Английская Википедия:Convection–diffusion equation

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Шаблон:Short description The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation,[1] or (generic) scalar transport equation.[2]

Equation

General

Файл:ADE contamination plume.gif
A contaminant plume moving in porous media via the Convection-Diffusive Equation(also called Advective-Diffusive Equation) (time stamps of 1 day, 85 days, 462 days and 674 days) with advection as primary transport mechanism.

The general equation in conservative form is[3][4] <math display="block">\frac{\partial c}{\partial t} = \mathbf{\nabla} \cdot (D \mathbf{\nabla} c - \mathbf{v} c) + R</math> where

For example, if Шаблон:Mvar is the concentration of a molecule, then Шаблон:Mvar describes how the molecule can be created or destroyed by chemical reactions. Шаблон:Mvar may be a function of Шаблон:Mvar and of other parameters. Often there are several quantities, each with its own convection–diffusion equation, where the destruction of one quantity entails the creation of another. For example, when methane burns, it involves not only the destruction of methane and oxygen but also the creation of carbon dioxide and water vapor. Therefore, while each of these chemicals has its own convection–diffusion equation, they are coupled together and must be solved as a system of simultaneous differential equations.

Understanding the current density terms involved

The convection-diffusion equation is a particular example of conservation equation. A conservation equation has the general form: <math display="block">\frac{\partial c}{\partial t} + \mathbf{\nabla} \cdot \mathbf j_c = R</math> Where jc is the current density term associated to the variable of interest Шаблон:Mvar.

In a convection-diffusion equation, the current density of the quantity Шаблон:Mvar is the sum of two terms: <math display="block">\mathbf j_c = - D \mathbf{\nabla} c + \mathbf{v} c</math>

  • The first, Шаблон:Math, describes diffusion according to Fick's law. Imagine that Шаблон:Mvar is the concentration of a chemical. When concentration is low somewhere compared to the surrounding areas (e.g. a local minimum of concentration), the substance will diffuse in from the surroundings, so the concentration will increase. Conversely, if concentration is high compared to the surroundings (e.g. a local maximum of concentration), then the substance will diffuse out and the concentration will decrease. The net diffusion is proportional to the Laplacian (or second derivative) of concentration if the diffusivity Шаблон:Mvar is a constant.
  • The second contribution, Шаблон:Math, describes convection (or advection). For example, in the continuity equation it is present only this term in the current density. Imagine standing on the bank of a river, measuring the water's salinity (amount of salt) each second. Upstream, somebody dumps a bucket of salt into the river. A while later, you would see the salinity suddenly rise, then fall, as the zone of salty water passes by. Thus, the concentration at a given location can change because of the flow.

Common simplifications

In a common situation, the diffusion coefficient is constant, there are no sources or sinks, and the velocity field describes an incompressible flow (i.e., it has zero divergence). Then the formula simplifies to:[5][6][7] <math display="block">\frac{\partial c}{\partial t} = D \nabla^2 c - \mathbf{v} \cdot \nabla c. </math>

In this form, the convection–diffusion equation combines both parabolic and hyperbolic partial differential equations.

In this case the equation can be put in the simple convective form: <math display="block">\frac{D c}{D t} = D \nabla^2 c </math>

where the derivative of the left had side is the material derivative of the variable c. In non-interacting material, Шаблон:Mvar (for example, when temperature is close to absolute zero, dilute gas has almost zero mass diffusivity), hence the transport equation is simply the continuity equation: <math display="block">\frac{\partial c}{\partial t} + \mathbf{v} \cdot \nabla c=0. </math>

Using Fourier transform in both temporal and spatial domain (that is, with integral kernel <math>e^{j\omega t+j\mathbf{k}\cdot\mathbf{x}}</math>), its characteristic equation can be obtained: <math display="block">j\omega \tilde c+\mathbf{v}\cdot j \mathbf{k} \tilde c=0 \rightarrow \omega=-\mathbf{k}\cdot \mathbf{v}, </math> which gives the general solution: <math display="block">c = f(\mathbf{x}-\mathbf{v}t), </math> where <math>f </math> is any differentiable scalar function. This is the basis of temperature measurement for near Bose–Einstein condensate[8] via time of flight method.[9]

