Finite Element-Difference Simulations of Isentropic Non-Newtonian Compressible Second-Grade Fluid

This study investigates the unsteady compressible flow of a second grade non-Newtonian fluid in a bounded two-dimensional domain Ω ⊂ R 2 over a time interval [0, T], motivated by applications in engineering problems where compressibility and non-Newtonian behavior may be significant. The governing momentum–constitutive equation is coupled with an isentropic state equation to eliminate the pressure variable. The resulting formulation is third order in space and first order in time. To obtain a computationally tractable weak form, the system is reformulated via suitable transformations into equivalent second-order, after which a weak (variational) formulation is derived. The spatial discretization is performed using finite element methods, whereas the temporal derivative is approximated by a generalized scheme that merges the second-order Gear’s method and backward Euler method. The resulting nonlinear discrete system is linearized using the Newton–Raphson iteration, leading at each time step to a system of linear algebraic equations. The numerical implementation is validated in a limiting case that recovers the compressible Navier–Stokes equations. Relative error norms are computed, and convergence is assessed through systematic refinement of the time step and mesh size. The convergence behavior of backward Euler is compared with that of the second-order Gear’s method. Finally, simulations for the full second-grade compressible case are presented through contour plots of the solution fields, showing the influence of flow parameters on the density and velocity components under the imposed domain and boundary conditions.