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The ultimate website for understanding granular flows | |
![]() An Overview of Granular Theories |
You will find the basic facts about Granular Flows - no details -. Detail is a matter of my current Ph.D. research and I will not show that here. If you wanna know more just email me or feel free to ask in the Volcano Discussion Forum. This general overview should help you to understand the modeling results and their interpretations that will be presented in this Granular Volcano Group Web Site. I purposely erased all the bibliographical references and detailed equations to keep the text simple and easy to read. If you have reached this page, please, be aware that this whole site may be better seen at the following urls: Those new urls will lead you to a faster, non-commercial, and pop-up free website. These are our official url addresses. Please, update your bookmarks. If you wish, you may see this specific page on the new website here: __________________________ You will find herewith: Please, dont forget to sign our Granular Volcano Guest Book. Done and updated by Sébastien Dartevelle, The WebMaster, June 29th, 2002. |
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.
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![]() [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:
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. II. Frictional Stress Formulation
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,
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. III. Kinetic-Collisional Stress Formulation
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 (
where
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![]() [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, III.2. Granular kinetic Theory Facts: Granular Pressure and Granular Viscosity
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, 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,
3-
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
(
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. III.3. Conservation of the Granular Temperature (or the Granular Fluctuating Energy)
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
(
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:
3. represents the loss of granular heat due to the inelastic nature of grain collisions.
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
III.4. Kinetic-collisional Stress Tensor
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,
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.
The total stress tensor will be the sum of the kinetic, collisional and frictional contributions:
Hence P and
Now, we can understand in reading these equations altogether the effect of grain size and grain concentration upon granular flow dynamics. V. Practical Application to Granular Flow: Grain concentration and Grain-size effect
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
(
Lets analyze first the effects of increasing grain concentration:
1- In a very dilute medium, we have
2- If the solid concentration increases, the rheology becomes more complicate.
A new term tend to decrease
3- Of course, the kinetic granular dynamic viscosity
(
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
(
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
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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