Stationary version

The stationary convection–diffusion equation describes the steady-state behavior of a convective-diffusive system. In a steady state, Шаблон:Math, so the equation to solve becomes the second order equation: <math display="block"> \nabla \cdot (- D \nabla c + \mathbf{v} c) = R.</math>

One dimensional case

In one dimension the spatial gradient operator is simply:

<math display="block"> \nabla = \frac d {dx}</math>

so the equation to solve becomes the single-variable second order equation: <math display="block"> \frac d {dx} \left(- D(x) \frac {dc(x)}{dx} + v(x) c(x) \right) = R(x)</math>

Which can be integrated one time in the space variable x to give:

<math display="block"> D(x) \frac {dc(x)}{dx} - v(x) c(x) = - \int_x R(x') dx'</math>

Where D is not zero, this is an inhomogeneous first-order linear differential equation with variable coefficients in the variable c(x):

<math display="block">y'(x) = f(x) y(x) + g(x).</math> where the coefficients are: <math display="block">f(x) = \frac{v(x)}{D(x)}</math> and: <math display="block">g(x) = - \frac 1 {D(x)} \int_x R(x') dx'</math>

This equation has in fact a relatively simple analytical solution (see the link above to first-order linear differential equation with variable coefficients).

On the other hand, in the positions x where D=0, the first-order diffusion term disappears and the solution becomes simply the ratio:

<math display="block"> c(x) = \frac 1 {v(x)} \int_x R(x') dx'</math>

Derivation

The convection–diffusion equation can be derived in a straightforward way[4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: <math display="block"> \frac{\partial c}{\partial t} + \nabla\cdot\mathbf{j} = R, </math> where Шаблон:Math is the total flux and Шаблон:Mvar is a net volumetric source for Шаблон:Mvar. There are two sources of flux in this situation. First, diffusive flux arises due to diffusion. This is typically approximated by Fick's first law: <math display="block">\mathbf{j}_\text{diff} = -D \nabla c</math> i.e., the flux of the diffusing material (relative to the bulk motion) in any part of the system is proportional to the local concentration gradient. Second, when there is overall convection or flow, there is an associated flux called advective flux: <math display="block">\mathbf{j}_\text{adv} = \mathbf{v} c</math> The total flux (in a stationary coordinate system) is given by the sum of these two: <math display="block">\mathbf{j} = \mathbf{j}_\text{diff} + \mathbf{j}_\text{adv} = -D \nabla c + \mathbf{v} c.</math> Plugging into the continuity equation: <math display="block"> \frac{\partial c}{\partial t} + \nabla\cdot \left(-D \nabla c + \mathbf{v} c \right) = R. </math>

Complex mixing phenomena

In general, Шаблон:Mvar, Шаблон:Math, and Шаблон:Mvar may vary with space and time. In cases in which they depend on concentration as well, the equation becomes nonlinear, giving rise to many distinctive mixing phenomena such as Rayleigh–Bénard convection when Шаблон:Math depends on temperature in the heat transfer formulation and reaction–diffusion pattern formation when Шаблон:Mvar depends on concentration in the mass transfer formulation.

Velocity in response to a force

In some cases, the average velocity field Шаблон:Math exists because of a force; for example, the equation might describe the flow of ions dissolved in a liquid, with an electric field pulling the ions in some direction (as in gel electrophoresis). In this situation, it is usually called the drift–diffusion equation or the Smoluchowski equation,[1] after Marian Smoluchowski who described it in 1915[10] (not to be confused with the Einstein–Smoluchowski relation or Smoluchowski coagulation equation).

Typically, the average velocity is directly proportional to the applied force, giving the equation:[11][12] <math display="block">\frac{\partial c}{\partial t} = \nabla \cdot (D \nabla c) - \nabla \cdot \left( \zeta^{-1} \mathbf{F} c \right) + R</math> where Шаблон:Math is the force, and Шаблон:Mvar characterizes the friction or viscous drag. (The inverse Шаблон:Math is called mobility.)

