  This work introduces a new higher-order accurate super compact (HOSC) finite
difference scheme for solving complex unsteady three-dimensional (3D)
non-Newtonian fluid flow problems. As per the author's knowledge, the proposed
scheme is the first ever developed finite difference scheme to solve
three-dimensional non-Newtonian flow problem. Not only that, the proposed
method is fourth-order accurate in space variables and second-order accurate in
time. Also, the proposed scheme utilizes only seven directly adjacent grid
points, at the $(n+1)^{th}$ time level, around which the finite difference
discretization is made. The governing equations are solved using a
time-marching methodology, and pressure is calculated using a
pressure-correction strategy based on the modified artificial compressibility
method. Using the power-law viscosity model, we tackle the benchmark problem of
a 3D lid-driven cavity, systematically analyzing the varied rheological
behavior of shear-thinning ($n=0.5$), shear-thickening $(n=1.5)$, and Newtonian
$(n=1.0)$ fluids across different Reynolds numbers $(Re= 1, 50, 100, 200)$.
Both Newtonian and non-Newtonian results are carefully investigated in terms of
streamlines, velocity variation, pressure distributions, and viscosity
contours, and the computed results are validated with the existing benchmark
results. The findings demonstrate excellent agreement with the existing
results. This extensive analysis, using the new HOSC scheme, not only increases
our understanding of non-Newtonian fluid behavior but also provides a robust
foundation for future research and practical applications.
