SUB-DIFFUSIVE AND SUB-ADVECTIVE VISCOUS FLUID FLOWS IN DEBRIS AND POROUS MEDIA DISSERTATION SUBMITTED FOR THE PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE MASTER OF PHILOSOPHY (M. Phil.) DEGREE IN MATHEMATICS BY KHIM BAHADUR KHATTRI

In geophysical and industrial mass flows, both the solid and fluid phases may be involved. Following the general two-phase debris flow model by Pudasaini (2012) [77], and neglecting the solid deformation, inertial forces, and lateral variation of fluid velocity results in a novel sub-diffusive and sub-advective (SD-SA) equation (Pudasaini, 2014) [78] for fluid flows through general porous landscape and debris materials. The sub-diffusion and sub-advection model describes the fluid flows through inclined porous landscape and debris material with systematically derived nonlinear (quadratic) diffusion and advection fluxes. It appears that, in general, it is not possible to find the exact analytical solutions to the full sub-diffusion and sub-advection model. Pudasaini (2014) [78] constructed some exact solutions and numerical solutions to the sub-diffusion and sub-advection model. Here, further advancement is made by constructing some additional exact solutions to the model equation for the sub-diffusion and sub-advection fluid flows in porous landscape and debris materials. However, some new exact solutions for the reduced SD-SA process is presented. To further construct analytical solutions for the SD-SA model, some special mathematical techniques, including, reduction to classical advection-diffusion equation, separation of variables and linearization are employed. Some advanced exact solutions are constructed by using Bring ultra-radical and hyper-geometric functions. In doing so, we also generate the Abel and Lineard equations in canonical form. The full SD-SA equations are integrated numerically by applying the high-resolution shock capturing TVD-NOC schemes. Exact and numerical solutions are compared for the sub-diffusive fluid flow, which shows that the numerical solutions match almost exactly with the exact solutions, thus, demonstrating the very high accuracy and performance of the numerical method and the computational code. It is revealed that the sub-diffusion and sub-advection processes are fundamentally different as compared to the classical diffusion-advection process where the entire fluid pocket advects in the main flow direction, which at the same time also diffuses with spreading Gaussian profile. In contrast to the classical advection-diffusion of fluid where the tail of the initial substance distribution also advects in the downslope direction, for the fluid flow through porous media, the tail remains in its original position. Unlike the front of the classical diffusion-advection solution where the front is diffused, the front of the sub-diffusive and sub-advective fluid flow in porous media successively forms a downslope propagating strong bore, the diffusion is strongly constrained by a compact support. New insights into the proposed sub-diffusion and sub-advection equation, its exact and numerical solutions, underlying physics and applicabilities are discussed in detail. The results may help in constructing early warning and mitigation strategies in potentially catastrophic failures of landslides, reservoir dams and embankments in geo-disaster-prone areas such as the Himalayas, Alps and the Andes.