GG170L: RHEOLOGY Handout (Material to read instead of a Lab Book chapter you might want to print this out and bring it to lab)

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1 GG170L: RHEOLOGY Handout (Material to read instead of a Lab Book chapter you might want to print this out and bring it to lab) Rheology is the study of how materials flow, and includes two main properties. The first is viscosity (which tells you how fast something will flow), and the second is yield strength (which tells you if something will flow). Rheology is very important in volcanology. For example, the combined viscosity and yield strength of a magma will determine if gas bubbles can rise through the magma chamber to escape peacefully out the top, or if they will be trapped, allowing the pressure in the magma chamber to increase to the point of an explosive eruption. Geologists also study the rheology of glaciers to understand what makes them flow rapidly or slowly, the rheology of lahars to understand how they pick up and carry large boulders and trees, the rheology of the Earth s mantle to understand how it convects, and much more. Although rheology is important, it is not so easy to understand, especially when people use a lot of math to describe it. In the Rheology lab we will take a mostly qualitative but partially quantitative approach to illustrate rheology, concentrating on viscosity. I VISCOSITY AND CYLINDERS OF SHAMPOO AND OIL Viscosity is the resistance to flow (it is the opposite of fluidity). Viscosity is a property of a substance that relates a known amount of deformation (for example stirring force) to the rate that the substance deforms (flows). In other words, how fast will something flow for any given amount of stirring. For example, say you push with a certain force on a spoon in a bowl of water. You will get reasonably fast flowing. However, if the bowl is full of cookie dough the same force on the spoon produces very slow movement. The other way you could think about it would be to see how much force on the spoon was required to give you one turn around the bowl full of water every 2 seconds, and compare this to how much force was required to give you the same stirring rate for the bowl of cookie dough. In either case, you would determine that the water has a lower viscosity (it deforms quickly with only a small force), and the cookie dough has a higher viscosity (it deforms slowly with the same amount of force or requires a greater force to deform at the same rate as water). There are lots of ways to measure viscosity, including attaching a torque wrench to a paddle, using a spring of known strength to push a rod into the fluid, or measuring how fast a fluid pours through a small hole. The method we will use is one of the oldest, and it involves measuring how fast a sphere of known size and density sinks through a fluid. If we know the size of the sinking sphere and its density relative to that of the fluid, we can determine the amount of sinking force it produces. Thus, if the sphere sinks quickly then the viscosity is low because it is offering only a little resistance. If the sphere sinks slowly the viscosity is high because it is offering a lot of resistance. 1

2 This lab will also let you practice measuring lots of things, practice converting units, show you how scientific results are never identical, and let you see how seemingly small measurement mistakes can sometimes turn out to be kind of significant. The formula that allows you to calculate viscosity with a sinking sphere is a modification of Stoke's Law, named for George Gabriel Stokes ( ): viscosity = η = 2( ρ)gr2 9v η = viscosity (what you want; units are Pa s) ρ = the difference between the density of the sphere and the density of the fluid (kg m-3) g = the acceleration due to gravity, which is constant on Earth ( 9.8 m s -2 ) r = the radius of the sphere (m) v = the velocity that the sphere sinks (m s -1 ) Note that all are in SI (Système International) units. In the lab, you will drop spheres of known radius and density (determined by you by measuring their mass and diameter) through a fluid of known density (determined by you by measureing its mass and volume). You will therefore know ρ and r 2. By calculating how fast the spheres sink a known distance through the fluid, you will determine the velocity, v, and at that point you can calculate the viscosity of the fluid. II UNITS Units: It is important to keep track of the units that go along with the numbers you are using. The most important reason why you do this is that if the ending units don t make sense, probably somewhere you have made a math mistake and your answer isn t correct. This is captured in the saying that goes something like Units are like sports officials nobody pays much attention to them until something goes wrong. For example, say you are in Samoa and you know that the money exchange rate is 3 Tala per dollar (3T/$). You have $30 and you want to check into a hotel that costs 70T per night. Can you do it? First, you divide 30$ by 3T/$ and you get 10. You are bummed. But did you do the calculations right? The key to using units is that you do the same operations on the units as you do to the numbers. So, if you divided the amount of dollars you have (30) by the amount of the conversion factor (3) then you have to divide the units of the dollars you have ($) by the units of the conversion factor (T/$). The results of doing this is $ 2 /T, which doesn t make sense. That is a pretty darn good hint that the math part is not correct either. If instead you multiplied 30$ by 3T/$, then the number is 90. And if you do the same thing to the units, you end up with $T/$, which cancels out to just T, which makes sense. You won t 2

