SCALING AND SELF-SIMILARITY IN MULTI-DIMENSIONAL FLOW THROUGH UNSATURATED POROUS MEDIA

Date

2026-2-27

Author

Basa, Mustafa Mazhar

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Researchers have been exploring scaling and self-similarity within hydrologic and hydraulic processes governed by nonlinear partial differential equations to better understand and model their complex nature. In this study, a theoretical scaling framework is developed using the Lie group of point transformations, under which one-, two-, and three-dimensional unsteady flow in unsaturated porous media exhibits self-similarity. This work extends scaling and self-similarity analysis to the initial and boundary value problem of unsteady flow in multi-dimensional unsaturated porous media, linking a critical gap between classical hydrodynamic scaling and porous media flow. Lie group theory, originally formulated by Sophus Lie in the nineteenth century for solving differential equations, provides a powerful mathematical foundation for identifying symmetries and invariance properties of the governing equations in complex physical processes. Main contribution of this study is the derivation of scaling conditions that simultaneously preserve form of the governing equations in scaled domain and associated initial and boundary conditions, yielding self-similarity in multi-dimensional unsteady unsaturated flow. Numerical simulations are used to demonstrate the existence of self-similar solutions across multiple space-time domains under the derived scaling transformations. Performed numerical examples confirm that self-similarity across multiple space-time domains of unsaturated flow can be achieved under the proposed scaling conditions. In engineering practice, configuration of experimental studies is constrained by factors such as the available laboratory space, accessible materials, and feasible experimental durations. The derived scaling relations enhance spatial, temporal, and economic flexibility in design and implementation of physical models, offering new insights into unsteady unsaturated porous media flow.

Subject Keywords

Scaling laws, Self-similarity, Unsaturated porous media, Unsteady multi-dimensional flow, Lie Group transformations, Ölçekleme yasaları, Özbenzerlik, Doymamış gözenekli ortamlar, Kararsız çok boyutlu akış, Lie Grubu dönüşümleri

URI

https://hdl.handle.net/11511/118996

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Graduate School of Natural and Applied Sciences, Thesis