At present, the majority of fluid mechanisms are multi-field coupling mechanisms, and their function is also achieved in the flow field. Therefore, calculating the aerodynamic characteristics of the multi-field coupling mechanism in a symmetric flow field is very important. However, at present, the strong coupling algorithm in the domain has the problems of low accuracy and computational efficiency, which make it more difficult to solve the coupling problem. This article obtains the vector potential of the law of conservation of momentum using the tensor analysis method in a Cartesian coordinate system. Meanwhile, the generalized operator of Navier–Stokes equations and the fundamental solution of the generalized operator are obtained on this basis. Then, this article proposes the boundary integral equation of the Navier–Stokes equations by combining the fundamental solution of the Laplace equation with generalized potential theory. Based on this boundary integral equation, this article has developed a new calculation method that can help achieve integral calculation without domains, greatly reducing the problem’s difficulty. Finally, by comparing the ellipsoid example solution with the experimental results, the algorithm’s reliability in solving the incompressible problem is verified.

Transition is the basic phenomena of complex flow regime changes in fluid mechanics. So far, the axiomatic expression and rigorous mathematical description of transition and separation are important basic scientific problems, which are also of great significance to the development of Applied Science. This article obtaining the basic conditions and general conditions for transition through the excited state theory of flowfield, which is derivation based on the N-S equations, combined with Euler’s description point of view, applied tensor analysis and with the help of the basic principles of superposition states of quantum mechanics. The research of this paper provide a theoretical view for the study of turbulence mechanism.

During the development of fluid mechanics, fluid separation is an important issue. So far, there is no mathematical formula to reveal and describe the essence of fluid separation. At the same time, due to the high cost and limitation of the experimental method, another method is urgently needed to predict the separation position of the fluid. After axiomatizing fluid mechanics and combining the principle of excited state of quantum mechanics, this paper reveals that fluid separation is a special form of fluid in an excited state, and deduces the state conditions of fluid separation. The research results of this paper provide new ideas for solving problems in fluid separation and engineering applications.