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An Overview of Granular Theories
You will find the basic facts about granular theories - 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.
For the granular phase, it is clear that any mathematical model pretending modeling a granular flow must account for the following effects, at any time and anywhere within the flow (see Figure 1):
- the flow can display a large span of grain concentrations, therefore,
- in the dilute part of the flow, grains randomly fluctuate and translate, this form of viscous dissipation and stress is named kinetic,
- at higher concentration, in addition to the previous dissipation form, grains can collide shortly, this gives rise to further dissipation and stress, named collisional,
- at very high concentration (more than 50% in volume), grains start to endure long, sliding and rubbing contacts, which gives rise to a totally different from of dissipation and stress, named frictional.
[Figure 1: The three main forms of viscous dissipation within a granular flow: kinetic, kinetic+collisional and frictional]
It is expected that the momentum and energy transfer will be different according to the granular regimes. A mathematical model that pretends to describe such granular flow requires an comprehensive unified stress tensor able to adequately describe stress within the flow for any of these regimes, and this without imposing what regime will dominate over the others (as often done in geophysics and volcanology). The ideal stress tensor:
where the total stress tensor of the solid phase (T) is the sum of the kinetic, collisional and frictional tensors, the superscript "f" stands for frictional and "k/c" for kinetic-collisional. The kinetic and collisional contributions will be defined based on Boltzmanns statistical approach as done in the gas kinetic theory, while the frictional contribution will be defined using the Plastic-Potential flow theory. Those two stress tensors have a deep difference in their nature: is a rate dependent stress tensor (i.e., dependent of the rate-of-deformation) and is a rate independent stress tensor (i.e., independent of the or rate-of-deformation). may be regarded as a pure viscous (dynamic) stress due to the momentum transfer during grain random motions and their collisions, which corresponds to the grain-inertia regime of Bagnold. This kinetic/collisional tensor is only important for "diluted", fluidized flows at high rate-of-strains. At very high concentrations and low rate-of-strain, collisions cannot be seen as instantaneous anymore, since grains suffer longer, permanent contacts in rubbing, rolling on each other. In this latter case, only the frictional stress tensor, , will be dominant.
In the following paragraphs, I shall define an comprehensive "unified" stress tensor for granular flows that accounts for frictions, collisions and kinetic. In doing so, I shall first define each stress tensor separately, and then combine them. I assume that compressive stress, compressive strain and their rates are taken positive.
Since grains suffer long and permanent contacts in rubbing, rolling on each other, a kinetic-collisional stress model based on the Boltzmanns integro-differential equation is irrelevant (this model assumes that binary collisions are instantaneous). Hence a stress tensor based on the mechanical law of friction must be developed. The main approach of such frictional granular flows is done through the application of the concepts of plasticity theory and critical state theory. The Mohr-Coulomb/von Mises yield function can be used in the Plastic Potential theory but, as demonstrated in A Review of Plastic-Frictional Stress, the Mohr-Coulomb/von Mises law only describes the onset of yielding, and is inadequate alone for describing the deformation of granular materials. In addition, the Mohr-Coulomb law can lead to physical inaccuracies, such as a infinite unbounded dilatancy phenomenon.
Assuming we know how to calculate the frictional stress terms (Pressure and viscous stress), then an ad hoc simple total frictional stress tensor would be:
where the superscript "f" stands for "frictional", I is the unit stress tensor, is the divergence of the velocity, is an isotropic normal stress (i.e., frictional pressure), D the rate-of-strain tensor of the solid phase, is the spherical part of the rate of-strain tensor (which represents all the deformations associated to the change of volume) and is the deviatoric part of the rate-of-strain tensor (which represents all the shear deformations), are the bulk and shear viscosities, both of them are a strong function of the granular concentration, the angle of internal friction, and the frictional Pressure. It is clear that the frictional stress will become important whenever grain contacts start to be long and permanent, at very high concentration. This threshold concentration is at about 50%-55% (% of the volume occupied by the grains). It is worth noting that the viscosity terms must be -somehow- a function of D-1. This is required since the frictional stress tensor must be independent of the rate-of-strain tensor (D) as required by the frictional theory. Depending on the chosen mathematical model for frictional flow, the bulk viscosity in Eq.2 may be equal to zero. See A Review of Plastic-Frictional Stress for more details on this and a complete demonstration of how to calculate the different terms in the frictional theories.
For now lets assume we know how to calculate all the terms in Eq.2, and lets move on to the kinetic-collisional part of the granular stress.
