# An SPH Approach for Non-Spherical Particles Immersed in Newtonian Fluids

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## Abstract

**:**

## 1. Introduction

## 2. Theory of Solid Particle Motion in Single-Phase Newtonian Fluid

#### 2.1. Balance Equations for Single-Phase Fluid Flow

#### 2.2. Introduction to Motion of Solid Particles Immersed in a Fluid

## 3. Numerical Modeling of Fluid Flow with Suspended Particles Using SPH

#### 3.1. SPH for Single-Phase Fluid (Case A and B)

#### 3.2. SPH for Suspension Flow (Case C)

#### 3.3. Solid–Solid Interactions in SPH (Case D)

## 4. Validation of Flow with Suspended Particles

#### 4.1. Details of the Implementation

#### 4.2. Validation of Immersed Particle Flow

**c**. Thus, the rotation of the spheroid over its angular velocity is plotted in Figure 7. As displayed there, the measurement of the rotation in our simulation starts at a point with an angle of $\phi =\pi /2$ and then rotates towards $\phi =0$ which equals $\phi =2\phantom{\rule{0.166667em}{0ex}}\pi $. After that, the particle continues to rotate to an angle of $\phi =\pi $. Even in one single period, the simulation results are in good agreement with the theoretical solution. Differences, especially in the first quarter of rotation ($0<\phi <\pi $), are expected with regard to transient, i.e., instationary phenomena, and due to the fact that the solid particle does not only rotate but also moves translatoric due to solid–fluid interaction at the beginning of the process. Moreover, considering the factor of time (indicated by black arrows), the simulation starts at $\phi =\pi /2$ and the particle is accelerated first until it reaches the theoretically rotation velocity.

**a**and

**b**we observe a rotational motion as well (see Figure 8). Since Jeffery’s approach does not map particle rotation, when the rotation axis of the object is perpendicular to the shear direction ($\theta =0$), a different representation of the rotation was used. Thus, the evolution of the coordinate in ${\mathbf{e}}_{z}$-direction (z) of a surface point normalized by the channel height h was plotted over the time t. As solid particles were not placed right in the center of the channel, the particles move translatoric as well. Figure 8 shows not only the (harmonic) motion of the material point at the surface of the particle, but also the (subtracted) translatoric motion of the center of mass. It could be clearly observed, that there is a stable periodic motion for spheroids with different aspect ratios where the initial state sets the a-axis parallel to the ${\mathbf{e}}_{y}$-axis ($\theta =0$). In addition to (stationary) Jeffery orbits, the numerical simulations predict also transient effects if moderate Reynolds number have been chosen and the reference configuration of the center of mass of the immersed particle is not in the vertical symmetry plane.

**c**). A detailed investigation of the Reynolds number-dependent motion of particles would be computationally challenging for Re < 1 regarding to the explicit nature of the presented SPH code.

## 5. Numerical Analysis of Solid Particles Immersed in a Fluid

## 6. Discussion of Relevant Model Parameters

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Representation of the kernel function W with compact support $\kappa h$ and dependent on particle distance ${r}_{ij}$.

**Figure 2.**Discretization of the fluid (blue) and solid (dark grey) phase using Smoothed Particle Hydrodynamic (SPH) particles and occurring short-range particle interaction forces related to the four introduced cases (A–D).

**Figure 3.**Forces acting on solid body ${P}^{\mathfrak{s}}$ and resulting force ${\mathbf{F}}_{\mathfrak{s}}$ and moment ${\mathbf{M}}_{\mathfrak{s}}$.

**Figure 4.**Workflow of SPH implementation. Blue background indicates own implementation within the HOOMD-blue library [23].

**Figure 5.**Sketch of validation test case with definition of half-axis and rotation angles as presented by Mueller [29].

**Figure 6.**Initial configurations and rotation axis and direction of the three considered cases (

**a**–

**c**).

**Figure 7.**Normalized angular velocity $\dot{\phi}$ over rotation angle $\phi $. Comparison of SPH results (simulation

**c**) with analytical solution by Jeffery [9] for ${r}_{e}=0.3$. Black arrows additionally show the parameter of time from the start of the simulation at ${t}_{0}$ to the end ${t}_{n}$.

