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What is a Granular Medium?
You will find the basic facts about granular medium - no details -. If you wanna know more just email me or feel free to ask in the Discussion Forum. I purposely erased all the bibliographical references and detailed equations to keep the text simple and easy to read.
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Done and updated by Sébastien Dartevelle, The WebMaster, Saturday November 22 2003.
What is a granular flow? Well, I guess that is a good question there are different ways to define a granular flow, depending on what you aim exactly, what you wanna study, and who you are (physicist, engineer, geophysicist, so forth). This question is far from being naive. Have you ever poured sugar on your table? Do it. You will be amazed If you watch the pouring sugar from a certain distance, it looks like a white liquid flowing down to the table, from that perspective flowing sugar seems to behave like milk. Now, if you take a closer look, what you see is no more than solid grains falling down in the air, like gas-molecule objects. If you stop pouring down, you are gonna have an immobile pile of grains, which acts like a solid material. As a matter of fact, granular media behave like any fluid and/or like any solid material. That is the key aspect of a granular medium, it acts like another intermediary physical state, neither solid, nor liquid, nor gaseous, but rather granular.
Sooooo what? Well, you see why, it would be useful to define what is a granular medium. You should not be surprised if those media share common properties with gas, water and solid
The first thing useful to say is that they are many, many definitions of granular matter. A geophysicist (thats me), or a chemical engineer, or a pure physicist will explain and define with different words what is a granular flow. So, we must acknowledge that the way we see a granular flow depends mostly on your background, and on what you are interested in. So, lets see how *I* see it
The simplest way (the one I much prefer) is to say that a granular flow is a flow with grains. The only problem is that we dont say much here, and therefore, it doesnt help much. We need to constrain a little bit more this definition if we want to make it useful for our computer modeling. But, I agree, sometimes, the least we say, the safer we feel.
A second way and still very simple, is to say that a granular flow is a flow of powder in a vacuum, which means there is no fluid (i.e., gas or water) to support the particles. This definition is simple because we only deal with one phase in this flow, i.e., the solid phase. Of course in Nature and in Volcanology, this is an extreme case (practically never found). But this definition will be highly useful for us anyway. That is the first definition ever used to define the kinetic-collisional rheologies of granular matter (early in the 1980s).
The third way and the most challenging and complicate way is to say that a granular flow is a mixture of grains and a fluid phase (e.g., gas, liquid). The grains are called the dispersed phase, while the fluid phase is the carrier phase supporting the grains. Typically, in volcanology, the carrier fluid is the gas phase. Now, the problem here is that we must deal with multi-phase flow theory hence, we must also deal with multi-phase flow turbulence. This leads to huge theoretical development always difficult to deal with. The good thing of this third definition is that it is so powerful that we can basically describe anything we want. Indeed, the gas phase is always there, even in highly concentrated granular flows therefore, for granular flows found in Nature, we must always have a continuum between highly concentrated flow (where the dispersed phase dominates) to highly diluted flow (where the gas phase dominates).
If we consider the two end-members of granular flows, highly concentrated to highly diluted, it should be possible to define in meaningful ways the constitutive equations of such end-member flows along with equations to connect those end-member rheologies. Knowing, understanding, and defining those equations are the necessary steps for carrying out super-computer modeling of granular flows.
One the main and most important aspect in those equations is to understand the distribution of all the forces acting on the granular fluid However, instead of analyzing those forces, we prefer to deal with forces acting on a surface of a fluid parcel. Force acting on a surface is called stress, e.g., normal stress (e.g., pressure), and shear (tangential) stress. If we understand how stress is distributed within the granular flow, we made 80% of the job for modeling granular flows. In addition, understanding the distribution of stress within granular flows is a key aspect for understanding the fundamental differences between granular fluid and any other fluid (such as gas or water).
Lets begin with stress within a highly concentrated granular flows.
This is equivalent to say that the carrier fluid is less important and grains endure permanent contact with their neighbors, e.g., grains do not collide but instead roll, rub and scrape all together. Such kind of stress is called frictional (opposed to collisional and kinetic, which are the other cases in a more dilute situation).
