Electro-osmotic transport and heat transfer of cross fluid model in an inclined contracting-expanding elastic microchannel: numerical and improved physics-informed modelling

The present investigation develops a computational framework for nonlinear electro-osmotic transport of a Cross fluid through an inclined deformable microchannel with contracting-expanding walls. Although Physics-Informed Neural Networks (PINNs) have emerged as an effective approach for solving nonlinear differential equations, their application to the coupled problem involving electro-osmosis, Cross-fluid rheology, deformable walls, mixed convection, wall slip, and channel inclination has not been comprehensively investigated. By invoking the long-wavelength and low-Reynolds-number approximations, the governing equations are reduced to a system of nonlinear ordinary differential equations. The obtained boundary-value problem is then numerically solved by means of a collocation-based method named BVP4C, a standard PINN, and a Bayesian-optimized PINN (BO-PINN) with Gaussian process-based hyperparameter tuning. Based on the available literature, this is the first research to integrate Bayesian hyperparameter optimization with PINNs for electro-osmotic transport of Cross fluids in inclined deformable microchannel with contracting-expanding walls. The proposed framework automatically determines the optimal network architecture and training hyperparameters, resulting in improved convergence and predictive accuracy compared with the baseline PINN. The effects of electro-osmotic forcing, wall slip, mixed convection, channel inclination, and the shear-thinning behaviour of the Cross fluid on the transport characteristics are systematically investigated. The optimized BO-PINN reduces the relative L2 error to below 10−3[jls-end-space/], demonstrating excellent agreement with the numerical solution. The proposed computational framework provides an efficient and accurate approach for analyzing coupled transport phenomena in deformable microfluidic systems and has the potential to support the design and optimization of practical lab-on-chip devices, artificial micro pumps, microfluidic bioreactors, and other microfluidic systems employing deformable micro-channels.