**Figure 8.**Motion of ${\mathbf{e}}_{z}$-coordinate over time t for two different aspect ratios (case

**a**: ${r}_{e}=0.2$, case

**b**: ${r}_{e}=5.0$).

**Figure 9.**3-dimensional view of the simulation results showing streamlines of the velocity ${v}_{x}$ in the channel. Prolate spheroid is aligned with ${\mathbf{e}}_{x}$-axis and does not perform any rotation for a Reynolds number of Re = 10.

**Figure 10.**Sketch of simulated boundary value problem (BVP) of shear induced suspension flow in a channel of height h in ${\mathbf{e}}_{z}$-direction. The simulation domain contains periodic boundary conditions in direction of ${\mathbf{e}}_{x}$ and ${\mathbf{e}}_{y}$. Upper and lower wall particles are moved with a constant velocity ${v}_{x}=0.005\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}$ corresponding to an input shear rate of $\dot{\gamma}=2\left|{v}_{w}\right|/h=0.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$.

**Figure 11.**Resulting profile of velocity in ${\mathbf{e}}_{x}$-direction over coordinate ${\mathbf{e}}_{z}$ for spherical solids (

**left**) and ellipsoidal solids (

**right**). While the black line represents the analytical solution for simple shear flow of a Newtonian fluid, the red line represents the mean velocity of a simulation of shear flow without solid aggregates (only fluid). Colored markers and and the same colored line represent SPH particle velocities and mean particle velocity, respectively. Investigated cases are shear flow with two spherical (pink,

**top left**), two ellipsoidal (brown,

**top right**), four spherical (blue,

**bottom left**) and four ellipsoidal (green,

**bottom right**) solid bodies.

**Figure 12.**Motion of the center of masses over time for simulations containing two and four solid particles. Solid and dashed lines represent spherical particles and non-spherical particles, respectively.

**Figure 13.**Streamlines of the velocity ${v}_{x}$-direction at various time steps considering the ${\mathbf{e}}_{x}$-${\mathbf{e}}_{x}$-plane for simulation containing four ellipsoidal solid particles.

**Figure 14.**3-dimensional view of streamlines at timestep $t=8.5\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$.

**Figure 15.**Velocity trajectories of ${v}_{x}$ for the evolution of contact between two solid particles over time (top left to bottom right). Contact occurs at $t=1.91\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ and again at $t=2.60\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. Due to hydromechanical and hydrodynamical forces, particles stay close to each other even after contact.

${\mathit{r}}_{\mathit{e}}$ | Re | ${\mathit{\theta}}_{0}$ | ${\mathit{\phi}}_{0}$ | |
---|---|---|---|---|

a | 0.2 | 10.0 | 0 | undefined |

b | 5.0 | 10.0 | 0 | undefined |

c | 0.3 | 1.0 | $\pi /2$ | 0 |

**Table 2.**Summary of parameters of solid aggregates for the four different simulation scenarios. Spherical solids are defined by their radius r while ellipsoidal solids are defined by their half-axes a, b and c (dimension in ${\mathbf{e}}_{x}$-, ${\mathbf{e}}_{y}$- and ${\mathbf{e}}_{z}$-direction, respectively).

Simulation with | Parameter of Aggregates [mm] | |||
---|---|---|---|---|

2 spherical solids | r = 3.0 | r = 4.0 | ||

2 elliptical solids | a = 3.0 b = 2.0 c = 1.0 | a = 2.2 b = 4.0 c = 3.0 | ||

4 spherical solids | r = 2.6 | r = 2.6 | r = 3.0 | r = 3.0 |

4 elliptical solids | a = 2.0 b = 1.0 c = 3.0 | a = 3.0 b = 4.0 c = 2.0 | a = 3.0 b = 2.0 c = 1.0 | a = 2.0 b = 1.0 c = 3.8 |

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**MDPI and ACS Style**

Kijanski, N.; Krach, D.; Steeb, H.
An SPH Approach for Non-Spherical Particles Immersed in Newtonian Fluids. *Materials* **2020**, *13*, 2324.
https://doi.org/10.3390/ma13102324

**AMA Style**

Kijanski N, Krach D, Steeb H.
An SPH Approach for Non-Spherical Particles Immersed in Newtonian Fluids. *Materials*. 2020; 13(10):2324.
https://doi.org/10.3390/ma13102324

**Chicago/Turabian Style**

Kijanski, Nadine, David Krach, and Holger Steeb.
2020. "An SPH Approach for Non-Spherical Particles Immersed in Newtonian Fluids" *Materials* 13, no. 10: 2324.
https://doi.org/10.3390/ma13102324