Refined Optimal and Modified-Homotopy Perturbation Method for Direct Solution of Incompressible Navier-Stokes Equations with Pressure Correction
Year: 2026
Type: preprint
Abstract: Motivation and Method Development: Direct semi-analytical treatment of the full two-dimensional (2D) incompressible Navier–Stokes equations (INSE) with simultaneous determination of velocity and pressure fields, remains a challenging problem in fluid mechanics, particularly for Newtonian and non-Newtonian internal flows. To address this challenge without simplifying the governing PDEs or reducing them to ODEs, a Refined Optimal and Modified Homotopy Perturbation Method (ROM-HPM) is developed. The framework provides systematic guidelines for selecting auxiliary linear operators and constructing initial approximations for PDEs. Arbitrary functions arising in the auxiliary solutions are represented by polynomials with unknown coefficients determined directly from the prescribed initial and boundary conditions. A pressure-correction algorithm is incorporated to overcome the inability of the standard OM-HPM formulation to adequately treat pressure-coupled incompressible flows. Necessary and sufficient conditions for convergence are established, together with truncation error, stability, and computational complexity analyses. Application to Newtonian and Non-Newtonian Microchannel Flows: ROM-HPM is applied directly to the full 2D INSE governing Newtonian and non-Newtonian flows in microchannels with non-unifrom injection-suction and slip boundary conditions under both prescribed and unknown pressure fields. The resulting semi-analytical velocity and pressure solutions are validated against available exact solutions and established computational fluid dynamics (CFD) results. Beyond reproducing benchmark flow fields, the method captures the transient development of coherent flow structures and resolves their evolution from elongated and partially developed vortices to fully developed vortical patterns. Flow Physics, Validation and Computational Performance: Comparisons between continuous and discrete implementations of ROM-HPM and conventional CFD demonstrate strong agreement in velocity, pressure, and associated flow structures while revealing advantages in computational efficiency, solution accuracy, convergence behaviour, and extension of the computational domain. The results demonstrate that ROM-HPM provides a direct semi-analytical framework for resolving pressure velocity coupling in the full 2D INSE while preserving the underlying flow physics, offering an alternative computational route for investigating complex incompressible internal flows.
Source: Research Square
Authors: Tapas Roy, Dilip Kumar Maiti, Palash Mondal
Institution: Vidyasagar University
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204841622