The kinetic granular theory is based on a deep analogy with the kinetic theory of dense gas (see also What is a Granular Medium?). It allows to define for a given solid component all the physical properties that owns any gas; namely, a shear viscosity (), a bulk viscosity (), a "thermodynamic pressure" (), a viscous stress tensor (), a free mean path (, where d is the diameter of the particles and is the volumetric concentration), so forth. The basic governing idea in the granular kinetic theory is that the grains are in a continuous and chaotic restlessness within the fluid. This chaotic random motion exists at very low concentration (due to friction between gas and particles, to gas turbulence, to pressure variation in the fluid, so forth) or at higher concentration (due to grain collisions). Taking the analogy of the gas, we may define a "temperature", , said, granular temperature, proportional to the mean quadratic velocity of the random motion of the grains:
where is the fluctuating energy per unit of mass due to the granular random motion and C is the velocity of this chaotic motion of grains. The symbol, , means an ensemble average of the square of the fluctuating velocity is considered. The analogy with the gas kinetic theory is striking (see also What is a Granular Medium?). Indeed, the same relation applies for defining the temperature of the gas phase (the ratio of the Boltzmann constant, kB, to the grain mass, ms, is equal to unity in standard granular kinetic theory, ). As done for the gas, it is assumed that particles oscillate about mean value in a very chaotic and isotropic manner. Hence, the velocity distribution follows a normal distribution about its mean (or mode), this velocity distribution profile is also called Maxwellian distribution. This is the reason of the number 3 in the definition of the granular temperature (average in the three directions of space). The general principle of dissipation in the granular kinetic theory is presented in Figure 2:
[Figure 2: Dissipation principles of Energy within a fluidized granular medium]
The production of this granular random motion is done mostly through the granular viscous dissipation. Afterwards, the inelastic nature of grain collisions will dissipate the granular fluctuating energy into enthalpy (or conventional "thermal heat"). Therefore, represents a transient energy state in the dissipation processes of fluidized granular materials. It is worth saying at this stage, that the kinetic granular theory applies as well for diluted system as highly loaded one. In general, the granular temperature is maximal in rather dilute situations. This indicates that we cannot fully approach granular system (dilute or not) without an appropriate granular model as presented here. Even though, the granular theory leads to modeling difficulties that must be overcome, no computer modeling can properly approach granular flows (even diluted ones) without an appropriate granular model. Also, It should be beard in mind that the granular temperature is not a measurement of the magnitude of collisions. It only measures the velocity of the random motion. Hence, it is only equal to zero (hence, granular kinetic theory is not valid in that case) when the granular system is "frozen", i.e., at very high concentration when there is no possibility for grains to oscillate. Theoretically, this can only happen close to maximum compaction concentration (about 64% for a single size solid component). Though, at a concentration of about 50% and higher, the granular temperature decreases as grains have less and less "room" to oscillate, at such high concentration another approach is needed, i.e., a frictional model.
As done for the gas, we simply define a "thermodynamic" pressure as proportional to the bulk density times the mean quadratic fluctuating velocity. Hence, it should be (only valid for a dilute system):
where and are respectively the volumetric granular concentration and the density of the grains. The bulk density (in kg/m3) is simply defined as:
And, in the same vein, we define the granular shear viscosity as (only valid for a dilute system):
You may notice that those definitions are the same as for a perfect gas. However, things are not that simple as some problems arise for fluidized granular flows. Indeed, the grains do not have a punctual negligible volume. They occupy a volumetric fraction of the Control Volume (which is not the case for low density gas molecule). Hence, the grain diameter can have an equivalent scale as their mean free path. Moreover, the collisions might be inelastic, enough to deeply modify the nature of the whole granular flow. Also, the gas phase can have some effects on this random motion of grains and on the granular viscosity. Therefore, we need to account these effects and particularly to prevent over-compaction. So, we must modify all these equations (static pressure, viscosities, granular heat conduction, so forth) by some functions (see Sébastien Dartevelles Ph.D.). These corrections are done through the following terms:
1- the radial distribution function, g0, which describes the probability of finding two particles in close proximity. It also corrects the probability of a collision for the effects of non-negligible volumes occupied by the particles. Its main goal is to prevent over-compaction of granular matter 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 increases for highly concentrated particulate system. Unfortunately this function is not very well-known for granular matter. Hence, there are many possible definitions.
