The Perturbation Curvature and its Geometric Interpretations
Authors: Wei Leo 
Published: July 2023

Abstract
The perturbation curvature in the Navier-Stokes equations influences the pressure field through its impact on the pressure gradient, conservation of mass (continuity equation), pressure-velocity coupling, and turbulence generation. Understanding and analyzing the influence of perturbation curvature on the pressure field is crucial for comprehending the behavior and characteristics of fluid flows and their interactions with the underlying dynamics. The perturbation curvature, represented by the tensor c_{i j}=\frac{\partial^2 u_i}{\partial x_j} \partial x_j, has a significant influence on various fields in the Navier-Stokes equations, including the pressure field. The pressure field in the Navier-Stokes equations acts as a driving force that accelerates or decelerates the fluid flow. The perturbation curvature affects the pressure field through its impact on the momentum equation, \operatorname{div}(u)=\sum_{i=1}^n\left(\frac{\partial u_i}{\partial x_i}\right)=0. (1.0 (1)) The perturbation curvature influences the pressure gradient term by contributing to the second derivative of the velocity components . Changes in the perturbation curvature can alter the pressure gradient and subsequently affect the flow behavior. The perturbation curvature also indirectly affects the pressure field through the continuity equation, C_{i j}=\frac{\partial^2 u_i}{\partial x_j} \partial x_j (1.0 (2)) The continuity equation ensures that the divergence of the velocity field is zero, indicating the conservation of mass.When the perturbation curvature is present, it affects the velocity field components \frac{\partial U_i}{\partial x_i} , which are involved in the continuity equation. The changes in the perturbation curvature can modify the velocity field, leading to variations in the divergence and, consequently, the pressure field. The pressure field and the velocity field are inherently linked in the Navier-Stokes equations through the pressure-velocity coupling terms. The perturbation curvature influences this coupling by modifying the velocity gradients \frac{\partial U_i}{\partial x_i} and their second derivatives \frac{\partial^2 u_i}{\partial x_j} \partial x_j in the momentum equation.Changes in the perturbation curvature can affect the distribution of pressure forces, which, in turn, influences the acceleration or deceleration of fluid elements. This coupling between pressure and velocity, mediated by the perturbation curvature, plays a crucial role in determining the overall flow behavior. In turbulent flows, the perturbation curvature is particularly significant as it contributes to the generation and amplification of turbulence. Turbulence is characterized by complex, unsteady flow patterns with high levels of vorticity and fluctuations. The perturbation curvature affects the velocity gradients and their second derivatives, which are crucial for the development and sustenance of turbulence. Variations in the perturbation curvature can lead to changes in the vorticity dynamics, energy transfer mechanisms, and turbulent structures, ultimately influencing the pressure field in turbulent flows. The perturbation curvature serves as the foundation for future construction of a “von Neumann Program” operated based on the perturbation curvature evolution equation, which will be presented in Paper II.