3 have to sleep outside under a coconut tree after all, and you ll even have some money left for a Vailima! The Stokes law equation has lots of units in it and they actually help us understand the eventual SI units of viscosity, Pascal-seconds (Pa s): η = 2( ρ)gr 2 9v In units, the part to the right of the = is: kg x m x m2 m m 3 s 2 s You can also write this as: (kg)(m)(m 2 )(s) (m 3 )(s 2 )(m) As we saw above, you can cancel and combine units in the same way that you can cancel and combine numbers. For example, you can combine (kg)(m) into what is called a Newton (N), a unit of force. (s2) This leaves you with (N)(m 2 )(s) which reduces down to (N)(s) (m 3 )(m) (m 2 ) (N) (m2) is force per area, or pressure, and in SI units is called a Pascal (Pa) So...for viscosity you are left with a unit called a Pascal-second (Pa s), a unit of pressure times a unit of time. Although this is the official SI unit of viscosity, it is not obviously intuitive that it should represent the resistance of a fluid to flowing. Let s play with the units a little more. If you start with the starting units: (kg)(m)(m2)(s) (m 3 )(s 2 )(m) and cancel everything you can, you are left with (kg) (m)(s) 3

4 This is a unit of mass (in kg) divided by length (in m) and time (in s). This makes a little more sense. Consider the same sphere (same mass) sinking for the same period of time, in other words (kg) is constant. (s) Then, if the distance the sphere sinks is big, (m is big), the fraction gets small. This makes sense - if the same sphere sinks a long distance in the same amount of time, the viscosity is low. On the other hand, if the sphere sinks only a short distance, (m is small), then the fraction gets big. This makes sense too - if the sphere sinks only a short distance in the same amount of time the viscosity is high. One way that viscosity is commonly illustrated uses a graph of the deforming pressure that you apply to a fluid plotted against the rate that the fluid deforms (Figure 1). Figure 1: qualitative plot of applied deformation pressure vs. strain rate for three hypothetical Newtonian fluids. Notice that for the same applied pressure, you get a low strain rate for the high viscosity fluid and a high strain rate for the low viscosity fluid. Or, you can say that in order to get the same strain rate, in a low viscosity fluid you need a low applied pressure but in a high viscosity fluid you need a high applied pressure. 4

5 III YIELD STRENGTH Yield strength is another important aspect of rheology, and it tells you how hard you have to push on a substance in order to get it to flow. It is important to recognize that yield strength is a property of some fluids, but not all. If you push on water, for example, no matter how softly you push, the water will move. Thus, water has no yield strength. We call substances that have no yield strength Newtonian. However, for many substances, tiny pushes don't produce any movement and maybe even medium pushes don't either. For these, there is a certain amount of force you have to exert on the substance before it will flow. That certain amount of force is called the yield strength of that substance. Once the substance starts to move, then the rate at which it moves is a function of its viscosity. Yield strength is very important when studying lava flows, glaciers, salt domes, and many other geological substances that flow. In nature, the most common source of a deforming force is gravity. Gravity causes lava flows and glaciers to flow downhill and to spread laterally, it causes stones to sink into lahars and glaciers, as well as many other geological motions. It also turns out to be much easier to determine yield strength than viscosity. That is because for yield strength, all you have to do is figure out if something flowed, and this can often be done long after the flowing has stopped. For viscosity, you have to somehow figure out how fast something flowed, which is difficult even if you are watching it happen, let alone if you come along after the flowing is pau. One of the easiest ways to visualize how yield strength works is to consider a blob of fluid (lava, glacial ice, cake batter) on a horizontal surface (Figure 2). Gravity will cause the fluid to try to spread out laterally. The fluid s viscosity will control how fast it spreads but here we are concerned with the yield strength, and whether it will allow the fluid to spread at all. The stress that is deforming the blob of fluid depends on gravity, the density of the lava, and the height of the fluid. As long as the combination of height, density, and gravity is large enough, it can overcome any internal strength (the yield strength) that the lava dome has. Gravity is constant so that doesn t change. We can also assume that the density of the fluid doesn t change either. As the blob spreads laterally, however, its height decreases (Figures 2 B and C), and Figure 2: a spreading blob of fluid 5

6 there will come a point where height, density, and gravity are no longer able to overcome the yield strength the fluid will have achieved an equilibrium shape. It won t matter how long we wait, the fluid will not be able to overcome its yield strength. This means that we can come along long after a lava dome has solidified, for example, measure its density, height, and width, and calculate what the yield strength must have been while it was actively flowing. The simplest formula for calculating yield strength utilizes this balance between the deforming forces and resisting forces: YS = ρgh 2 w YS = yield strength (in units of pressure, N m -2 ) ρ = density of the fluid (kg m -3 ) h = height (or thickness) of fluid (m) w = width of fluid at its base (m) So you see that as a lava dome spreads laterally, h decreases while w increases. If the dome can spread a lot, the h 2 /w part of the equation will be small and the yield strength will likewise be small. On the other hand, if the dome stops spreading while h is still high and w has not increased very much, the h 2 /w part of the equation will be large and the yield strength will likewise be large. IV MORE UNITS The units of the above yield strength equation are: (kg)(m)(m2) (m 3 )(sec2)(m) again, you can take out the (kg)(m) (sec 2 ) which is a Newton (N) all the leftover m's reduce down to 1 (m2) so, the whole thing becomes N, (m2) which is our old friend the Pascal (Pa), a unit of pressure, or stress. 6

