The diffusion equation assumptions and applications

Fick s first law of diffusion [56], which arises as a consequence of random unbiased movement of particles, states that the mass flux of particles in a continuous concentration field is proportional to the negative of the spatial concentration gradient. Fick s first law in a general form is [39] 

Here f is the mass flux density and c(x, t) is the concentration of a solute, continuously distributed in the spatial field x. For this general anisotropic case B is the positive definite diffusion matrix.1 

Off-diagonal entries of B arise from the fact that the coordinate system in which Equations (8.1) and (8.2) are expressed does not necessarily coincide with the principle directions of anisotropic diffusion defined by B. The principle diffusion directions correspond to the eigenvectors of B, with the highest rate of diffusion occurring in the direction associated with the largest eigenvalue of B. To see this we introduce the coordinate transformation = Rx (or = RijXj). Application 

1 Compare Equation (8.1) to Equation (3.49), which applies to the isotropic case. In the case of Equation (8.1), diffusive transport proceeds at different rates in different directions. 

2 Equation (8.2) is also called the heat equation and was introduced by Fourier to simulate the distribution of temperature in solids. In mathematics it is characterized as a parabolic differential equation. 

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