# Research Progress and Engineering Applications of Viscous Fluid Mechanics
## 2 Advances in the Study of the State of Motion of Viscous Fluids
### Experimentation and Research  
#### 2.1.1 Experiments and Studies on the Permeability of Media to Viscous Fluids  
Experiments and studies on the permeability of media to viscous fluids usually focus on investigating the permeability properties of fluids in porous media and their dynamic behavioral properties. Such studies have important applications in petroleum engineering, geology, civil engineering, and other fields [17, 18, 19]. The following are some common related contents and methods: (1) Porous media and permeability: porous media can be rocks, soils, rocks, or other materials in petroleum reservoirs. Their pore structure and  
Figure 1: Velocity gradient diagram of liquid interlayer motion. The lower plate in the channel is fixed, and the upper plate shifts to the right to drive the liquid movement. The proportional coefficient in the formula is called the dynamic viscosity coefficient, or the internal friction coefficient, the force direction of the viscous force \(F\) is the negative direction of the \(X\)-axis in the figure, and \(A\) is the unit area of the liquid. connectivity determine the permeability of the media to the viscous fluid. (2) Permeability experimental methods: (i) the pressure-driven method (conventional method) involves applying a pressure difference to push the fluid through the porous medium, measuring the rate of fluid passage and pressure changes to calculate the permeability; (ii) the constant pressure method involves maintaining a constant pressure difference, measuring the rate of fluid permeation and permeability; and (iii) the resistance method involves determining the permeability properties by measuring the resistance of the fluid to passage through the porous medium. (3) The influence of the permeability of viscous fluids: the permeability properties of viscous fluids (e.g., petroleum, polymer solutions, etc.) are usually different from those of nonviscous fluids because viscosity affects the flow rate and distribution of the fluid. (4) Experimental equipment and techniques: permeability determination equipment (e.g., permeameters, permeability test benches, etc.) is used, usually in conjunction with pressure transducers, flow meters, and other instruments for accurate measurements. (5) Numerical simulation: Computational fluid dynamics (CFD) and numerical simulation methods are used to simulate the permeability behavior of viscous fluid in porous media and predict the permeability and fluid distribution of the medium to the viscous fluid [20; 21; 22; 23]. These studies are important for optimizing oil extraction, groundwater resource management, environmental pollution prevention, and other fields and can help engineers and scientists to better understand and control the behavior of viscous fluids in complex media [24]. Currently, [PERSON]'s law and Forchheimer's law are the main laws for determining the permeability of viscous fluids. [PERSON]'s law describes the relationship between the rate of permeation of a viscous fluid in a porous medium and the gradient of osmotic pressure under steady-state seepage conditions [25]. However, this law is mainly used for the determination of macroscopic permeability k under constant conditions, and it is questionable whether the fine channel changes under microscopic conditions affect this law. [PERSON]'s law as a model for describing [PERSON]'s flow is based on the overall average of the porous media model and does not take into account the specific effect of the microstructure inside the medium, which may limit its accuracy in some delicate porous media [26; 27]. Both of these problems have been explored by researchers worldwide. In this regard, Prof. [PERSON] et al. [28] of Princeton University carried out an experimental study on the pressure-driven passage of viscous fluids through a superelastic porous membrane.