In this study, a modified Taylor-Galerkin/pressure-correction finite-element approach has been developed to analyze non-isothermal, compressible, non-Newtonian, and inelastic fluid flows. In many cases, the coupling of viscosity and density to pressure and temperature is critical
for accurate simulation of complex industrial and natural flows. The governing equations include the compressible Navier-Stokes system, the viscosity power-law model (for fluid rheology), and the Tait equation of state (to relate pressure and density). This important development is
based on the integration of a two-step Lax-Wendroff method into the energy equation solver, enabling accurate thermal coupling and ensuring numerical stability. The novelty of this work lies in being the first to integrate compressibility, non-Newtonian viscosity, and non-isothermal effects
into the Taylor-Galerkin/pressure-correction framework developed for power-law fluids. This integrated approach allows the simulation of high-fidelity, strongly coupled flow cases that are intractable for standard solvers. This provides a robust, approximate, and generalized method
for higher-order problems in fluid mechanics, applicable to engineering or geophysical environments. The method’s performance is assessed by investigating the influence of the power-law index, the consistency coefficient, the Prandtl number, and the thermal conductivity on the flow variables. The results show that, for a shear-thickening fluid, increasing the power-law index from 0.5 to 1.5 reduces the outlet velocity by 10%; by contrast, increasing the consistency coefficient from 1 (base case) to 20 raises the outlet velocity by up to 35% in both shear-thinning and shear-thickening scenarios. The centerline pressure increases with the consistency coefficient, and the density shows a similar trend. Thermal analysis indicates 22% increase in wall temperature under conditions of low thermal conductivity when the Prandtl number decreases from 1.4 to 0.6; however, the wall temperature increases rapidly in compressible, shear-thickening flows. Shear and normal stresses, which are elevated relative to those in a Newtonian fluid, can increase substantially (by 50%) as the power-law index and the consistency coefficient increase,
indicating strong nonlinear rheological effects.