for electricity is invisible; yet the science of the flow of electricity has never faltered and a host of volumes are required for its exposition, while the science of the flow of matter has lagged a t the starting post. In fact, were one to speak of the science of matter as we speak of the science of electricity, no one would know what was being referred to. So an entirely new term for the science of flow was suggested a t the Third Plasticity Symposium, the name rheology, which I am adopting here. It needs to be pointed out a t once that ordinary flow in streams and pipes is indeed hydraulic and it is this type of flow which engaged much of the attention of the early investigators, including many of the most celebrated, such as Galileo, Newton, Napier, ,and Bernouilli. They observed that hydraulic flow is turbulent flow and quite complex in character but it was only in 1846 that the simpler linear fluifflow was demonstrated by the Frenchman, Poiseuille. He was a physiologist interested in the flow of blood in the capillaries of the body but most of his experiments were made with water in glass capillaries and he thus discovered by accident that flow in very narrow channels or at low velocities is linear in character, giving a simple type of flow. This type of flow had been suggested by Newton but not demonstrated. The difference between hydraulic flow and fluid flow may be stated very simply, for the resistance to hydraulic flow increases nearly as the square of the velocity but the resistance to linear fluid flow increases merely as the first power of the velocity. This law is to the flow of matter what Ohm's Law is to the flow of electricity, i. e., it is simple and exact and upon it rests the development of the saence of fluid flow. It may be stated in the "elegant language of mathematics" as:
In the above formula o is the velocity imparted to one layer in referenceto another at a distance r, by the shearing stress F.
where 7 is defined as the viscosity and q as the fluidity. If we consider a sea of fluid of indefinite area but of unit depth subjected to a unit shearing stress, the velocity imparted to the surface is a measure of the fluidity. This is the formal definition of the unit of fluidity, for which the name rhe has been suggested. The poise is the name of the reciprocal unit of viscosity, but the centipoise is more familiar. This is preferred because the viscosity of water a t 20°C. is almost exactly one centipoise and therefore all absolute viscosities expressed in centipoises are really specific at the same time. In order to make the argument clearer, I will here present some simple demonstrations. A tubulated bell-jar was supported in an inverted position a t a little distance above the table (Figure 1). By experiment it was found that 50 ~ mof .water ~ ran out of the tubulure in a single second, but it may be surprising to learn that on substituting linseed oil, which is comparatively viscid, the time of efflux of 50 cm.$ still turned out to be one second; but this is in .--accordance with the principle of hydraulic flow that the . - -- rate of flow is nearly independent of the viscosity of the liquid. However on placing in the tubulure of the container a rubber stopper carrying a narrow capillary 18 cm. long and repeating the experiment, very different results were obtained. The time of efflux for 10 ~ m . ~ of water a t 26'C.,was 3.7 minutes while that of linseed oil a t the same temperature was 140 minutes. Since the fluidity of water a t 26*C. is 114 and of linseed oil 2.6, these rough measurements demonstrate that in this second type of flow, the rate of effluxis directly proportional to the fluidity, the flow of the water being about plCURE forty times as fast as the linseed oil. By using an efflux tube 4.4 mm. in diameter and 4.5 cm. long an intermediate result was obtained, the water flowingonly 5 times as fast as the linseed oil. Sir Osborne Reynolds and others have studied carefully the transition from linear to turbulent flow and using an instrument like that shown in Figure 2 one can make quite precise measurements. Reference has been made to the resemblance of the law of NewtonPoiseuille to Ohm's law. A few experiments will help to bring out this similarity. In electricity we learn that when conductors are in series, their resistances are additive but when the conductors are in parallel their conductances are additive. Similarly, in fluid flow when fluids flow side by side as in true solution the fluidities are additive, but when the fluidsflow in series, as in certain emulsions or poorly mixed liquids, the viscosities are additive. We will consider first the simple case of linseed oil and water which are
,\
immiscible. Adding 5 ~ mof .oil~and an equal volume of water to the simple viscometer used above, we find the time for the two liquids to flow out the one after the other to be 80 minutes. From the earlier measurements, we
140 or a little over 70 minutes. Since the 2 times of efflux have been shown to be proportional to the viscosity in these experiments, we conclude that the viscosities are additive2 or calculate the value to be 3'7
+
=
or
+ n~
If the linseed oil and water had gone into solution the time of efflux would have been quite different and we can calculate what it should have been although we cannot verify the value by measurement with this particular pair of liquids. Since the fluidity is inversely proportional to the time of efflux and 9 = PA
+
VB