Derivation of Einstein relation

Шаблон:Main

When the force is associated with a potential energy Шаблон:Math (see conservative force), a steady-state solution to the above equation (i.e. Шаблон:Math) is: <math display="block">c \propto \exp \left( -D^{-1} \zeta^{-1} U \right)</math> (assuming Шаблон:Mvar and Шаблон:Mvar are constant). In other words, there are more particles where the energy is lower. This concentration profile is expected to agree with the Boltzmann distribution (more precisely, the Gibbs measure). From this assumption, the Einstein relation can be proven:[12] <math display="block">D \zeta = k_\mathrm{B} T.</math>

Smoluchowski convection-diffusion equation

The Smoluchowski convective-diffusion equation is a stochastic (Smoluchowski) diffusion equation with an additional convective flow-field,[13] <math display="block">\frac{\partial c}{\partial t} = \nabla \cdot (D \nabla c) - \mathbf{\nabla} \cdot (\mathbf{v} c) - \nabla \cdot \left( \zeta^{-1} \mathbf{F} c \right)</math> In this case, the force Шаблон:Math describes the conservative interparticle interaction force between two colloidal particles or the intermolecular interaction force between two molecules in the fluid, and it is unrelated to the externally imposed flow velocity Шаблон:Math. The steady-state version of this equation is the basis to provide a description of the pair distribution function (which can be identified with Шаблон:Mvar) of colloidal suspensions under shear flows.[13]

An approximate solution to the steady-state version of this equation has been found using the method of matched asymptotic expansions.[14] This solution provides a theory for the transport-controlled reaction rate of two molecules in a shear flow, and also provides a way to extend the DLVO theory of colloidal stability to colloidal systems subject to shear flows (e.g. in microfluidics, chemical reactors, environmental flows). The full solution to the steady-state equation, obtained using the method of matched asymptotic expansions, has been developed by Alessio Zaccone and L. Banetta to compute the pair distribution function of Lennard-Jones interacting particles in shear flow[15] and subsequently extended to compute the pair distribution function of charge-stabilized (Yukawa or Debye–Hückel) colloidal particles in shear flows.[16]

As a stochastic differential equation

The convection–diffusion equation (with no sources or drains, Шаблон:Math) can be viewed as a stochastic differential equation, describing random motion with diffusivity Шаблон:Mvar and bias Шаблон:Math. For example, the equation can describe the Brownian motion of a single particle, where the variable Шаблон:Mvar describes the probability distribution for the particle to be in a given position at a given time. The reason the equation can be used that way is because there is no mathematical difference between the probability distribution of a single particle, and the concentration profile of a collection of infinitely many particles (as long as the particles do not interact with each other).

The Langevin equation describes advection, diffusion, and other phenomena in an explicitly stochastic way. One of the simplest forms of the Langevin equation is when its "noise term" is Gaussian; in this case, the Langevin equation is exactly equivalent to the convection–diffusion equation.[12] However, the Langevin equation is more general.[12]

Numerical solution

Шаблон:Main The convection–diffusion equation can only rarely be solved with a pen and paper. More often, computers are used to numerically approximate the solution to the equation, typically using the finite element method. For more details and algorithms see: Numerical solution of the convection–diffusion equation.

Similar equations in other contexts

The convection–diffusion equation is a relatively simple equation describing flows, or alternatively, describing a stochastically-changing system. Therefore, the same or similar equation arises in many contexts unrelated to flows through space.

