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**Unformatted text preview: **In this manner, we arrive at the basic conservation law relating the heat energy density ε and the heat flux vector w . As in our one-dimensional model, cf. (4.3), the heat energy density ε ( t, x, y ) is pro- portional to the temperature, so ε ( t, x, y ) = σ ( x, y ) u ( t, x, y ) , where σ ( x, y ) = ρ ( x, y ) χ ( x, y ) (12 . 9) is the product of the density and the heat capacity of the material at the point ( x, y ) ∈ Ω. Combining this with the Fourier Law (12.6) and the energy balance equation (12.9) leads to the general two-dimensional diffusion equation ∂u ∂t = 1 σ ∇ · ( κ ∇ u ) (12 . 10) governing the thermodynamics of an isotropic medium in the absence of external heat sources or sinks. In full detail, this second order partial differential equation is ∂u ∂t = 1 σ ( x, y ) bracketleftbigg ∂ ∂x parenleftbigg κ ( x, y ) ∂u ∂x parenrightbigg + ∂ ∂y parenleftbigg κ ( x, y ) ∂u ∂y parenrightbiggbracketrightbigg . (12 . 11) Such diffusion equations are also used to model movements of populations, e.g., bacte- ria in a petri dish or wolves in the Canadian Rockies, [ 99 , 103 ]. Here the solution u ( t, x, y ) represents the population density near position ( x, y ) at time t , which diffuses over the do- main due to random motions of the individuals. Similar diffusion processes model the mixing of chemical reagents in solutions. On the other hand, convection due to fluid mo- tion combined with chemical reactions lead to the more general class of reaction–diffusion and convective–diffusion equations , [ 128 ]....

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