the calculated time of flow is 7 minutes. As a matter of fact a fine emulsion in which the drops did not completely close the capillary flowed through the capillary in 20 minutes. The flow of solids gives a third type of flow, known as plastic flow. It is characteristic of solids that they are able to maintain their shape indefinitely under small shearing stresses but that they can be deformed more or less readily by stresses which exceed a certain critical value, known as the yield value. Since fluids all yield to any stress, no matter how small, this distinction between solids and fluids is fyndamental. It is on the basis of this distinction that one is able to say that pitch and glass are really very viscous liquids and that paint and butter are soft solids. The discussion of this type of flow will be postponed. The question proposed a t the beginning of this article as to why the study of the flow of matter has lagged, we are now prepared to answer. 1. The hydraulicians were not aware that there is more than the one type of flow, and the type with which they were occupied did not help much in solving problems concerning the constitution of matter. 2. The analogy of Poiseuille's law to Ohm's law pointed out above has been overlooked, with the result that it has been incorrectly assumed that the viscosities of true solutions should be additive. This error would have been quickly corrected by a study of experimental results, but unfortunately on mixing many pairs of liquids there results some sort of chemical combination or dissociation with a concomitant change in the fluidity. But this seeming evil will in time prove to be a blessing, since the fluidity method is ideally suited to follow these chemical changes. 2 The discrepancy here may he reduced by a more carefully designed apparatus. For example, the ohrewed time in this experiment is a little low because the film of water facilitates, i. e., "lubricates," the flow of the oil.
3. Perhaps it is only to be expected that so long as there was uncertainty concerning the fundamental theory, there would be little incentive to exact measurement. It is a misfortune that much of the data in the literature is of such a low order of accuracy, due to omission of corrections necessary in precise measurements. As an indication of the lack of importance still attached to this property, a recent examination of some'thirty texts on physics and physical chemistry failed to disclose one which described a method for the exact determination of the property. Very unfortunately the texts either do not mention the corrections referred to above or else they make the misleading statement that they are negligible, when as a matter of fact it bas been shown that they may amount to over ten per cent. I t is so very simple to test one's results by the use of several shearing stresses that there will doubtless be a rapid improvement in.our data in the next few years. The greatest difficulty in the exact measurement of fluidity is in the precise measurement of the average diameter of a capillary tube (Figure 2). As the easiest way to solve the Alexandrian knot was to cut it, so the simplest way is to adopt as standard the average value for water at, let us
say, 20°C., which would be nearly 100 rhes. Even now workers assume some substance as standard for the purposes of calibration but the trouble is that they do not agree as to the value for the standard, which is an entirely unnecessary cause of inaccuracy. The value could be arrived a t by a commission and adopted for all time as in the derivation of the weight of the kilogram from the cubic decimeter of water, or the value could be changed from time to time as are the atomic weights. We may now briefly consider how fluidity data may further our knowledge, and point out a t once that in the hands of Maxwell the viscosity of gases very early served as a principal buttress to the kinetic theory of gases. On the basis of the theory, Maxwell predicted that the viscosity of a gas would prove to be nearly independent of the pressure and when this proved to be the case, it was regarded as a strong proof because the result was so unexpected. The viscosity of air was used by J. J. Thomson and Millikan in the determination of the charge on the electron. An understanding of fluidity is necessary in the consideration of conductivity of solutions, catapboresis, diffusion, rate of reaction, rate of crystallization, etc., but all of these important subjects will be left out of
consideration here in order to focus our attention upon what seems to be the most fundamental fluidity relationship. For many years evidence was accumulating to prove that in liquids, in contrast with gases, the fluidity is closely related to the volume. Thus in warming a liquid its flnidity increases, on putting it under pressure its fluidity decreases, and on mixing with another liquid if the volume decreases, the fluidity also in general decreases. The Russian chemist Batschinski first showed that in unassociated liquids there is a linear relationship between the two properties. As the volume is reduced, either by temperature or other means, if we name the particular molar volume V a t which the fluidity would become zero the limiting volume, designated with the symbol W, then we may state the law that the fluidity of a liquid is directly proportional to its free volume, or
