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| '''Particle-laden flows''' refers to a class of [[Two-phase flow|two-phase fluid flow]], in which one of the phases is continuously connected (referred to as the continuous or carrier phase) and the other phase is made up of small, immiscible, and typically dilute particles (referred to as the dispersed or particle phase). Fine aerosol particles in air is an example of a particle-laden flow; the aerosols are the dispersed phase, and the air is the carrier phase.
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| The modeling of two-phase flows has a tremendous variety of engineering and scientific applications: pollution dispersion in the atmosphere, fluidization in [[combustion]] processes, aerosol deposition in spray medication, along with many others.
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| == Governing equations ==
| | The European Way of Life<br><br>Mountain biking can be a popular sport that involves a specialized bicycle designed for the rough issues that exist on off-road trails. This sport pits one cyclist against a field of other contenders all aiming to complete the course at the fastest possible speed. The terrain of your area features a major effect on the rate these cyclists can travel at. The conditions is frequently dangerous and hairy, therefore it is crucial that you mtb riders to understand that their mtb wheels are made to handle the impacts and abuse they are subjected to.<br><br>1. Get the right bike -- exactly what are you gonna use this bike for? Going on the trails within your local forest, residing in the town on paved roads all the time or maybe even going on a very light off-road track? Mountain bikes are popular however they are not the top pick for all. If you need a city bike obtain one, should you prefer a mtb obtain that instead, just avoid getting the incorrect bicycle.<br><br>Once you have chosen an appropriate bike, a cycling helmet which fits properly will probably be your most vital part of biking equipment to ensure safety, followed by clothing that will make you plainly visible to drivers, should you be cycling at night. When you are certain that there is a appropriate biking equipment, you will have to make certain that it's kept in proper condition. Maintenance of your bike will incorporate: maintaining proper inflation of tires and examining tires for damage, inspection of brake function and cleanliness, as brakes which might be covered in mud might not function properly. Be sure that your bike chain is clean and not obstructed.<br><br>Not only that, biking helps preserve nature. You are not using any varieties of gas therefore, you aren't polluting the surroundings. Biking enables you to talk with people. You go round the neighborhood biking and even meet other people who share the same passion. This is ideal for people that want to speak with people.<br><br>4. Cycling is therapeutic and might relieve stress. Riding bicycles is proven to be a therapeutic activity and much more and more people are taking up cycling for any serene, more peaceful form of exercise. Studies are beginning to show that cycling might help to decrease stress - particularly if driving beautiful scenic areas.<br><br>In case you loved this post and you wish to receive more details about hanging Bike Rack assure visit the web site. |
| The starting point for a mathematical description of almost any type of fluid flow is the classical set of [[Navier–Stokes equations]]. To describe particle-laden flows, we must modify these equations to account for the effect of the particles on the carrier, or vice versa, or both - a suitable choice of such added complications depend on a variety of the parameters, for instance, how dense the particles are, how concentrated they are, or whether or not they are chemically reactive. In most real world cases, the particles are very small and occur in low concentrations, hence the dynamics are governed primarily by the continuous phase. A possible way to represent the dynamics of the carrier phase is by the following modified Navier-Stokes momentum equation: | |
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| :<math> \frac{\partial \rho u_i}{dt} + \frac{\partial \rho u_i u_j}{\partial x_j} = - \frac{\partial P}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} + S_i, </math>
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| where <math>S_i</math> is a momentum source or sink term, arising from the presence of the particle phase. The above equation is an [[Eulerian reference frame|Eulerian]] equation, that is, the dynamics are understood from the viewpoint of a fixed point in space. The dispersed phase is typically (though not always) treated in a Lagrangian framework, that is, the dynamics are understood from the viewpoint of fixed particles as they move through space. A usual choice of momentum equation for a particle is:
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| :<math> \frac{d v_i}{dt} = \frac{1}{\tau_p} (u_i - v_i),</math>
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| where <math> u_i </math> represents the carrier phase velocity and <math> v_i </math> represents the particle velocity. <math> \tau_p </math> is the particle relaxation time, and represents a typical timescale of the particle's reaction to changes in the carrier phase velocity - loosely speaking, this can be thought of as the particle's inertia with respect to the fluid with contains it. The interpretation of the above equation is that particle motion is hindered by a drag force. In reality, there are a variety of other forces which act on the particle motion (such as gravity, Basset history and added mass) – as described through for instance the [[Basset–Boussinesq–Oseen equation]]. However, for many physical examples, in which the density of the particle far exceeds the density of the medium, the above equation is sufficient.<ref>{{cite journal|last=Maxey|first=M. R.|coauthors=J. J. Riley|journal=Phys. Fluids|year=1983|volume=26|pages=883–889}}</ref> A typical assumption is that the particles are spherical, in which case the drag is modeled using [[Stokes' law|Stokes drag]] assumption:
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| :<math> \tau_p = \frac{\rho_p d_p^2}{18 \mu}. </math>
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| Here <math>d_p</math> is the particle diameter, <math>\rho_p</math>, the particle density and <math>\mu</math>, the dynamic viscosity of the carrier phase. More sophisticated models contain the correction factor:
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| :<math> \tau_p = \frac{\rho_p d_p^2}{18 \mu} (1 + 0.15 Re_p^{0.687})^{-1}, </math>
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| where <math>Re_p</math> is the particle Reynolds number, defined as:
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| :<math> Re_p = \frac{| \vec{u} - \vec{v} | d_p}{\nu}. </math>
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| == Coupling ==
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| If the mass fraction of the dispersed phase is small, then ''one-way coupling'' between the phases is a reasonable assumption; that is, the dynamics of the particle phase are affected by the carrier phase, but the reverse is not the case. However if the mass fraction of the dispersed phase is large, the interaction of the dynamics between the two phases must be considered - this is ''two-way coupling''.
