## 2 Working Principle and Kinematic Modeling of Extension-Dominated Mixer  
### Working Principle  
The internal meshing single-screw device primarily consists of a stator with a spiral cavity, created by rotating the inner surface of an elliptical long hole with a specific pitch, and an eccentrically meshing screw that conjugates with the stator's profile surface featuring a spiral cavity, as illustrated in Figure 1. In the stator's cross-section with the spiral cavity, the movement track's center of the eccentric meshing screw forms a circle's diameter, traversing a linear distance. This implies that the eccentric screw reciprocates along a straight line between the spiral cavity's two centers, executing pure rolling within the stator's spiral cavity. Throughout this process, the material, observed from a cross-sectional perspective, is driven by the eccentric screw to oscillate within the stator's spiral cavity, undergoing stretching, folding, and rotation within the flow field. Differing from the performance criteria of NETZSCH pumps (Figure 1a), which predominantly emphasize transmission efficiency, the internal meshing screw device (Figure 1b) aims to enhance tensile mixing performance and augment disturbance. To achieve this, flat and spiral sections are alternately arranged on both the internal meshing screw and its corresponding stator. Additionally, the pitch and eccentricity are reduced, gradually decreasing along the axis. The incorporation of spiral and straight sections facilitates material volume deformation during transportation, leading to periodic volume stretching [15]. The meshing stator profile with a spiral cavity and the eccentric screw profile in the internal meshing single-screw device is designed based on the internal cycloid. Generally, when a circle moves within a fixed circle, the trajectory formed by any point \(M\) on the moving circle is an end cycloid [33; 34], shown in Figure 2a. The radius of the moving circle \(R_{2}\) is one-third of the radius of the fixed circle \(R_{1}\). In a specific scenario, if the radius of the moving circle is half of the radius of the fixed circle, and the point lies on the moving circle's circumference, the trajectory of the point is the fixed circle's center diameter. Building upon this principle, any section of the rotor in the axial direction forms a circle of radius. During operation, the rotor's motion state depicts rotation around center B, which is the rotation center positioned at a distance e eccentricity from the geometric center A, while simultaneously rotating around the stator's center (also the center of revolution, denoted as center C), maintaining the same eccentricity e from point B. According to hypocycloid theory [35], the rotor's geometric center A's motion state involves rolling along the stator's linear bone line. Since the rotor's linear shape is a conjugate curve, resulting in a circular shape in any section, and the stator's linear shape is formed by two identical semi-circular arcs and two straight lines, its shape remains consistent across sections. The linear shape of the fixed rotor merely rotates to different angles across various sections. The formation principle of the internal meshing screw mixer and stator profile [36] in this paper is depicted in Figure 2.  
Figure 1: Internal meshing screw (eccentric rotor) structure: (**a**) The structure of the NETZSCH pump screw; (**b**) The structure of internal meshing screw mixer; (**c**) The stator and the rotors with different cross-sections. Figure 2: (**a**) Formation principle of internal cycloid; (**b**) Formation principle of Internal meshing screw mixer and stator profile. Figure 3 illustrates the positional diagram of each point at different time intervals during the rotor's movement.