Now, lets imagine a minute that our granular material does not move or flow at all. In this case, we would imagine that the only force acting on grains is the gravity force. Hence, as for any fluid in the world, the total stress at the bottom of a granular pile of height h would be:
where g is the acceleration of gravity and is the bulk density of the granular material. We see that the total stress at the bottom of this pile would be a normal stress along the vertical direction (h), and therefore would be an isostatic Pressure (simply due to the weight of the granular pile). This is true for any fluid BUT granular fluid. Indeed, for a sufficient height of the granular column, the pressure reaches a maximum value and does not increase no matter how high is the pile of grains. This is because granular material supports frictional shear stress as well, even if there is no motion. In addition, if side walls are present (in a container), they can also support the extra weight of the granular column. Therefore, the total stress at the bottom of a granular pile must be a combination of normal and shear stress.
Fortunately, the relation between shear and normal stress is well known and is even quite simple, i.e., it is linear. This linear relation is often referred as the Mohr-Coulomb relation for frictional stress. In its simplest form in 2D, we have:
where S is the frictional shear stress, N is the normal frictional stress, k is a known material property (describing the cohesive state of grains, and is therefore a cohesive shear), and is the angle of repose (or the angle of internal friction of the material). Eq. 1 is described on the following figure:
[Figure 1: At yield, the higher the frictional pressure, the higher the shear stress. The more sticky the particles (the more cohesive) and/or the higher the angle of friction, the higher the shear]
The Mohr-Coulomb law described by Eq.2 and shown on Fig. 1 is a yielding law which asserts that a material will yield by shearing on a surface element if S attains a critical value given by Eq. 2. Therefore this linear relationship is sometimes called the "yield line". Below this yield line, the material response will be rigid or elastic and does not suffer any strain or only elastic strain (however elastic strain is negligible for granular matter). If the shear stress is increased for a given normal stress so that the stress state of the material is just on that line, then plastic strain or yielding will results. It is impossible to have a state of stress above this yield Mohr-Coulomb line. When the yield stress is reached then particles will slide over one another.
The angle of repose (or angle of internal friction) can be easily understood with the following figure:
[Figure 2: Internal angle of friction of a pile of non moving grains]
This angle of repose is low when grains are smooth, coarse or rounded, and, it is high for sticky, sharp, irregular, or very fine particles. Typically, it is between 15o and 50o.
Now, those generic relations hold valid for non-moving granular material but are not quite useful in our case since we want to model flowing or deforming (flowing means deforming) granular media for highly concentrated flow, with possibly some gas between grains. Believe it or not, this is tremendously challenging Indeed, what is the relationship between moving frictional grain and the static frictional pile of grain described by (Eq. 2)? Actually, this relationship is not easy to find. The Mohr-Coulomb law cannot be used straightforwardly for flowing granular medium. The only way to do the connection between this yielding law and a flowing granular medium is with the help of the Plastic Potential and critical state theories developed in Soil Mechanics and Engineering Sciences. The complete demonstration can be found in the A Review of Plastic-Frictional Stress (Part. 1) course (in all, it has 4 parts).
Now, as I explained in the previous section, highly-concentrated granular flow can be seen as mono-phase flow where the gas phase is negligible relative to the solid phase. But this case is an extreme case rarely seen in Nature. What I meant is that we have to deal with a carrier continuous phase. Ouch!! That is bad, really. Now we must write our constitutive equations for a multi-phase flow system, which is, most of the time, very painful to do (at least, if you wanna work seriously). Again, I am not gonna write those constitutive equations on this page. What I am gonna do instead is to explain the deep philosophy of multi-phase flow. Indeed, my goal here is to understand the key differences between a granular multi-phase flow and a simple "normal" non-granular mono-phase flow (e.g., an easy gas flow).
Multiphase flow arises when the averaged motion of one material is distinctly different from that of another: droplets of rain falling in air, sand moving along a stream bed under the force of running water, products of coal combustion, man made dust deflagration and detonation, or ash clouds and flows emitted by volcanoes are a few common examples.
