Some Exact Solutions of Unsteady Incompressible Coupled Stress Fluids with Topology Conservation of Vorticity Vector Field by Ansatz Method

Objectives: The primary objective of this study is to obtain exact analytical solutions for unsteady incompressible couple stress fluid flows using the Ansatz method. The study is motivated by the need to better understand the influence of microstructural effects present in couple stress fluids, which arise in various engineering and biological applications. In addition, the work aims to establish and verify the topology conservation equation associated with the vorticity vector field in order to investigate the structural and rotational characteristics of the flow. Method: The governing equations for unsteady incompressible couple stress fluids are formulated based on the fundamental principles of fluid mechanics. From the vorticity transport equation, a topology conservation equation related to the vorticity vector field is derived. The resulting higher-order nonlinear partial differential equations are then analyzed using the Ansatz method by assuming an appropriate vector potential. This approach simplifies the complex system of equations and allows the derivation of exact analytical solutions for specific flow configurations. Furthermore, graphical representations are employed to examine the influence of viscosity and couple stress parameters on the velocity and vorticity fields. Findings: The study successfully derives exact analytical solutions for selected unsteady flow configurations of couple stress fluids. The obtained expressions for the velocity and vorticity fields are shown to satisfy the derived topology conservation equation, confirming the mathematical validity and consistency of the model. The graphical analysis illustrates that variations in viscosity and couple stress parameters significantly influence the structure and magnitude of the velocity field as well as the behavior of the vorticity distribution within the fluid. Novelty: The novelty of this work lies in the application of the Ansatz method to solve higher-order nonlinear equations governing unsteady incompressible couple stress fluid flows while incorporating the topology conservation of the vorticity vector field. Unlike many existing studies that rely primarily on numerical or approximate techniques, this research provides explicit analytical solutions together with graphical analysis. The results contribute to a deeper theoretical understanding of the structural properties and dynamic behavior of couple stress fluids under unsteady flow conditions. Keywords: Unsteady incompressible couple stress fluid, topology conservation equation, couple stress parameters