Multiple regression and stability analysis of Powell-Eyring fluid flow over a stretching/shrinking surface in uniform shear flow

The present study investigates the transpiration effects on flow over a stretching/shrinking surface in a uniform shear flow of Powell-Eyring fluid with thermal radiation. The governing partial differential equations are highly nonlinear, making analytical solutions difficult to obtain. Therefore, a Lie group analysis is conducted to derive appropriate similarity transformations that reduce the system to ordinary differential equations. Numerical solutions of the resulting ODEs are obtained using the MATLAB bvp4c solver. Notably, dual solution branches are identified for both stretching and shrinking cases across various material and transpiration parameter values. These dual solutions bifurcate from a critical point, with no solutions existing below this threshold. While both solutions are mathematically valid, they exhibit distinct characteristics in their velocity and temperature profiles. A linear temporal stability analysis is employed to determine the physically reliable solution, revealing that the first solution branch is physically meaningful. Additionally, multiple linear regression analysis is performed to establish relationships between physical quantities and governing parameters. The analysis demonstrates exceptional predictive capability with $R^2 > 0.96$ for all cases, confirming strong statistical correlations. The effects of key parameters on various physical quantities are explored and presented through graphical and tabular representations.