Typically we are interested only in the averaged behavior of the flow system, as opposed to wanting to know the exact history of a particular grain within a gas cloud. Indeed, the solution of the equations of motion for a single particle is generally inadequate for saying how the system works as a whole. Hence, we can characterize the multiphase flow problem as one for which we want to know the averaged behavior of a system having a large number of degrees of freedom, and in only knowing the behavior of a small number of the components making up the system (gas, particle, water drops, so forth). Such numerical approach will not give any detailed information about how a specific given grain behaves within the flow, neither it will give information about the sedimentary features of the resulting deposits. Rather these average modeling techniques display information on the global flow behaviors, information on the average effect of the small-scale (i.e., grains) upon the global flow.
We have here the essence of what this project is all about:
- How can we model by average ensemble techniques the consequence of small-scale phenomena on the global resulting flow?
- In a nutshell, what are the effects of grain concentrations or sizes within the flow?
- What are the effects of the local initial conditions (e.g., mass flux at volcano, vent diameter, initial grain concentration, so forth)?
- What are effects of gas chemistry on the global flow dynamic (e.g., presence of water vapor, of CO2, humidity of the atmosphere, etc.)?
Those small-scales features (small because it is only either local or no larger than the molecule or the grain sizes) might look like trifling. And as a matter of fact, they have been so far disregarded by volcanologist modelers. Though, huge financial and scientific efforts have been undertaken to understand the role of the small-scales upon the global flow dynamic those efforts are lead by leader national laboratories (in the U.S., the Los Alamos National Laboratory, the Jet Propulsion Laboratory and the National Energy Technology Laboratory and in Europe, the Laboratory of Geophysical and Industrial Flows and the European Research Community on Flow, Turbulence and Combustion) In the same vein, the development of granular theories over those last 20 years is precisely motivated by the small-scale granular behaviors. For example (one example among many others), the primary feature of granular media differentiating them from simple molecules is the nature of the collision, i.e., inelasticity. Such a first order difference explains why a granular media cannot be model as a simple gas as essentially done today in volcanology. The fact that grain collisions are an inelastic phenomenon will cause an enormous change in the energetic balance and dynamical behavior of the flow. Geophysicists and volcanologists cannot ignore those research on the key roles of small-scales on the dynamic of a flow. Our project is strictly in line with those current modern researches.
Those small-scale motions might have a dramatic role upon the bulk flow behavior. In physics, those small-scale motions are more often named Turbulence. As matter of fact, studying the dissipative role of grains within a flow is just a "simple" matter of turbulence. Generally speaking, the problem of fluid turbulence is similar to that of multiphase flow, in that we are likewise interested in the averaged dynamics of a system having a large (in fact infinite) number of degrees of freedom. In the case of a pure fluid we can likewise write down the equations of motion, and solve them, for a given region of space. The obvious difficulty is that one may be interested in a region of space that is very large compared to the smallest scales of motion that the fluid will exhibit; and the averaged motion may be greatly influenced by the details of these small scales. So once again, the modeler interested in the average technique modeling must devise a means of connecting the exact motion, which is known on a small scale, to the averaged motion on a larger scale.
As pointed out by Los Alamos National Laboratory researchers, there is no multiphase flow that does not exhibit multiphase turbulence. Even a single perfect sphere, settling in the middle of a perfectly quiet and idle ocean can exhibit multiphase turbulence. In other words, we can never neglect the small-scales, viz -in our case- the grains. Sadly, we must humbly acknowledge that what volcanologist modelers have done so far is precisely disregarding the role of the grains within the flow, which somehow dooms all the results obtained from their previous modeling. One may pretend their results looks similar to what is understood from the observations of Natural phenomena, but this only holds valid knowing these models only adjust a set of free unphysical parameters to make sure their results approached Natural phenomena. Such modeling way is disregarded in our project. We prefer to aim a more challenging if not tormented way, but aiming high will always be rewarded at the very end.
As stated before, we aim for the first time to understand the effects of those small-scales motions on the global flow behavior. Those small-scale effects will be modeled through the stress tensors of the different phases within the flow as there is an urgent necessity to account for the particulate stresses in granular multi-phase flow.
It is a well known fact that modeling particle (granular) pressure has always been a problem. Therefore, some scientists simply ignore it (after all, why bother for just a few grains in the fluid) or others make it equal to the gas pressure. But on the other hand the behavior of granular pressure can have significant effects on the behavior of multiphase flow systems, which is consistent with the previous paragraph. It has been shown that the stability of a fluidized bed is greatly affected by the dependence of the granular pressure on the void fraction. Therefore, instabilities may grow or be damped through the forces transmitted within the solid particulate phase. This indicates that in more dilute situation (true multi-phase flow), we must understand with great care granular stress, and more specifically granular pressure.
