Coupled non-linear diffusion under periodic boundary conditions

  • Roger Shepherd

    Student thesis: Doctoral Thesis

    Abstract

    A mathematical investigation of the simultaneous diffusion of heat and moisture in the non-linear medium of soil is presented. In such a porous medium, the evolution of each diffusing agent is determined both by its own momentary distribution and by the distribution of each associated diffusing agent. Mutual destructive and constructive feedback occurs. Understanding of this complex system assists in the management of soil in adverse conditions. Equally, the governing equations contain features central to the concerns of contemporary mathematics.

    The controlling equations are derived from basic physical principles. The resulting simultaneous, non-linear partial differential equations have formed the basis of recent work. Relevant research is reviewed in the areas of laboratory and field experimentation, computer simulation, and analytical and numerical approaches. The equations have been supposed analytically intractable for any generality of coupled flow. On the contrary, simulation indicates that a phenomenon of coupled entrainment occurs in which initial profiles are subsumed into the periodicity of the dominant input variation. The propensity of the system to establish quickly such an asymptotic steady state allows analytical accounts to be formed. Concurrence between simulation, analytical solutions and field observation has been achieved.

    Understanding of the general state is acquired through the following intermediate stages. The linear diffusion of a system of n coupled variables under both Dirichlet and Neumann conditions is considered. In the latter case, it is shown that the n variables behave as a composition of n2 component waves. Periodicity is preserved. For each variable, the amplitude, phase and frequency of each of the n input fluxes is carried into the medium by n wave fronts whose rate of decay and speed of penetration depends solely on the coupling of an individual variable. An alternative derivation of this result using semi-group theory is given. The complementary situation of an individual variable diffusing under non­ linear gradients is next addressed. Approximate, but applicable, series solutions may be formed. The non-linearity is accommodated within a stable period by the addition of severely decaying higher harmonics. The heating phase is accelerated whilst the cooling phase is retarded. An increase in the mean oscillation occurs.

    Simulation indicates that the full system has an analogous qualitative stability. Input periodicity is retained and no subharmonic, chaotic or catastrophic responses develop. The rigorous analytical establishment of this property is apparently beyond the current state of knowledge of non­ linear evolution operators in Banach space. Despite the underlying theoretical uncertainty, series accounts may be formed which concur with both simulation and data from field experiments. A possibility of coupled resonance between the variables is restrained by the exponentially decaying nature of the responses. The agreement between simulation, analysis and field data is illustrated by surface and contour plots of the behaviour for a range of surface conditions and controlling parameters.
    Date of AwardJun 1994
    Original languageEnglish

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