## 1 Introduction
### Movement of Non-Spherical Particles within Polymer Composite  
The anisotropic particles are generally considered to be ellipsoid form with the following aspect ratio:  
\[E=\frac{d_{II}}{d_{\perp}} \tag{10}\]  
Figure 1: Drag coefficient for a smooth solid sphere at various Reynolds numbers for incompressible Figure 1. Reprinted from [8] with permission from Elsevier. where \(d_{II}\) and \(d_{\perp}\) are the parallel and normal diameters, corresponding to the semi-minor and semi-major axis lengths respectively. The ellipsoid's volume is proportional to the diameters of the three major axes, so the effective diameter is given by:  
\[d=d_{\perp}E^{1/3}=d_{II}E^{-2/3} \tag{11}\]  
\(E=1\), it reverts the geometry to a sphere. Regular-shaped particles generally include simple cross-sections, which are made up of connecting orthogonal rectangles, which are traced back to crystallization. In contrast, irregular-shaped particles are from random coagulation or also generated from the fragmentation of bigger particles [23; 24; 25]. The drag of irregular particles generally depends on the degree of non-sphericity as well as their orientation to the flow. The Stokes correction for non-spherical particle ratio could be defined as:  
\[f_{E}=\frac{F_{D}\big{(}E,R_{ep}\to 0\big{)}}{3\pi d\mu_{f}w} \tag{12}\]  
Thus,\(f_{E}\) approaches unity as \(E\) approaches unity. The variations in \(E\) from 0.01 to 100 result in moderate changes in the overall drag correction. This arises because increases in friction drag for longer bodies are approximately balanced by decreases in pressure drag, while the reverse is true for shorter bodies [26; 27]. Drag force for particles whose aspect ratio is unity can often be reasonably estimated by simply using Stokesian drag and the volume-based diameter. If the initial orientation is parallel or perpendicular to the relative velocity, no torque exists since the drag acts in opposite direction to the gravitational force. If the initial orientation is neither parallel nor perpendicular, the net drag force may be obtained based on a simple combination of the individual components,  
\[F_{D}=-3\pi\mu_{f}(w_{II}f_{EII}+w_{\perp}f_{E\perp})\text{ for }R_{ep}\ \ll\ 1 \tag{13}\]  
where \(w_{II}\) and \(w_{\perp}\) are the components of relative velocity, which are parallel and perpendicular to the particles axis of symmetry. If both velocity components are non-zero, drag force is not parallel to \(w\) so that particles will have an angle with respect to \(g\). If the particles have three axis of direction, becomes orthotropic in nature, the drag force generates from combination of all the components is: [28]  
\[F_{D}=-3\pi\mu_{f}(w_{1}f_{E1}+w_{2}f_{E2}+w_{3}f_{E3}),\text{ for }R_{ep}\ \ll\ 1 \tag{14}\]  
In Brownian motion, all orientations are assumed equally possible since molecular interaction is random. By integrating all factors for average force, the drag one could be represented by:  
\[\frac{3}{\langle f_{E}\rangle}=\frac{1}{f_{E1}}+\frac{1}{f_{E2}}+\frac{1}{f_{ E3}}\text{ for }R_{ep}\ \ll\ 1 \tag{15}\]  
The drag coefficients for irregular particles are shown in Figure 2. In addition to the orientation of the filler particles, the minor effect of roughness and turbulence affects significantly on particles boundary layer to convert into turbulence flow than a steady flow. As a result, a drag crisis occurs at lower Reynolds numbers in the presence of turbulence, due to the effect of particles roughness [29; 30]. The drag coefficient for regular and irregular shapes is based on the Reynolds numbers. The aspect ratio plays a crucial role in determining the shape of the particles; thus, the surface area ratio is the optimum for regular particles, whilst max-med-min ratio is considered for irregular particles [31; 32]. Adhesion is defined as the strength of a bond uniting two different components often made of two different materials.