The particle pressure may be thought as the force per unit area exerted on a surface by the grains of a multiphase mixture. The striking fact here is that there is a deep analogy with gas pressure in which the pressure of gas acting on a surface is visualized as a result of the impacts of molecules. What we do in granular kinetic theory is that we replace gas molecules by grains. However, there are two main differences with simple gas molecules. The solid grains in addition to short-duration collisional impacts can also transmit a force via long-duration (frictional) contact with a surface. In this latter case, the particle pressure reflects the forces exerted across their contact points. The other difference with simple gas molecules is the nature of the collision between grains as it is inherently inelastic and some energy is systematically lost in each collision. This will cause interesting and intriguing features only found in granular media, and will deeply modify the thermo-dynamical balance and behavior in granular multiphase flow. For instance, an individual marble dropped onto a glass plate, may bounce for quite a while. But if you drop at once a set of identical marbles on the same glass plate, they will stop dead once hitting the plate, none of those marbles will bounce. This strikingly different collective behavior can only be explained by the exceedingly large number of rapid inelastic collisions among neighboring marbles. This inelastic nature of grain collisions will cause grain clustering within the granular multiphase flow. This clustering effect is subject to huge scientific research, but has not yet been recognized in experimental, theoretical and field volcanology. This is unfortunate, because granular flows subject to clustering effect will have very different behavior than any simple flow. The distribution of stress, grain concentration, so forth will be totally affected by such key phenomenon. No one can pretend to model granular flow, no matter how dilute, without an accurate multiphase flow granular model.
As I indicated earlier, it is possible to define a Pressure in the highly concentrated frictional case (see A Review of Plastic-Frictional Stress page), but what about the more dilute one? In this more dilute case, the granular pressure will be caused by the random motion and collision of grains within the gas phase. This kinetic/collisional granular Pressure can be somehow seen as a true thermodynamical Pressure. But before defining it, it would be interesting to recall some fundamental theoretical aspects from gas kinetic theory.
It is more than useful to recall how classical gas kinetic theory defines gas Pressure and Temperature. This will help us immensely to understand how we define the granular (kinetic/collisional) Pressure and Temperature in a granular system.
Lets assume that the gas molecules are in a continuous state of restlessness, those molecules are small, hard, elastic spheres acting on each other during short duration collisions. This random fluctuating motion of molecules can be characterized by a random velocity profile, i.e., by a statistical velocity distribution function. We further assumed that (i) at any spatial location the distribution of molecular velocities is independent of time, (ii) after a large number of collisions the random velocity components are statistically independent, (iii) the distribution function is isotropic (therefore there is no preferred direction) and is independent of the orientation of the coordinate system, (iv) when two sphere collides, the direction after collision is distributed with equal probability over all solid angles. This full set of assumptions are known as the Maxwells assumptions.
The fluctuating random velocity distribution function is called the Maxwellian distribution. The Maxwellian velocity distribution function f(c) can be seen as the probability of finding a randomly selected molecule with a random fluctuating velocity c. Therefore, over the whole probability domain we must have:
where f(c) will be defined with a Gauss-Laplace function (the full demonstration will be done later, when I will find time for it) over the full 3D domain of velocity (c is the vector velocity of the fluctuating random motion of the molecule, and is indeed the random variable). Please keep in mind that Eq. 3 is a vector integral and that you must integer three times over the 3 velocity components of the 3 directions of space (this is a triple integral). f(c) is given by:
where "m" is the mass of the molecule, "k" is the Boltzmann constant (we will understand its role here below on this page), "T" is the temperature of the gas, is the standard deviation of f(c) and is also known as the Lagrange multiplier. Those last relations will make a direct relationship between microscopic molecular quantities (c and m) and macroscopic ones (T and P). However, we must also know how to calculate the statistical moments of this Maxwellian distribution. The first moment, which is the average fluctuating velocity of the molecule will be:
which is a vector equation with three velocity components. The variance will be given by:
For instance, if we consider the average velocity and the variance of the velocity in the X-direction only (cx), we have:
It is also easy to show that the average kinetic energy in the cx direction using the Maxwellian distribution function of the molecular fluctuating velocity would be (I used Eq.9):
Similar expressions can be found for the mean kinetic energy in cy and cz directions as well. Therefore, under equilibrium conditions, and since there is no preferred directions in the entire phase space, we can conclude that the average kinetic energy is the same in every direction, i.e., . The mean kinetic energy over the whole space will be then:
This set of equations suggests that a mono-atomic gas in equilibrium and following Maxwells assumptions has kinetic energy equally shared between the three directions of space, this is called the equipartition of energy in the 3-Dimensional space. We see the role of the Boltzmann constant k (= 1.3805x10-23 J/K) which converts kinetic energy (J) into temperature (degree K), it is simply a scale factor.
