Many microfluidics-based applications involve fluid–structure interaction (FSI) of flexible membranes. Thin flexible membranes are now being widely used for mixing enhancement, particle segregation, flowrate control, drug delivery, etc. The FSI simulations related to these applications are challenging to numerically implement. In this direction, techniques like immersed boundary method (IBM) have been successful. In this study, two-dimensional numerical simulation of flexible membrane fixed at two end points in a rectangular channel subjected to uniform fluid flow is carried out at low Reynolds number using a finite volume based IBM. A staggered Cartesian grid system is used and SIMPLE algorithm is used to solve the governing continuity and Navier–Stokes equations. The developed model is validated using the previous research work and numerical simulations are carried out for different parametric test cases. Different membrane mode shapes are observed due to the complex interplay between the hydrodynamics and structural elastic forces. Since the membrane undergoes deformation with respect to inlet fluid conditions, a variation in flowrate past the flexible structure is confirmed. It is found that, by changing the membrane length, bending rigidity, and its initial position in the channel, flowrate can be controlled. Also, for membranes that are placed at the channel midplane undergoing self-excited oscillations, there exists a critical dimensionless membrane length condition L ≥ 1.0 that governs this behavior. Finally, an artificial neural network (ANN) model is developed that successfully predicts flowrate in the channel for different membrane parameters.