Asymptotic Theory of Classical Impurity Transport in Inhomogeneous Media. Fermat’s Principle

Abstract

An asymptotic theory of impurity transport based on advection–diffusion in media with large-scale inhomogeneities is developed. The expression for the concentration is reduced to one-dimensional integrals along the characteristic line called the trajectory of the concentration signal. The trajectory itself is determined from the variational principle—an analog of Fermat’s principle in geometrical optics—which leads to a first-order ordinary differential equation for the unit tangent vector to the trajectory. The asymptotic theory is applicable at distances from the impurity source much larger than the size of the main distribution region of the impurity.

APPENDIX

The purpose of this section is to find, in the Laplace representation, the impurity concentration c_p(r) satisfying Eq. (1) with the initial condition (3) for coordinate-independent diffusivity and advection velocity:

$$D({\mathbf{r}}) = D(0),\quad {\mathbf{u}}({\mathbf{r}}) = {\mathbf{u}}(0).$$

(A.1)

The additional transition to the Fourier representation

$${{c}_{{p{\mathbf{k}}}}} = \int {{{d}^{3}}r{{c}_{p}}({\mathbf{r}}){{e}^{{ - i({\mathbf{kr}})}}}} $$

reduces the problem to an algebraic one, whose solution is given by the expression

$${{c}_{{p{\mathbf{k}}}}} = \frac{N}{{p + i({\mathbf{u}}(0){\mathbf{k}}) + D(0){{k}^{2}}}}.$$

(A.2)

Applying the inverse Fourier transformation to this expression gives

$${{c}_{p}}({\mathbf{r}}) = \int {\frac{{{{d}^{3}}k}}{{{{{(2\pi )}}^{3}}}}\frac{N}{{p + i({\mathbf{u}}(0){\mathbf{k}}) + D(0){{k}^{2}}}}{{e}^{{i({\mathbf{kr}})}}},} $$

(A.3)

hence, after passing to the new integration variable k′ = k + iu(0)/2D(0), we obtain

$$\begin{gathered} {{c}_{p}}({\mathbf{r}}) = \int {\frac{{{{d}^{3}}k{\kern 1pt} '}}{{{{{(2\pi )}}^{3}}}}} \frac{N}{{p + {{{\left( {\frac{{{\mathbf{u}}(0)}}{{2D(0)}}} \right)}}^{2}} + D(0)k{{'}^{2}}}} \\ \times \exp \left[ {\frac{{({\mathbf{u}}(0){\mathbf{r}})}}{{2D(0)}} + i({\mathbf{k}}{\kern 1pt} '{\kern 1pt} {\mathbf{r}})} \right]. \\ \end{gathered} $$

(A.4)

Integrating first with respect to the angular coordinates of the vector k and then, using the residue theory, with respect to the absolute value of this vector, we obtain

$$\begin{gathered} {{c}_{p}}({\mathbf{r}}) \\ = \,\frac{N}{{4\pi D(0)r}}{\text{exp}}\left[ {\frac{{({\mathbf{u}}(0){\mathbf{r}})}}{{2D(0)}}\, - \,r\sqrt {{{{\left( {\frac{{{\mathbf{u}}(0)}}{{2D(0)}}} \right)}}^{2}}\, + \,\frac{p}{{D(0)}}} } \right]. \\ \end{gathered} $$

(A.5)

Hence we find the preexponential factor in the expression for concentration under condition (A.1):

$${{A}_{p}}({\mathbf{r}}) = \frac{N}{{4\pi D(0)r}}.$$

(A.6)