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| A problem with the Lagrangian treatment of the dispersed phase is that once the number of particles becomes large, it may require a prohibitive amount of computational power to track a sufficiently large sample of particles required for statistical convergence. In addition, if the particles are sufficiently light, they behave essentially like a second fluid. In this case, an Eulerian treatment of the dispersed phase is sensible.
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| == Modeling ==
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| Like all fluid dynamics-related disciplines, the modelling of particle-laden flows is an enormous challenge for researchers - this is because most flows of practical interest are [[Turbulence|turbulent]].
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| [[Direct numerical simulation]]s (DNS) for single-phase flow, let alone two-phase flow, are computationally very expensive; the computing power required for models of practical engineering interest are far out of reach. Since one is often interested in modeling only large scale qualitative behavior of the flow, a possible approach is to decompose the flow velocity into mean and fluctuating components, by the [[Reynolds-averaged Navier–Stokes equations|Reynolds-averaged Navier-Stokes]] (RANS) approach. A compromise between DNS and RANS is [[large eddy simulation]] (LES), in which the small scales of fluid motion are modeled and the larger, resolved scales are simulated directly.
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| Experimental observations, as well as DNS indicate that an important phenomenon to model is preferential concentration. Particles (particularly those with Stokes number close to 1) are known to accumulate in regions of high shear and low vorticity (such as turbulent [[boundary layer]]s), and the mechanisms behind this phenomenon are not well understood. Moreover, particles are known to migrate down turbulence intensity gradients (this process is known as [[turbophoresis]]). These features are particularly difficult to capture using RANS or LES-based models since too much time-varying information is lost.
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| Due to these difficulties, existing turbulence models tend to be ''ad hoc'', that is, the range of applicability of a given model is usually suited toward a highly specific set of parameters (such as geometry, dispersed phase mass loading and particle reaction time), and are also restricted to low [[Reynolds number]]s (whereas the Reynolds number of flows of engineering interest tend to be very high).
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| <!--- Categories --->
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| == Further reading ==
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| * Mashayek, F. and Pandya, R. V. R. (1921), ''Progress in Energy and Combustion Science'' '''20''', 196, 196–212.
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| == References ==
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| <!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
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| {{Reflist}}
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| [[Category:Fluid mechanics]]
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| [[Category:Fluid dynamics]]
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The European Way of Life
Mountain biking can be a popular sport that involves a specialized bicycle designed for the rough issues that exist on off-road trails. This sport pits one cyclist against a field of other contenders all aiming to complete the course at the fastest possible speed. The terrain of your area features a major effect on the rate these cyclists can travel at. The conditions is frequently dangerous and hairy, therefore it is crucial that you mtb riders to understand that their mtb wheels are made to handle the impacts and abuse they are subjected to.
1. Get the right bike -- exactly what are you gonna use this bike for? Going on the trails within your local forest, residing in the town on paved roads all the time or maybe even going on a very light off-road track? Mountain bikes are popular however they are not the top pick for all. If you need a city bike obtain one, should you prefer a mtb obtain that instead, just avoid getting the incorrect bicycle.
Once you have chosen an appropriate bike, a cycling helmet which fits properly will probably be your most vital part of biking equipment to ensure safety, followed by clothing that will make you plainly visible to drivers, should you be cycling at night. When you are certain that there is a appropriate biking equipment, you will have to make certain that it's kept in proper condition. Maintenance of your bike will incorporate: maintaining proper inflation of tires and examining tires for damage, inspection of brake function and cleanliness, as brakes which might be covered in mud might not function properly. Be sure that your bike chain is clean and not obstructed.
Not only that, biking helps preserve nature. You are not using any varieties of gas therefore, you aren't polluting the surroundings. Biking enables you to talk with people. You go round the neighborhood biking and even meet other people who share the same passion. This is ideal for people that want to speak with people.
4. Cycling is therapeutic and might relieve stress. Riding bicycles is proven to be a therapeutic activity and much more and more people are taking up cycling for any serene, more peaceful form of exercise. Studies are beginning to show that cycling might help to decrease stress - particularly if driving beautiful scenic areas.
In case you loved this post and you wish to receive more details about hanging Bike Rack assure visit the web site.