A transform method for analyzing erosion of porous media in a channel with multiple cylinders

We present a transform method to analyze the erosion of a porous medium with multiple cylindrical bodies. Our study focuses on a two-dimensional channel geometry containing an array of cylinders of varying sizes and arbitrary locations. We solve the associated boundary value problem for the biharmonic equation using our transform method, which provides quasi-analytical solutions and leads to fast and accurate schemes for evaluating the solutions. Specifically, our model considers cases based on the threshold law, where erosion occurs when the total shear stress exceeds a specified critical value dependent on the material of the cylindrical bodies. This erosion process not only reduces the size of the cylindrical bodies but also alters their shapes, causing them to shrink and eventually vanish in finite time. We compute the shear stress on the cylinders and use it to determine the updated shapes of the eroded bodies.