# Insect-Inspired Micropump: Flow in a Tube with Local Contractions

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results and Discussion

**Figure 1.**Problem schematic given to mimic the insect’s main tracheal tube segment with two contractions [1] and the Stokeslets-mesh-free numerical setup: (

**a**) 3D tube with moving upper wall contraction profile $R(z,t)$; (

**b**) ${g}_{1}\left(t\right)$ and ${g}_{2}\left(t\right)$, the motion protocols assigned to the first and second contractions, respectively.

#### 3.1. Wall Profile, R(z,t)

#### 3.2. Algorithm for Finding Accurate Stokeslets Strengths

Algorithm 1 Algorithm for finding the strengths of each Stokeslet-source point. |

Input: $\mathbb{A},\mathbb{I}\in {M}_{3NX3N}\left(R\right)$, b$\in {R}^{3N}$, h, $tol\in R$ |

1. ${\alpha}^{\left(\mathbf{0}\right)}$ = 0, ${\mathbf{r}}^{\left(\mathbf{0}\right)}$ =b - ${\mathbf{f}}^{\left(\mathbf{0}\right)}$ =b |

2. Do while $\parallel \mathbf{r}{\parallel}_{\infty}\ge \mathbf{tol}$ |

3. ${\mathbb{A}}_{R}={\mathbb{A}}^{*}\mathbb{A}+{h}^{2}\mathbb{I}$ |

4. ${\alpha}^{\left(\mathbf{k}\right)}$= ${\alpha}^{(\mathbf{k}-\mathbf{1})}$+${{\mathbb{A}}_{R}}^{-1}{\mathbb{A}}^{*}{\mathbf{r}}^{(\mathbf{k}-\mathbf{1})}$ |

5. ${\mathbf{r}}^{\left(\mathbf{k}\right)}=\text{b}-\mathbb{A}{\alpha}^{\left(\mathbf{k}\right)}$ |

end Do |

Output: Stokeslets strengths: $\alpha ={\left[{\alpha}_{xj}{\alpha}_{yj}{\alpha}_{zj}\right]}^{T}$, $j=1:3N$ |

#### 3.3. Tube with Two Local Rhythmic Wall Contractions

**Figure 2.**Axial velocity contour lines: (

**a**) 3D computations at $t=T/4$, ${\theta}_{12}={30}^{\circ}$; (

**b**) 3D computations at $t=3T/4$, ${\theta}_{12}={30}^{\circ}$; (

**c**) 2D theory at $t=T/4$, ${\theta}_{12}={30}^{\circ}$; (

**d**) 2D theory at $t=3T/4$, ${\theta}_{12}={30}^{\circ}$.

**Figure 3.**Vertical velocity contour lines: (

**a**) 3D computations at $t=T/4$, ${\theta}_{12}={30}^{\circ}$; (

**b**) 3D computations at $t=3T/4$, ${\theta}_{12}={30}^{\circ}$; (

**c**) 2D theory at $t=T/4$, ${\theta}_{12}={30}^{\circ}$; (

**d**) 2D theory at $t=3T/4$, ${\theta}_{12}={0}^{\circ}$.

**Figure 4.**Pressure contour lines: (

**a**) 3D computations at $t=T/4$, ${\theta}_{12}={30}^{\circ}$; (

**b**) 3D computations at $t=3T/4$, ${\theta}_{12}={30}^{\circ}$; (

**c**) 2D theory at $t=T/4$, ${\theta}_{12}={30}^{\circ}$; (

**d**) 2D theory at $t=3T/4$, ${\theta}_{12}={30}^{\circ}$.

#### 3.4. Unidirectional Net Flow

**Figure 5.**Time-averaged net flow rate comparisons between the 2D analytical solution and 3D Stokeslets-mesh-free computations.

#### 3.5. Channel vs. Tube Simulations

## 4. Conclusions

## Conflicts of Interest

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**MDPI and ACS Style**

Aboelkassem, Y.
Insect-Inspired Micropump: Flow in a Tube with Local Contractions. *Micromachines* **2015**, *6*, 1143-1156.
https://doi.org/10.3390/mi6081143

**AMA Style**

Aboelkassem Y.
Insect-Inspired Micropump: Flow in a Tube with Local Contractions. *Micromachines*. 2015; 6(8):1143-1156.
https://doi.org/10.3390/mi6081143

**Chicago/Turabian Style**

Aboelkassem, Yasser.
2015. "Insect-Inspired Micropump: Flow in a Tube with Local Contractions" *Micromachines* 6, no. 8: 1143-1156.
https://doi.org/10.3390/mi6081143