A New Higher-Order Super Compact Finite Difference Scheme to Study Three-Dimensional Non-Newtonian Flows

TL;DR

This paper presents a novel, highly accurate finite difference scheme for 3D non-Newtonian fluid flows, achieving fourth-order spatial accuracy with minimal grid points, validated through benchmark simulations of shear-thinning, shear-thickening, and Newtonian fluids.

Contribution

The work introduces the first finite difference scheme specifically designed for three-dimensional non-Newtonian flows, combining high accuracy with computational efficiency.

Findings

The scheme achieves fourth-order accuracy in space and second-order in time.

Validated results show excellent agreement with benchmark data.

The method effectively captures complex rheological behaviors in 3D flows.

Abstract

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.