Numerical algorithm for nonlinear flow response of soft brush beds based on the homotopy method

Fuzzy flexible fibers attached to biological surfaces can reduce drag and influence fluid flow by altering their shape, an intriguing phenomenon that has garnered increasing attention. The complexity of fluid–structure interaction problems leads to highly nonlinear equations that describe this issue. Existing analytical solutions, such as perturbation methods and series expansion techniques, still do not perfectly align with experimental observations. To address this problem, this study explores numerical approaches. By introducing the method of homotopy transformation, we first tackle the convergence difficulties associated with the classical Newton–Raphson iterative method due to the a priori selection of initial values. Second, numerical results indicate that the proposed method aligns with experimental results across all velocity distributions, confirming previous conjectures regarding the −1/2 scaling rate at higher velocities. The developed homotopy numerical algorithm is compared with the shooting method provided by the MAPLE software, demonstrating superior convergence and computational results compared to existing numerical methods. Finally, we further investigate the deformation of elastic fibers under very large loads. The proposed method exhibits excellent convergence properties and can be utilized to explore numerical solutions for nonlinear equations, particularly in fluid–structure interaction problems, providing insights for future scientific exploration and engineering applications.