2- (1 + e) or simply "e", the inelasticity factor. It accounts for the inelasticity of collisions between grains. It is only valid for slightly inelastic collisions but it is a first order parameter since it will deeply modify the granular flow properties (mainly the rheology) at high concentration. Because the granular collisions are slightly inelastic, it will change the granular temperature to thermal temperature (through the granular temperature dissipation, ). For perfectly elastic collisions, "e" would be equal to unity and would be equal to zero (no granular temperature loss). For a "typical" granular matter, "e" is taken between 0.9 and 0.99.
3- , "adjustment functions". They serve as correction functions for the granular shear viscosity and the conductivity of the granular temperature because the collision between grains is an inelastic phenomenon and because grains have a non-zero volume. These functions are equal to unity for dilute system but they asymptotically tend to high values or infinity for highly concentrated systems (the exact trends depend on the chosen definition of g0).
4- Since, the mean-free-path scale might be somehow equivalent to the grain-diameter scale (e.g., at high concentration), we should expect that Stokes hypothesis would not be valid anymore. For concentrated flow, we must also account for a non-zero bulk viscosity (). The bulk viscosity is simply proportional to the shear viscosity.
5- In most practical situations, grains are within a gas. The gas phase may have some effects upon grain behaviors. The gas phase may enhance or decrease the granular temperature (the fluctuating motions of the grains). But there is an even more complicate issue. As two particles get closer and closer to each other, the resistance of the gas between those two near-colliding particles dramatically increase. It may prevent the collision between particles. This is important as the granular viscosity is somehow affected by the amount of collision. We should keep in mind that early granular theories were developed without taking into account the gas phase. Therefore, applying such early theories straightforwardly to geophysical granular system is simply not an option (at least, in most volcanological and geophysical applications), some modifications are required.
The conservation of the fluctuating granular energy for the granular matter is given by:
This equation shows all the dependencies of the granular temperature (or the fluctuating Energy). The first term on the Left Hand Side (LHS) represents the net rate of fluctuating energy increase, the second term on the LHS represents the net rate of fluctuating energy transferred by convection into a fixed Control Volume. Focusing on the Right Hand Side (RHS), we have:
1. represents the work done by the surface forces, i.e., the granular viscous dissipation (), always positive (irreversible work) and the reversible work done by the granular pressure (), where ,
2. conduction of the granular temperature. Where q is the vector flux of the granular temperature and will be fully defined by a Fourier type law: , where k is the conductivity of the granular temperature, which strongly depends on the grain concentration, grain size and collisional inelasticity. It can be simply written as a function of the granular shear viscosity. Generally speaking, the higher the grain size or the higher the grain concentration, the better the granular temperature conduction,
3. represents the loss of granular heat due to the inelastic nature of grain collisions. is always positive, hence it represents a net loss of granular temperature. This term causes a transformation of granular temperature to "conventional" heat temperature. Generally speaking, the higher the concentration, the higher . The smaller the grain size, the higher . Hence, small grains tend to have a smaller granular temperature (or equivalently tend to have a higher granular temperature loss). If the granular system is highly compressive (e.g., due to a shock wave), then it tends to increase (i.e., increases the granular temperature loss). This parameter is therefore a key one, since small grain tend to have a smaller granular temperature and thereby, a smaller granular viscosity. A pyroclastic flow made of very fine small grains will be less viscous than a coarse-grained pyroclastic flow. Consequently, for the same concentration conditions, fine-grained pyroclastic surges should be less viscous than coarse-grained pyroclastic flows. However, other processes might interfere with this scheme, as diluted surge may be subject to strong gas turbulences and then have a higher gas viscosity, hence increasing the overall bulk viscosity of the flow,
4. It represents the net rate of transfer of fluctuating energy between the gas and solid phases. This term doesnt exist in earlier granular theories, which do not consider the gas phase and its effects on the granular temperature of the solid phase. The first term in this parenthesis represents a gain of granular fluctuating energy from the gas molecular fluctuating turbulent energy, while the second term represents a loss of the granular fluctuating energy due to the aerodynamic gas-solid friction, where , C is the fluctuating random velocity of the grains and Cg is the fluctuating random velocity of gas molecules, K is the gas-solid drag function (always positive term). Unfortunately, it is difficult to calculate the term , a comprehensive gas-particle turbulence model is needed but does not exist at the present time. But it is very likely that such a term can be neglected for large and heavy particle, unaffected by the random motion of the gas molecules.
