Diffusion equations based on generalized continuum mechanics and numerical analysis

We propose that an increasing value in entropy inequality is not a local value of entropy but an average value in a mesodomain. From this standpoint, new balance equations are obtained by applying the concept of generalized continuum mechanics to mass transfer. In the present paper, it is clarified that the influence of microscopic gradient of mass concentration should be included in Fick's first law and Fourier's law. Even if thermal effects on diffusion are neglected, the diffusion equations obtained here are simultaneous differential equations with two undetermined values, which coincide with the conventional diffusion equation in a special case. The conventional diffusion equation describing infinite velocity of propagation associated with diffusion entails a contradiction. The velocity of propagation defined here is shown in the results of numerical analysis for an extreme initial state. Consequently, it is indicated that the present theory gives an acceptable solution to the above problem of the conventional theory.