7 This unit is a little bit more intuitive than the Pa s is for viscosity. This is because it makes sense that pressure is what causes a fluid to move. If the pressure is too low (below the yield strength) the fluid will not flow, no matter how long you wait. If the pressure is above the yield strength, the fluid will flow, and how fast it flows depends on its viscosity. Yield strength is also often illustrated graphically (Figure 3) Figure 3: qualitative plot of applied pressure vs. deformation rate for two hypothetical non- Newtonian fluids This graph is similar to Figure 2 except for one key feature. For fluids that have a yield strength (non-newtonian fluids), the graph of applied pressure vs. strain rate doesn t go through the origin. For Fluid B, for example, any applied pressure that is less than the yield strength produces no deformation, and therefore a deformation rate of zero. For Fluid A, an even higher applied pressure needs to be occur before there is deformation. Thus, the yield strengths are defined by the values of the y-intercepts. Both these fluids also have viscosities (called non-newtonian viscosities), which, as before, are determined by the slopes of the lines. 7

8 IV What does any of this have to do with Geology??? Figure 4 shows an outcrop of the 1868 Mauna Lava flow. The outcrop was divided into equal-sized boxes marked with chalk, and in each box the number of olivine crystals ~4 mm across was counted. The counts of olivines have been chalked next to the boxes. Figure 5 shows what we think is going on. Specifically, when a flow comes to a halt, there is an even distribution of olivines from top to bottom (Fig. 5a). The flow cools from the top down and from the bottom up, and the very top and very bottom cool quickly enough to preserve the initial olivine concentration (Fig. 5b). However, the middle of the flow takes a while to cool, and as long as they can escape the cooling front that is working its way down from the top, they will sink until they land on the cooling front that is working its way up from the bottom (Figs. 5c-d). The final part of Figure 5 shows the abundance of olivines with depth, characterized by a C-number, which is a measure of olivine concentration or depletion relative to the starting Figure Lava flow concentration. We need to do two things: 1) figure out how far they sank by assuming that at least some of them made it from just below the top to the depth where the greatest concentration is; and 2) figure out how much time was available for them to do this sinking by calculating how long it takes the lava to cool from the bottom up to the depth where the greatest concentration is. Once we know sinking distance and sinking time available, we can calculate a sinking velocity, and just like with the shampoo, calculate the viscosity of the lava using Stokes law. Figure 5. Olivine crystal sinking diagram 8

9 Figure 6 is based on data that were collected from temperature probes lowered into a lava lake that formed in Alae crater during an eruption of Kīlauea in What they did was stick a thermocouple probe through the lake surface as the lake cooled, and determined at what depth each temperature was with respect to time. What we care about, however, is the temperature variation from the base up, because it is the lava cooling from the bottom that produces a floor onto which the olivines sank. The geologists back in 1963 didn t measure from the bottom up. However, thermal theory tells us that a flow will cool from the bottom up at a rate ~75% of that at Figure 6. depth vs. time to cool to 1130º C which it cools from the top down. This allows us to produce Figure 6. It is a graph of height, in meters, up from the base of the lava lake (on the x axis) vs. the time, in seconds, that it took that level in the lake to cool to 1130º C. Even though 1130º C is still pretty hot, at this temperature the lava will have gained a yield strength that will prevent 4 mm diameter olivines from sinking. Diameter is in quotes because olivines aren t really shaped like spheres. So how do you use this graph? Let s say you had a flow that was 1 m thick in total, and you determined that the concentration of olivines was at 60 cm (0.6 m) below the surface of a flow. This means that the concentration of olivines is 40 cm (0.4 m) above the base of the flow. You know, therefore, that at least some of the olivines managed to sink 0.6 m in the time that it took the lava 0.4 m from the base to cool to 1130º C. Figure 6 tells you what this time was. Find 0.4 on the x-axis and go up until you hit the curve. Then go left, and you hit the y-axis at about 257,000 seconds. That s how long it took the lava at 0.4 m above the base to cool to 1130º C, and therefore how long the olivines had to collect there. You therefore know the distance they sank (0.6 m) and you know the time it took them to do the sinking (257,000 s), so you can calculate a sinking velocity of 2.3 x 10-6 m s -1, which is pretty darn slow! 9