where the free volume (V-TV) is the diierence between the molar volume at any given condition and the limiting volume and K is a constant for each liquid. Leaving the constant out of consideration, this law signifies that the flnidity is not dependent upon the temperature or the pressure but solely upon the volume not occupied by the molecules themselves. Since the molar volume and the fluidity are easily measured, there is placed a t our disposal a means for calculating the volumes of the molecules, provided the constant K can be evaluated, 6hich Batschinski has indicated c may be possible. I t strikes one as curious that simply observing the rate of flow through a capillary tube can give one valuable information in regard to the size of the molecules that make up the liquid, but the operations used in getting the vapor density are of the same order of complexity. As a matter of fact our whole body of knowledge about matter, both chemical and physical, is derived from the measurement of what may be called simple physical properties. However this may be, there is an obvious need for a convenient and reliable method for obtaining the molecular weights of pure liquids and of the compounds which they form with substances dissolved in them, for our laws of osmotic pressure, freezing point, conductivity, etc., all seem to break down and the modern physical chemist seems to be helpless before those bbtes noires of association and hydration. No one has yet made direct use of the Batschinski relation in obtaining the molecular weight. The author has shown that liquids are most comparable a t temperatures where the flnidities are identical; e. g., 200 or 300 rhes. The temperatures are additive; and the following atomic constants have been obtained:
Atom or grouping P = 200 Carbon.. . . . . . . . . . . . . . . . . . . . . . . . . . . . -95.7 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.2 Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24.2
Chlorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.9 -7.6
Iso
Double bond. . . . . . . . . . . . . . . . . . . . . . . . . 114.4
300 -110.2 67.8 27.1 131.7 -8.2 131.3
P =
As illustrations,the absolute temperature required to give heptane (C?Hl6) a fluidity of 200 rhes is 276.1' and the calculated value is (16 X 59.2) (7 X 05.7) = 277.3"; or methylpropylether requires an absolute temperature of 265.5' to give it a fluidity of 300 rhes and the calculated value is 27.1 - (4 X 110.2) = 264.3". It perhaps goes without (10 X 07.8) saying that the negative value of carbon applies o d y to compounds of carbon containing other elements with positive values. If for any reason the carbon atoms lack their full complement of satellite atoms, their ahsence must be compensated for and it is not without interest to note that the value of the "double bond" is only four degrees less than for two hydrogen atoms. The more symmetrical iso compound seems to require a slightly lower temperature to produce a given fluidity, but the effect is small. Whether the effect is real or not will be determined as soon as sufficient data is accumulated. Here again that bate noire of association rises before us, for there is a great class of substances which we have come to regard as associated which do not give close agreement between the observed and calculated temperature?% Taking as an extreme example the case bf water, the observed temperature required to give a fluidity of 200 is 328.9' K whereas the calculated is 24.2 = 142.8', but since the temperatures are additive only (2 X 59.2)
+
+
142 fir = 328 9,
so that the association factor x = 2.31, at 56°C. It is 2.20 at 85°C. This is but a single example of which many might be given. The important thing is to apply tests to the values obtained in order to prove whether they are correct or not. There are now several ways which have been proposed for calculating the molecular weight of associated liquids. I t can be said of the fluidity method that it agrees with the others as well as they agree with each other. This is valuable as confirmatory evidence hut the method must find its merit in its directness and simplicity. The very important conclusion here is that water is not made up exclusively of simple molecules of H 2 0 called hydro1 or even of trihydrol (Hz0)3. I t is more likely a mixture of the two in equilibrium, but we see that a change of thirty degrees only changes the association of water by five per cent. We can therefore consider liquid water as a substance with a molecular weight of roughly 41 instead of 18. This change in our conceptions will he highly advantageous.
VOL.6. No. 6
RHBOLOGY. I. THENATURE OF FLUID FLOW
1119
Since an unfamiliar subject strikes the reader with strangeness if not indeed dismay, it may be well before closing this part to present a photograph of a model of the fluidity pressure isothermals of carbon dioxide prepared by Mr. Baxter Lowe (Figure 3). The coordinates of pressures, fluidities, and temueratures are measured off on the boards painted black. The model was made from plaster of Paris as accurately as the available data would permit. The curve ABCD presents a close analogy to the well-known pressure-volume isothermal of carbon dioxide except that in the gaseous state CD there is a marked difference due to a new cause of viscous resistance (diffusional viscosity). In the liquid state the effect of pressure on the fluidity AB is seen to be very small. Above the critical temperature E G the effect of pressure on the fluidity is quite different, becoming less and less as the temperature is raised. One is even faced by the paradox of a gas at a very high temperature having a lower &idity, i. e., higher viscosity than it would have a t high pressure in the liquid state. The effect of temperature on the fluidity of liquids is to increase it in a nearly linear manner A E but the effect of temperature on gases is to decrease it DG. Since this model represents the fluidity model of any substance near its critical temperature F, much may be learned by a careful study of it.
