Coefficient de diffusion pdf

To learn more, view our Privacy Policy. To browse Academia. Log in with Facebook Log in with Google. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up. Download Free PDF. Fick's laws of diffusion. Embe Jule. A short summary of this paper. Fick's laws of diffusion Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in They can be used to solve for the diffusion coefficient, D.

Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation. Top: A single molecule Biological perspective moves around randomly. History Bottom: With an enormous number See also of solute molecules, randomness becomes undetectable: The solute Notes appears to move smoothly and References systematically from high- External links concentration areas to low- concentration areas.

Fick's first law Fick's first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient spatial derivative , or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low concentration across a concentration gradient.

It might thus be expressed in the unit m. D is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes—Einstein relation.

In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0. In chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential of this species.

When the Riccati system is used, conditions are established so that finite-time singularities might occur. We explore solution dynamics across multi-parameters. In the suplemen- tary material, we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions. Keywords and Phrases.

Introduction Most physical and biological systems are not homogeneous, in part due to fluctuations in en- vironmental conditions and the presence of nonuniform media. Reaction- diffusion equations play a fundamental role in a large number of models of heat diffusion and reaction processes in nonlinear acoustics [7], biology, chemistry, genetics and many other areas of research [1], [3], [13], [19], [59], [27] and [8]. The Riccati equations have played an important role in explicit solutions for Fisher and Burgers equations see [25], [45] and references therein.

In this paper, in order obtain the main results, we use a fundamental approach consisting of the use of similarity transformations and the solu- tions of Riccati-Ermakov systems with several parameters for the diffusion case [69] and [68].

Of Date: November 5, The consideration of parameters in this work is inspired by the work of E. The solutions presented by E. Ermakov systems for the dispersive case with solutions containing parameters [46] have been used successfully to construct solutions for the generalized harmonic oscillator with a hidden symmetry [47], [48], and they have also been used to present Galilei trans- formation, pseudoconformal transformation and others in a unified manner, see [48].

More recently they have been used in [49] to show spiral and breathing solutions and solutions with bending for the paraxial wave equation. One of the main results of this paper is to use similar techniques to provide solutions with parameters providing a control on the dynamics of solutions see Figures 1, 2 and 3 for reaction diffusion equations of the form GNLH and GBE. To this end, it is necessary to establish the conditions on the coefficients of the differential equations to satisfy Riccati-Ermakov systems for the diffusion case, which has different solutions compared to the dispersive case.

Once the transformations are established for standard models such as the Fisher equation or the KPP- equation, explicit global solutions proposed by Clarkson [14] can be used; these solutions include Jacobi elliptic functions and exponential and rational functions, see Table 1 for an extended list of examples.

Transformations to the Burgers equation will allow us to produce special exact solutions such as rational, triangular wave and N-wave type solutions. These results would help to test numerical methods in the study of numerical solutions and dynamics of singularities. Finite time blow-up for the nonlinear heat equation [28]- [39], [42], [52], [54] and pole dynamics for the Burgers equation have been studied extensively [16], [15], [20].

We will provide solutions with singularities, which should motivate further research in the dynamics of singularities. In general, the variable coefficients Burgers equation vcBE is not integrable, and there are not many exactly solvable models known for vcBE [60]- [66].

For an interesting application on nonlinear magnetosonic waves propagating perpendicular to a magnetic neutral sheet, see [61]. Examples of exact solutions include BE with time-dependent forcing [58] and with elastic forcing terms [55] and [24].

In this work we generalize the transformation presented in [24] where Langevin equations and the Hill equation were used to express the transformation. Instead, we will use what we have called the Riccati system; further, our solutions will show multiparameters. Table 2 shows several examples of families with explicit solutions and mutiparameters. This paper is organized in the following manner: In Section 2, we present Lemma 1 where con- ditions on the coefficients are established for equation GNLH to be transformed into the standard Fisher-KPP equation through a similarity transformation.

Further, conditions are given to obtain solutions with singularities or to avoid them. Examples of equations constructed using Ermakov systems with solutions without singularities are also explained. In Section 3, an alternative ap- proach to solve Riccati systems explicitly is presented.

Table 2 presents examples of equations with singularities. Conditions for the existence of solutions are presented. In Section 4, we will study explicit solutions with multiparameters for the variable coefficient Burgers equation GBE. Several examples with explicit solutions are presented in Tables 3 and 4.

We also provide an appendix where we recall solutions of Riccati-Ermakov systems for the diffusion case previously published [69]. Finally, we provide a supplementary file where our solutions are verified. Conditions are given to obtain solutions with singularities or to avoid them. Examples of equations constructed using Ermakov systems with solutions without singular- ities are also explained. Lemma 1. This system will be refered as the Riccati-Ermakov system.

Therefore, using the balance of the coefficients 2. Corollary 1. We use the remarkable solution of the equation 2. These toy examples show the control of the dynamics of the solutions as an application of our multiparameter approach. We have prepared a Mathematica file as supplemental material for this Section.

Also, all the formulas from the appendix have been verified previously in [44] and [68]. Exponential-type solutions for a nonautonomous reaction-diffusion model. Figure 1. Jacobi elliptic solutions for reaction diffusion equation 2.

Therefore, by Lemma 1 the differential equation 2. Parameter variation in the equation 2. If we make a change in the parameter in 2. However, equation 2. Our differential model Equation 2. In this section, we present an alternative approach to deal with the Riccati system and see how the dynamics of the solutions change with multiparameters. The solution of the particular Riccati system 3. Families of generalized Fisher-KPP equations with explicit solutions of the form 3.

Lemma 2. Examples of equations with singularities of the form 3. Remark 1. The solutions presented in Tables 1 and 2 are of the form 3. Jacobi elliptic-type solutions for a nonautonomous reaction-diffusion model.

Figure 2. Krischer, Kroll. Download references. You can also search for this author in PubMed Google Scholar. Reprints and Permissions. Perrin, B. Download citation. Issue Date : January Anyone you share the following link with will be able to read this content:.

Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative. Brown, R. Raharinaivo, A. Page, C. Download references. You can also search for this author in PubMed Google Scholar. Reprints and Permissions.

Hachani, L. Materials and Structures 24, — Download citation. Published : 01 May Issue Date : May Anyone you share the following link with will be able to read this content:.