# Research Progress and Engineering Applications of Viscous Fluid Mechanics
## 1 Introduction  
Viscous fluids are actual fluids with viscous properties, which are manifested by the nature of the fluid preventing its own flow, and this viscosity is caused by the interaction forces between the fluid molecules. As an important branch of fluid mechanics, viscous fluid mechanics studies the behavior and properties of fluids under the influence of viscous effects [1, 2, 3]. The most classical study is [PERSON]'s assumption that the shear stress inside the fluid is proportional to the velocity gradient, thus introducing the concept of kinetic viscosity, which is applicable to many fluids in practical applications, such as water and air [4, 5]. The formula and schematic diagram are shown below (Figure 1).  
\[F=\mu\frac{d_{x}}{dy}A, \tag{1}\]  
However, in some engineering and natural phenomena, the viscous effects of fluids may significantly affect the kinematic properties of fluids, such as viscous hysteresis phenomena, the deformation and diffusion behaviors of fluids, etc. [6, 7, 8, 9]. The study of viscous fluids is not limited to traditional Newtonian fluid models but also includes non-Newtonian fluids such as plastic, elastic, and viscoelastic fluids, which exhibit complex nonlinear behavior at high strain rates or high stresses. In addition to viscous fluids, viscous fluid theory also applies to fluids that are not viscous in nature (such as air, water, ethanol) which exhibit high viscosity when flowing in confined spaces (such as through membrane pores, porous materials, droplets hitting surfaces, etc.). Other relevant formulasand criteria for viscous fluid motion include the Froude criterion, Euler equation, N-S equation, Reynolds criterion, laminar flow, turbulence, turbulent flow, boundary layer theory, etc. [10,11].  
In a wide range of fields of scientific exploration and engineering, many problems involve the permeability of viscous fluids moving on the surface or inside various materials and their specific modalities. For example, in the field of aircraft design, an in-depth understanding of the turbulence characteristics on aerodynamic surfaces and the potential impact of air viscosity on aircraft performance is essential [12,13]; in river engineering practice, an accurate assessment of the erosion of water flow on the riverbed and riverbank structures is equally indispensable [14]. In the pipeline transportation of chemical viscous liquids such as oil and liquefied natural gas, it is necessary to fully consider the potential impact of the viscosity of these liquids on the transportation efficiency [15]. To solve these problems, it is indispensable to use experiments and theoretical calculations to accurately determine various key parameters of viscous fluids [16]. In this paper, the basic theory, mathematical models, and practical applications of viscous fluid dynamics are systematically explored. This article summarizes the recent progress in the direction of experimental exploration and simulation research on the state of motion of viscous fluids, including the current problems and limitations. Meanwhile, the future development trend is envisioned, aiming to provide guidance and reference for research in the field of viscous fluid kinematic states.