Transition and separation are difficult but important problems in the field of fluid mechanics. Hitherto, separation and transition problems have not been described accurately in mathematical terms, leading to design errors and prediction problems in fluid machine engineering. The nonlinear uncertainty involved in separation and transition makes it difficult to accurately analyze these phenomena using experimental methods. Thus, new ideas and methods are required for the mathematical prediction of fluid separation and transition. In this article, after an axiomatic treatment of fluid mechanics, the concept of an excited state is derived by generating a fluctuation velocity, and it is revealed that fluid separation and transition are special forms of this excited state. This allows us to clarify the state conditions of fluid separation and transition. Mathematical analysis of the Navier–Stokes equations leads to a general excited state theorem suitable for flowfields. Finally, the conditions of separation and transition are derived, and the corresponding general laws are established. The results presented in this article provide a foundation for future research on the mechanism of turbulence and the solution of engineering problems.

This work presents a mathematical framework based on uncertain numbers to address the inherent uncertainty in nonlinear systems, a challenge that traditional mathematical frameworks often struggle to fully capture. By establishing five axioms, a formal system of uncertain numbers is developed and embedded within set theory, providing a comprehensive characterization of uncertainty. This framework allows phenomena such as infinity and singularities to be treated as uncertain numbers, offering a mathematically rigorous analytical approach. Subsequently, an algebraic structure for uncertain numbers is constructed, defining fundamental operations such as addition, subtraction, multiplication, and division. The framework is compatible with existing mathematical paradigms, including complex numbers, fuzzy numbers, and probability theory, thereby forming a unified theoretical structure for quantifying and analyzing uncertainty. This advancement not only provides new avenues for research in mathematics and physics but also holds significant practical value, particularly in improving numerical methods to address singularity problems and optimizing nonconvex optimization algorithms. Additionally, the anti-integral-saturation technique, widely applied in control science, is rigorously derived within this framework. These applications highlight the utility and reliability of the uncertain number framework in both theoretical and practical domains.

In this paper, the boundary element method is applied to determine the characteristics of distributed flow around the airfoil group. Because the boundary integral equation can describe various initial boundary value problems in mechanics and has many advantages such as good convergence, it can be used as a tool to establish appropriate mathematical models and numerical realization methods, so as to comprehensively study the distribution kinematics and dynamics of the airfoil group motion system in the viscous fluid flow co-flow process. In this paper, some numerical results are presented to determine the aerodynamic distribution characteristics of the viscous flow field around the airfoil group in the control area.

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.

In this paper, we derive and propose basic differential operations and generalized Green’s integral theorems applicable to multidimensional spaces based on Cartesian tensor analysis to solve some nonlinear problems in smooth spaces in the necessary dimensions. In practical applications, the theorem can be applied to numerical analysis in the conservation law, effectively reducing the dimensions of high-dimensional problems and reducing the computational difficulty, which can be effectively used in the solution of complex dimensional mechanical problems.

The rational mechanic’s research method is synthesized with the help of mathematical means such as generalized function analysis and tensor analysis. The basic definition of the fluctuation velocity generation is based on phenomenological physics in this article. The basic control equations of general excited state fluid dynamics applicable to the flowfield are obtained based on the basic principles of quantum mechanical superposition states. The simplified basic equations of the excited State are finally obtained through time and space discretization. The basic theory of excited state fluid dynamics is established, providing new ideas for the innovation and application of flow control, fluid mechanical engineering design, and other aspects of research work.