  • It is formally identical to the Fokker–Planck equation for the velocity of a particle.
  • It is closely related to the Black–Scholes equation and other equations in financial mathematics.[17]
  • It is closely related to the Navier–Stokes equations, because the flow of momentum in a fluid is mathematically similar to the flow of mass or energy. The correspondence is clearest in the case of an incompressible Newtonian fluid, in which case the Navier–Stokes equation is: <math display="block">\frac{\partial \mathbf{M}}{\partial t} = \mu \nabla^2 \mathbf{M} -\mathbf{v} \cdot \nabla \mathbf{M} + (\mathbf{f}-\nabla P)</math>

where Шаблон:Math is the momentum of the fluid (per unit volume) at each point (equal to the density Шаблон:Mvar multiplied by the velocity Шаблон:Math), Шаблон:Mvar is viscosity, Шаблон:Mvar is fluid pressure, and Шаблон:Math is any other body force such as gravity. In this equation, the term on the left-hand side describes the change in momentum at a given point; the first term on the right describes the diffusion of momentum by viscosity; the second term on the right describes the advective flow of momentum; and the last two terms on the right describes the external and internal forces which can act as sources or sinks of momentum.

In semiconductor physics

Файл:Diffusion center.gif
As carriers are generated (green:electrons and purple:holes) due to light shining at the center of an intrinsic semiconductor, they diffuse towards two ends. Electrons have higher diffusion constant than holes leading to fewer excess electrons at the center as compared to holes.

In semiconductor physics, this equation is called the drift–diffusion equation. The word "drift" is related to drift current and drift velocity. The equation is normally written:[18] <math display="block">\begin{align} \frac{\mathbf{J}_n}{-q} &= - D_n \nabla n - n \mu_n \mathbf{E} \\ \frac{\mathbf{J}_p}{q} &= - D_p \nabla p + p \mu_p \mathbf{E} \\ \frac{\partial n}{\partial t} &= -\nabla \cdot \frac{\mathbf{J}_n}{-q} + R \\ \frac{\partial p}{\partial t} &= -\nabla \cdot \frac{\mathbf{J}_p}{q} + R \end{align}</math> where

The diffusion coefficient and mobility are related by the Einstein relation as above: <math display="block">\begin{align} D_n &= \frac{\mu_n k_\mathrm{B} T}{q}, \\ D_p &= \frac{\mu_p k_\mathrm{B} T}{q}, \end{align}</math> where Шаблон:Math is the Boltzmann constant and Шаблон:Mvar is absolute temperature. The drift current and diffusion current refer separately to the two terms in the expressions for Шаблон:Math, namely: <math display="block">\begin{align} \frac{\mathbf{J}_{n,\text{drift}}}{-q} &= - n \mu_n \mathbf{E}, \\ \frac{\mathbf{J}_{p,\text{drift}}}{q} &= p \mu_p \mathbf{E}, \\ \frac{\mathbf{J}_{n,\text{diff}}}{-q} &= - D_n \nabla n, \\ \frac{\mathbf{J}_{p,\text{diff}}}{q} &= - D_p \nabla p. \end{align}</math>

This equation can be solved together with Poisson's equation numerically.[19]

An example of results of solving the drift diffusion equation is shown on the right. When light shines on the center of semiconductor, carriers are generated in the middle and diffuse towards two ends. The drift–diffusion equation is solved in this structure and electron density distribution is displayed in the figure. One can see the gradient of carrier from center towards two ends.

See also

Шаблон:Div col

Шаблон:Div col end

References

Шаблон:Reflist

Further reading

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ 1,0 1,1 Шаблон:Cite journal See equation (312)
  2. ↑ Шаблон:Cite book
  3. ↑ Шаблон:Cite book
  4. ↑ 4,0 4,1 Шаблон:Cite web
  5. ↑ Шаблон:Cite book
  6. ↑ Шаблон:Cite book
  7. ↑ Шаблон:Cite book
  8. ↑ Шаблон:Cite arXiv
  9. ↑ Шаблон:Cite journal
  10. ↑ Шаблон:Cite journal
  11. ↑ Шаблон:Cite web
  12. ↑ 12,0 12,1 12,2 12,3 Шаблон:Cite book
  13. ↑ 13,0 13,1 Шаблон:Cite book
  14. ↑ Шаблон:Cite journal
  15. ↑ Шаблон:Cite journal
  16. ↑ Шаблон:Cite journal
  17. ↑ Шаблон:Cite journal
  18. ↑ Шаблон:Cite journal
  19. ↑ Шаблон:Cite journal
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