en>Reynolds stress equation model |
|
| Line 1: |
Line 1: |
| The '''Boltzmann-Matano method''' is used to convert the [[partial differential equation]] resulting from [[Fick's law of diffusion]] into a more easily solved [[ordinary differential equation]], which can then be applied to calculate the [[diffusion coefficient]] as a function of concentration.
| | If you are looking for a specific plugin, then you can just search for the name of the plugin. Good luck on continue learning how to make a wordpress website. Change the site's theme and you have essentially changed the site's personality. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility. <br><br>These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>But before choosing any one of these, let's compare between the two. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. If you have any thoughts with regards to in which and how to use [http://topne.ws/WordpressBackupPlugin465043 wordpress backup], you can call us at the web-page. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. The animation can be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as an alternative gives some viewing enjoyment for the visitor of the internet site. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>It is the convenient service through which professionals either improve the position or keep the ranking intact. * Robust CRM to control and connect with your subscribers. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. By using Word - Press MLM websites or blogs, an online presence for you and your MLM company can be created swiftly and simply. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Your topic is going to be the basis of your site's name. |
| | |
| [[Ludwig Boltzmann]] worked on [[Adolf Eugen Fick|Fick]]'s second law to convert it into an ordinary differential equation, whereas [[Chujiro Matano]] performed experiments with diffusion couples and calculated the diffusion coefficients as a function of concentration in metal alloys.<ref>Matano, Chujiro. On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). Japanese Journal of Physics. Jan. 16, 1933.</ref> Specifically, Matano proved that the diffusion rate of A atoms into a B atom crystal lattice is a function of the amount of A atoms already in the B lattice.
| |
| | |
| The importance of the classic '''Boltzmann-Matano method''' consists in the ability to extract diffusivities from concentration-distance data. These methods, also known as ''inverse methods'', have both proven to be reliable, convenient and accurate with the assistance of modern computational techniques.
| |
| | |
| == Boltzmann’s Transformation ==
| |
| Boltzmann’s Transformation converts Fick's second law into an easily solvable ordinary differential equation.
| |
| Assuming a diffusion coefficient ''D'' that is in general a function of concentration ''c'', Fick's second law is:
| |
| :<math>\frac{\partial c}{\partial t} = \frac{\partial}{\partial x}\overbrace{\left[ D(c)\frac{\partial c}{\partial x} \right]}^\text{Flux}</math>
| |
| where ''t'' is time and ''x'' is distance.
| |
| | |
| Boltzmann's transformation consists in introducing a variable ''ξ'', defined as a combination of ''t'' and ''x'':
| |
| :<math>\xi=\frac{x}{2 \sqrt{t}}</math>
| |
| | |
| The partial derivatives of ''ξ'' are:
| |
| :<math>\frac{\partial \xi}{\partial t} = -\frac{x}{4 t^{3/2}} = -\frac{\xi}{2t}</math>
| |
| :<math>\frac{\partial \xi}{\partial x} = \frac{1}{2 \sqrt{t}}</math>
| |
| | |
| To introduce ''ξ'' into Fick's law, we express its partial derivatives in terms of ''ξ'', using the [[chain rule]]:
| |
| :<math>\frac{\partial c}{\partial t} = \frac{\partial c}{\partial \xi} \frac{\partial \xi}{\partial t} = -\frac{\xi}{2 t} \frac{\partial c}{\partial \xi}</math>
| |
| :<math>\frac{\partial c}{\partial x} = \frac{\partial c}{\partial \xi} \frac{\partial \xi}{\partial x} = \frac{1}{2 \sqrt{t}} \frac{\partial c}{\partial \xi}</math>
| |
| | |
| Inserting these expressions into Fick's law produces the following modified form:
| |
| :<math>-\frac{\xi}{2 t} \frac{\partial c}{\partial \xi} = \frac{1}{2 \sqrt{t}} \frac{\partial}{\partial x} \left[ D(c) \frac{\partial c}{\partial \xi} \right]</math>
| |
| Note how the time variable in the right-hand side could be taken outside of the partial derivative, since the latter regards only variable ''x''.
| |
| | |
| It is now possible to remove the last reference to ''x'' by using again the same chain rule used above to obtain ''∂ξ/∂x'':
| |
| :<math>-\frac{\xi}{2 t} \frac{\partial c}{\partial \xi} = \frac{1}{4 t} \frac{\partial}{\partial \xi} \left[ D(c) \frac{\partial c}{\partial \xi} \right]</math>
| |
| Because of the appropriate choice in the definition of ''ξ'', the time variable ''t'' can now also be eliminated, leaving ''ξ'' as the only variable in the equation, which is now an ordinary differential equation:
| |
| :<math>-2\xi \frac{\mathrm{d} c}{\mathrm{d} \xi} = \frac{\mathrm{d}}{\mathrm{d} \xi} \left[ D(c) \frac{\mathrm{d} c}{\mathrm{d} \xi} \right]</math>
| |
| This form is significantly easier to solve numerically, and one only needs to perform a back-substitution of ''t'' or ''x'' into the definition of ''ξ'' to find the value of the other variable.
| |
| | |
| === The Parabolic Law ===
| |
| Observing the previous equation, a [[trivial solution]] is found for the case d''c''/d''ξ''=0, that is when concentration is constant over ''ξ''.
| |
| This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (<math>x \propto \sqrt t</math>), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance (<math>t \propto x^2</math>); the square term gives the name ''[[Parabola|parabolic]] law''.<ref>See [http://dragon.unideb.hu/~zerdelyi/Diffusion-on-the-nanoscale/node4.html an animation of the parabolic law]</ref>
| |
| | |
| == Matano’s method ==
| |
| Chuijiro Matano applied Boltzmann's transformation to obtain a method to calculate diffusion coefficients as a function of concentration in metal alloys.
