Residual, tolerance, and mesh sensitivity analysis for ciliary-driven Johnson-Segalman fluid flow in a divergent wavy channel with double slip effects

The coordinated motion of motile cilia generates metachronal waves, which are vital for fluid transport in biological systems. These cilia facilitate physiological functions such as motility, feeding, and fluid pumping. This study examines the dynamics of a non-Newtonian Johnson-Segalman (J-S) fluid in a divergent wavy channel embedded in a porous medium, incorporating first- and second-order slip conditions. The governing nonlinear fourth-order boundary value problem is solved numerically using MATLAB's bvp4c solver. The solution is rigorously validated through residual testing, tolerance sensitivity analysis, and mesh convergence studies to ensure numerical accuracy. Limiting cases confirm the model's consistency with established fluid models, including Newtonian, Maxwell, Oldroyd-B, and Corotational/Jaumann model scenarios. Graphical results are presented to illustrate the effects of key parameters on velocity profiles, shear stress distributions, pressure gradients, frictional forces, and streamline patterns. The findings demonstrate that the divergent nature of the channel, the wavy wall structure, and the slip conditions significantly influence the flow behavior of the non-Newtonian fluid. A special case of an exact solution is also presented.