The stress tensor that accounts for the stress induced by the collision and the kinetic random motion of the grains is defined as:
where I is the unit tensor, is the granular "thermodynamical" Pressure, is the granular viscous stress tensor, D is the rate-of-strain tensor of the granular phase, is the spherical part of the rate of-strain tensor (which represents all the deformations associated to the change of volume) and is the deviatoric part of the rate-of-strain tensor (which represents all the shear deformations). The term 4 represents viscous normal isotropic stress associated with any change of volume (no shear), term 5 is the pure viscous shear stress (no change of volume, only proportional to the deviator of the rate-of-strain tensor which is therefore traceless), and the whole term 3 represents the mechanical Pressure. The latter is not equal to the thermodynamical pressure represented by the term 1, since term 4 is not necessarily equal to zero.
It should be kept in mind this is only valid when collisions are short and the mean free path is large enough to allow grain to oscillate. is a strong function of the grain size (d), grain concentration () and also the granular temperature, which is also a strong non-linear function of many other parameters (see Eq. 7).
The total stress tensor will be the sum of the kinetic, collisional and frictional contributions:
Hence P and represent the whole contribution from collisional, kinetic and frictional effects on momentum. Notice that the granular temperature is not affected by frictions; also, dissipation effects due to friction forces are directly converted to thermal internal energy. As soon as the granular concentration hits 50%, friction starts to kick in alongside with kinetic and collisional effects. However, very quickly as the concentration increases (), the kinetic and collisional effects dwindle to nothing (because the granular temperature decreases as the mean free path goes to zero), while friction solely becomes predominant.
Now, we can understand in reading these equations altogether the effect of grain size and grain concentration upon granular flow dynamics.
Since many variables play at the same time, the reading of these equations might turn out to be difficult. Hence, for only a demonstrative purpose, lets imagine that the granular fluid is incompressible (, hence the granular bulk viscosity does not matter anymore) and made of only one grain size. The collisions are slightly inelastic (e<1).
Lets analyze first the effects of increasing grain concentration:
1- In a very dilute medium, we have and (see Eq. 7). Then, the only source for the granular temperature is the viscous dissipation () and increases as long as , which means that only the conduction of the granular heat can "cool" the granular fluid. In very dilute medium, the granular temperature can be very high, higher than a more concentrated medium. Consequently, the viscosity of such a fluid might be unexpectedly high, no matter how diluted the medium is. This indicates that even in dilute medium the granular rheology is not necessarily negligible.
2- If the solid concentration increases, the rheology becomes more complicate. A new term tend to decrease : the inelastic collisional energy dissipation function, . A competition starts between the source term (viscous dissipation) and the sink one (). For a concentration close to the maximal allowed (64%), will be equal to zero. Notice that the conduction of the granular temperature becomes more efficient as grain concentration increases.
3- Of course, the kinetic granular dynamic viscosity () depends not only of the granular temperature but also it depends on the grain concentration. Hence, the dynamic viscosity might display a complicate non-linear behavior:
4- The effect of the kinetic viscosity is predominant at moderately high concentration (due to the grain collisions) and at low concentration as well (due to the magnitude of the mean free path). Hence, it exists a range of intermediary concentration where the effects of viscosity are minimum (around 10%-15% ??).
5- Therefore, the less viscous pyroclastic flows are not necessarily the dilute ones but rather those who have higher intermediary values (). This might be an extremely important result in terms of pyroclastic flow dynamic that still needs to be demonstrated.
6- At a grain concentration of about 50%, friction slowly starts to kick in. With increasing concentration, the granular temperature decreases, and the kinetic viscosity dies away. But the nature of the stress totally changes as the granular flow moves from a rate-of-strain dependent stress to a rate-of-strain independent stress. The granular frictional pressure and viscosity eventually becomes infinite close to the maximum allowed concentration, at this stage we form a granular deposit rather than a granular flow.
Lets now analyzed the effects of grain size on the kinetic-collisional stress contribution:
1- For a same set of conditions, the smaller the grains, the higher . Therefore, the granular temperature tend to be lower for a flow made of very fine particles. In addition to this, the dynamic kinetic viscosity is lower for small particles. Consequently, a pyroclastic flow made of small particles is less viscous than a coarse-grained one.
2- The conduction of the granular temperature is less efficient in a fine-grained flow (in other words, it might cool or heat by conduction less efficiently in a fine grained-flow).
Clearly, according to the granular kinetic theory, the dynamic of any pyroclastic flow will strongly depend on both grain concentration and grain size.
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