| |
| Two alloys with different concentration would be put into contact, and [[Annealing|annealed]] at a given temperature for a given time ''t'', typically several hours; the sample is then cooled to ambient temperature, and the concentration profile is virtually "frozen". The concentration profile ''c'' at time ''t'' can then be extracted as a function of the ''x'' coordinate.
| |
| | |
| In Matano's notation, the two concentrations are indicated as ''c<sub>L</sub>'' and ''c<sub>R</sub>'' (L and R for left and right, as shown in most diagrams), with the implicit assumption that ''c<sub>L</sub>'' > ''c<sub>R</sub>''; this is however not strictly necessary as the formulas hold also if ''c<sub>R</sub>'' is the larger one.
| |
| The initial conditions are:
| |
| :<math>c=c_L \qquad \forall x<0</math>
| |
| :<math>c=c_R \qquad \forall x>0</math>
| |
| Also, the alloys on both sides are assumed to stretch to infinity, which means in practice that they are large enough that the concentration at their other ends is unaffected by the transient for the entire duration of the experiment.
| |
| | |
| To extract ''D'' from Boltzmann's formulation above, we integrate it from ''ξ''=+∞, where ''c''=''c<sub>R</sub>'' at all times, to a generic ''ξ<sup>*</sup>''; we can immediately simplify d''ξ'', and with a change of variables we get:
| |
| :<math>-2\int_{c_R}^{c^*} \xi \mathrm{d}c = \int_{c=c_R}^{c=c^*} \mathrm{d}\left[ D(c)\left(\frac{\mathrm{d}c}{\mathrm{d}\xi}\right)\right]</math>
| |
| | |
| We can translate ''ξ'' back into its definition and bring the ''t'' terms out of the integrals, as ''t'' is constant and given as the time of annealing in the Matano method; on the right-hand side, extraction from the integral is trivial and follows from definition.
| |
| :<math>-\frac{1}{2 t} \int_{c_R}^{c^*} x \mathrm{d}c = \left[ D(c)\left(\frac{\mathrm{d}c}{\mathrm{d}x}\right)\right]_{c=c_R}^{c=c^*}</math>
| |
| | |
| We know that d''c''/d''x'' → 0 as ''c'' → ''c<sub>R</sub>'', that is the concentration curve "flattens out" when approaching the limit concentration value.
| |
| We can then rearrange:
| |
| :<math>D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}}</math>
| |
| | |
| Knowing the concentration profile ''c(x)'' at annealing time ''t'', and assuming it is invertible as ''x(c)'', we can then calculate the diffusion coefficient for all concentrations between ''c<sub>R</sub>'' and ''c<sub>L</sub>''.
| |
| | |
| === The Matano interface ===
| |
| The last formula has one significant shortcoming: no information is given about the reference according to which ''x'' should be measured.
| |
| It was not necessary to introduce one as Boltzmann's transformation worked fine without a specific reference for ''x''; it is easy to verify that the Boltzmann transformation holds also when using ''x''-''X<sub>M</sub>'' instead of plain ''x''.
| |
| | |
| ''X<sub>M</sub>'' is often indicated as the Matano interface, and is in general not coincident with ''x''=0: since ''D'' is in general variable with concentration ''c'', the concentration profile is not necessarily symmetric.
| |
| Introducing ''X<sub>M</sub>'' in the expression for ''D(c<sup>*</sup>)'' above, however, introduces a bias that appears to make the value of ''D'' completely arbitrary, depending on which ''X<sub>M</sub>'' we choose.
| |
| | |
| The values for ''X<sub>M</sub>'', however, are limited to a single one by physical meaningfulness. Since the denominator term d''c''/d''x'' goes to zero for ''c'' → ''c<sub>L</sub>'' (as the concentration profile flattens out), the integral in the numerator must also tend to zero in the same conditions, otherwise ''D(c<sub>L</sub>)'' would tend to infinity, which is not physically meaningful (note that, strictly speaking, this does not guarantee that ''D'' does not tend to infinity, but it is only a necessary condition).
| |
| The condition is then:
| |
| :<math>0 = \int^{c_L}_{c_R} (x-X_M) \mathrm{d}c</math>
| |
| :<math>X_M = \frac{1}{c_L-c_R} \int^{c_L}_{c_R} x \mathrm{d}c </math>
| |
| | |
| In other words, ''X<sub>M</sub>'' is the average position weighed on concentrations, and can be easily found from the concentration profile providing it is invertible to the form ''x(c)''.
| |
| | |
| == Sources ==
| |
| * M. E. Glicksman, ''Diffusion in Solids: Field Theory, Solid-State Principles, and Applications'', Wiley, New York, 2000.
| |
| * Matano, Chujiro. "On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System)". Japanese Journal of Physics. Jan. 16, 1933.
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| | |
| {{DEFAULTSORT:Boltzmann-Matano Analysis}}
| |
| [[Category:Diffusion]]
| |
If you are looking for a specific plugin, then you can just search for the name of the plugin. Good luck on continue learning how to make a wordpress website. Change the site's theme and you have essentially changed the site's personality. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility.
These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. Many times the camera is following Mia, taking in her point of view in almost every frame.
But before choosing any one of these, let's compare between the two. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. If you have any thoughts with regards to in which and how to use wordpress backup, you can call us at the web-page. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. The animation can be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as an alternative gives some viewing enjoyment for the visitor of the internet site. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them.
It is the convenient service through which professionals either improve the position or keep the ranking intact. * Robust CRM to control and connect with your subscribers. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL.
This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. By using Word - Press MLM websites or blogs, an online presence for you and your MLM company can be created swiftly and simply. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Your topic is going to be the basis of your site's name.