## 3 Theoretical Modeling  
Suction force directly determines the adhesion performance of wall-climbing cleaning robots, and the theoretical model of suction force plays a critical role in research. The geometric parameters of the suction unit are depicted in Figure 5. In order to establish a theoretical model for suction force, the following assumptions are made:  
Figure 4: Suction force testing platform. (**a**) Schematic diagram; (**b**) physical image. Figure 3: Schematic diagram of the cleaning unit structure. Figure 2: Schematic diagram of the suction unit structure. The continuum assumption, which posits that a fluid is composed of a continuous distribution of fluid particles. 2. The incompressible flow assumption, which holds when the Mach number is within the range of 0-0.3, and the density and temperature of the airflow within the adhesion chamber are considered constants. The maximum Mach number in this suction unit is 0.22. 3. Only the tangential air velocity generated by the fan rotation is considered, with the vertical and radial air velocities being neglected. Based on the assumptions stated above, and by neglecting the viscous and external force terms, the Navier-Stokes equations can be simplified to the following form:  
\[\frac{\rho u_{\alpha}^{2}}{r}=\frac{\partial P}{\partial r}, \tag{1}\]  
In this equation, \(\rho\) denotes the air density, \(u_{a}\) represents the circumferential velocity of the airflow, \(r\) signifies the radius, and \(P\) refers to the pressure. Within the suction unit, the airflow is not fully developed in the fan region, where \(u_{a}\) represents the tangential velocity at that location, caused by the rotation of the fan blades. In the regions between the fan blades and the shell surface, as well as between the fan blades and the working surface, the flow remains undeveloped and can be characterized by the \(1/n\) power law, as shown in Figure 6. This region is referred to as the rotational boundary layer. Some researchers have demonstrated that as the air flow velocity increases, the degree of turbulence intensifies, and, typically, \(n\) is assumed to be 7, which is considered a reasonable approximation. Additionally, [PERSON]'s [32] research indicated the presence of secondary flow between the blades, suggesting that the air velocity within this interval cannot reach the rotational speed of the blades, and the airflow in the z-plane does not undergo complete rotation. This paper proposes that the velocity distribution in the z-direction can be expressed as follows:  
\[u_{a}=\left\{ \begin{array}{cc}(1-e^{-\frac{N_{b}}{4}})(\frac{z}{R+R_{0}})^{ 1/n}\omega&(0\leq z<h+h_{0})\\ (1-e^{-\frac{N_{b}}{4}})\omega r&(h+h_{0}\leq z<h+H_{1}-h_{1})\\ (1-e^{-\frac{N_{b}}{4}})(\frac{H_{1}+h-z}{h_{1}})^{1/n}\omega r&(h+H_{1}-h_{1} \leq z<h+H_{1})\\ \end{array} \right., \tag{2}\]  
Figure 5: Geometric parameters of the suction unit. Figure 6: Velocity profile in the z-dimension. Substituting (2) into (1) and integrating (1) over \(0\leq z<h+H_{1}\) gives  
\[\frac{\partial P}{\partial r}=(1-e^{-\frac{N_{b}}{4}})(\frac{n}{n+2}+\frac{2}{n+ 2}\frac{H_{b}}{H_{1}+h})\rho\omega^{2}r=\varepsilon\sigma\rho\omega^{2}r \tag{3}\]  
In this equation, \(\varepsilon\) represents the degree of flow development within the z-plane of the suction unit, where \(\sigma\) represents the degree of flow development along the z-axis. It can be observed that both \(\varepsilon\) and \(\sigma\) are less than 1, indicating that the flow within the suction unit is not fully developed.