🔎Derivation of the Diffusion and Heat Equations from the Continuity Principle

Based on the derivation, the diffusion equation and the heat equation are fundamentally the same partial differential equation, both arising from applying a constitutive relationship to the continuity equation for the conservation of an intensive quantity $u$ (concentration or heat concentration). This relationship, known as Fick's first law (for diffusion) or Fourier's law (for heat), posits that the current density $\jmath$ is proportional to the negative gradient of the quantity ( $-\nabla u$ or $-\nabla T$ ), causing flow from regions of high concentration/temperature to low concentration/temperature. For homogeneous and isotropic materials, this yields the standard form $\partial_t u-D \nabla^2 u=\kappa$, where $D$ is the diffusivity or thermal diffusivity, and $\nabla^2$ is the Laplacian, modeling how a substance or heat spreads out over time. When the medium is in motion, the resulting current must also include a convective term, $J_{\text {convection }}=u v$, in addition to the diffusive/conductive current.

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Derivation of the Diffusion and Heat Equations from the Continuity Principle | Condensed Notesviadean on Notion