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Abstract

In this study, we investigate the rectilinear displacement and deformation of a highly viscous, miscible circular blob influenced by a less viscous fluid within a homogeneous porous medium featuring physically realistic no-flux boundaries. We utilize a fourth-order accurate compact finite difference scheme for the spatial discretization of the nonlinear partial differential equations that govern this phenomenon. The resulting semi-discrete equations are then integrated using the second-order Crank-Nicolson (CN) method. We conduct numerical simulations for a P\'eclet number ($Pe \leq 3000$) and a log-mobility ratio $0 \leq R \leq 7$, which reveal three distinct pattern formations: comet-shape, lump-shape, and viscous fingering instability. Our results demonstrate that the deformation, spreading, and mixing of the blob vary non-ideally with both $Pe$ and $R$, a behavior attributed to the blob's initial curvature. Consequently, enhanced mixing can be achieved at intermediate values of $Pe$ and $R$, suggesting the existence of an optimal mixing condition. These findings have significant implications for fields such as oil recovery, CO$_2$ sequestration, pollution remediation, and chromatography separation.