We may conclude that the temperature is nothing but a measurement of the state of restlessness of the gas molecules. The higher the average random fluctuating motion, the higher the gas temperature. Setting the temperature in terms of the molecular fluctuating velocity, we have:
Therefore the temperature of the gas phase is proportional to the mean quadratic fluctuating velocity of the random motion of the gas molecules.
In a similar way, we can define the thermodynamic Pressure as:
This equation says that the higher the kinetic energy transported by a number "n" of molecules per unit of volume, the higher the pressure. This equation can be readily changed into:
where is the number of moles, Na is the Avogadros number (6.0221x1023) and R is the universal gas constant (8.3145 J/mol K). Eq.15 is, of course, Boyles Perfect Law of gases.
Well, the same thing applies for granular medium. If grains can randomly move within the flow (as long as it is not too concentrated), and therefore are subject to a random fluctuating motion like the gas-molecules, then the same demonstration should apply for a granular flow. The only thing is that collisions are inelastic (and that is a *big* difference).
We define a temperature, , said granular temperature, that measures the fluctuating random motion of the grains within the fluid as:
where is the fluctuating energy per unit of mass due to the granular random motion and C is the velocity of this fluctuating motion of grains. The kinetic interpretation of the granular temperature is similar to the one of the gas but in addition to the granular temperature grains can have a proper thermal temperature, T. Therefore, you should not confused and T. Please visit A Review of Granular Theories page if you wish to understand the relationships between thermal Temperature, T, and granular Temperature, , for the dispersed solid phase.
As we have written a perfect gas law for the pressure, we can define such a thermodynamical Pressure for the granular medium in terms of the density of the solid phase () and the volumetric solid concentration ():
And, here again, the interpretation is the same as for the thermodynamical gas pressure. The higher the kinetic fluctuating energy of the grains, the higher the granular pressure. Though, the problem is that the granular concentration can be so high (higher than gas molecule) that we deal with numerous collisions and those collisions are inelastic. So, the granular pressure is modified to account for those collisional inelastic effects:
The first term between brackets represents the kinetic part of the pressure (as for the gas molecules), while the second term is the collisional contribution to the Pressure. The latter term is negligible for highly diluted granular flow. In this equation, "e" is the inelasticity coefficient, and g0 is the radial distribution function, it prevents over-compaction, as it acts as a repulsion function between grains when they are close to each other. This function is equal to unity for very low concentration but it asymptotically tends to infinity for highly concentrated particulate system.
Of course, if the granular concentration gets so high and close to the maximum limit allowed for a randomly packed structure (64%), the grains will not be allowed to randomly move, and therefore, the granular temperature will tend to zero. In this specific case, only frictions dominate, the collisional-kinetic part to the stress dwindle to nothing.
You might have noticed that I qualify this microscopic random motion as a "fluctuating" motion. As a matter of fact we could also qualify it as "Turbulent" motion, as it is often done in engineering literature. This is more than correct and consistent with the second paragraph and the multiphase granular flow turbulence. This is the whole goal, to understand the effects of small scale upon the average bulk flow.
Lastly, the kinetic-collisional Pressure is often named "thermodynamical" Pressure as it is indeed related to the random motions of the grains. The higher this chaotic random motions, the hotter the granular flow (in term of its granular temperature), the higher its granular thermodynamical Pressure.
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