# Analysis of Smoluchowski’s Coagulation Equation with Injection

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## Abstract

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## 1. Introduction

## 2. Smoluchowski’s Coagulation Equation with Injection

## 3. Exact Analytical Solutions for Steady-State Coagulation with Injection

#### 3.1. Integer Values of the Parameter $\nu $

#### 3.2. Half-Integer Values of the Parameter $\nu $

#### 3.3. Source Term with Sub-Linear Prefactor and Exponential Decay

## 4. Unsteady-State Smoluchowski’s Coagulation Equation

#### 4.1. Analytical Solution to the Coagulation Equation without Injection

#### 4.2. Approximate Solution to the Unsteady Coagulation Equation with Injection

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Theory (expression (22)) is compared with experimental distribution (Figure 3E in Ref. [10]) after 45 min internalization of low-density lipoprotein (LDL) for four different LDL concentrations: $\gamma =0.0015$ s${}^{-1}$; $K=0.00016$ s${}^{-1}$, ${x}_{0}=450$ FI; $J=546$ FI s${}^{-1}$ (blue solid line) and $J=300$ FI s${}^{-1}$ (red dashed line).

**Figure 3.**The unsteady-state density distribution functions $n(x,\tau )$ at different times accordingly to an analytical solution (37) (lines) and experimental data [10] (symbols). System parameters correspond to Figure 1 and $J=546$ FI s${}^{-1}$, ${b}_{0}^{\prime}={b}_{s}^{\prime}=5$ × ${10}^{5}$. The initial distribution function ${n}_{0}\left(x\right)$ was chosen at ${\tau}_{0}=3$ min.

$\mathit{\nu}$ | $\mathit{I}\left(\mathit{x}\right)$ | ${\mathit{n}}_{\mathit{s}}\left(\mathit{x}\right)$ | $\tilde{\mathit{I}}\left(0\right)$ | ${\mathit{I}}_{*}$ | $\mathit{q}\left(\mathit{\nu}\right)$ |
---|---|---|---|---|---|

$1/2$ | ${I}_{*}{x}^{-1/2}exp\left(-\frac{x}{{x}_{0}}\right)$ | Equation (24) | $\frac{2J}{{x}_{0}}$ | $\frac{2J}{\sqrt{\pi}{x}_{0}^{3/2}}$ | $\frac{4J}{K{x}_{0}^{3/2}}$ |

1 | ${I}_{*}exp\left(-\frac{x}{{x}_{0}}\right)$ | Equation (14) | $\frac{J}{{x}_{0}}$ | $\frac{J}{{x}_{0}^{2}}$ | $\frac{2J}{K{x}_{0}^{2}}$ |

2 | ${I}_{*}xexp\left(-\frac{x}{{x}_{0}}\right)$ | Equation (16) | $\frac{J}{2{x}_{0}}$ | $\frac{J}{2{x}_{0}^{3}}$ | $\frac{J}{K{x}_{0}^{3}}$ |

$5/2$ | ${I}_{*}{x}^{3/2}exp\left(-\frac{x}{{x}_{0}}\right)$ | Equation (26) | $\frac{2J}{5{x}_{0}}$ | $\frac{8J}{15\sqrt{\pi}{x}_{0}^{7/2}}$ | $\frac{4J}{5K{x}_{0}^{7/2}}$ |

3 | ${I}_{*}{x}^{2}exp\left(-\frac{x}{{x}_{0}}\right)$ | Equation (18) | $\frac{J}{3{x}_{0}}$ | $\frac{J}{6{x}_{0}^{4}}$ | $\frac{2J}{3K{x}_{0}^{4}}$ |

4 | ${I}_{*}{x}^{3}exp\left(-\frac{x}{{x}_{0}}\right)$ | Equation (20) | $\frac{J}{4{x}_{0}}$ | $\frac{J}{24{x}_{0}^{5}}$ | $\frac{J}{2K{x}_{0}^{5}}$ |

8 | ${I}_{*}{x}^{7}exp\left(-\frac{x}{{x}_{0}}\right)$ | Equation (22) | $\frac{J}{8{x}_{0}}$ | $\frac{J}{40,320{x}_{0}^{9}}$ | $\frac{J}{4K{x}_{0}^{9}}$ |

- | ${I}_{*}sin\left(\omega x\right)exp\left(-\frac{x}{{x}_{0}}\right)$ | Equation (29) | $\frac{({\omega}^{2}{x}_{0}^{2}+1)J}{2{x}_{0}}$ | $\frac{{({\omega}^{2}{x}_{0}^{2}+1)}^{2}J}{2\omega {x}_{0}^{3}}$ | - |

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**MDPI and ACS Style**

Makoveeva, E.V.; Alexandrov, D.V.; Fedotov, S.P. Analysis of Smoluchowski’s Coagulation Equation with Injection. *Crystals* **2022**, *12*, 1159.
https://doi.org/10.3390/cryst12081159

**AMA Style**

Makoveeva EV, Alexandrov DV, Fedotov SP. Analysis of Smoluchowski’s Coagulation Equation with Injection. *Crystals*. 2022; 12(8):1159.
https://doi.org/10.3390/cryst12081159

**Chicago/Turabian Style**

Makoveeva, Eugenya V., Dmitri V. Alexandrov, and Sergei P. Fedotov. 2022. "Analysis of Smoluchowski’s Coagulation Equation with Injection" *Crystals* 12, no. 8: 1159.
https://doi.org/10.3390/cryst12081159