# Distribution and Deposition of Cylindrical Nanoparticles in a Turbulent Pipe Flow
## 5 Results and Discussion
### Distribution of Particle Volume Concentration and Orientation at Different Axial Positions  
When the particles enter the tube, the particles show different distributions of the concentration and orientation at different axial positions with the development of time. The distributions of particle concentration and orientation can be obtained by solving Equations (10) and (7). The distributions of particle volume concentration on the cross-section at different axial positions are shown in Figure 4a. The uniform distribution of volume concentration on the cross-section at inlet becomes non-uniform along the flow direction because the distribution of particle number density is affected by the shear-induced force caused by the mean shear rate of the fluid, Brownian diffusion and turbulent diffusion as shown in Equations (10) and (12). The non-uniformity becomes obvious with the increase in the axial distance from the inlet. The particle volume concentration first decreased and then increased from the pipe center to the wall, and there was a minimum around \(r/R=0.95\). As we know, Brownian diffusion makes particles move from a high concentration area to a low concentration one, resulting in a more uniform distribution of the particle volume concentration. So, it can be deduced that the non-uniform distributions of particle volume concentration were caused by the shear-induced force and turbulent diffusion. The particles in the region around \(r/R\approx 0.95\) were subjected to the influence of larger shear-induced force and stronger turbulent diffusion because the mean shear rate and the turbulent diffusion coefficient \(v_{t}\) were large (\(v_{t}=0.09\)\(k^{2}/\epsilon\), while turbulent kinetic energy \(k\) is large there), hence were diffused to the region near the centerline and the region close proximity to the wall where the turbulent diffusion diminished to zero and particles accumulated in the viscous sublayer with low fluid velocity.  
The distributions of the mean orientation of a cylindrical nanoparticle at different axial positions are shown in Figure 4b where the mean orientation angle \(\phi\) was obtained by averaging the orientation angles on the cross-section at a fixed axial position. The particles with uniform distribution of mean orientation angle on the cross-section at inlet will change their orientation angles along the flow direction. The distributions of orientation angle become non-uniform, more and more particles aligned with their major axis near to the flow direction. As shown in Equations (13)-(15), the mean probability density function for particle orientation was affected by the Brownian diffusion, turbulent diffusion and the shear-induced force, and the former two made the distribution of particle orientation become more uniform. The shear-induced force resulted from the mean shear rate of the fluid and the overall effect can be decomposed in two torques, one making a particle rotate around the vorticity axis (i.e., axis \(\theta\) as shown in Figure 1), and another causing a particle spin around the flow direction, i.e., particles tended to align according to the mean shear of  
Figure 3: Penetration efficiency as a function of particle diameter (\(\lambda=1\), Re = 5120, \(L/D=375\)). \(\bullet\): numerical results and \(\bullet\): experimental results [25].  
the flow. The particles flowing through the pipe were always subjected to the influence of the mean shear of the flow, so align with the flow direction.