# Accurate Solutions to Non-Linear PDEs Underlying a Propulsion of Catalytic Microswimmers

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## Abstract

**:**

## 1. Introduction

## 2. Model and Governing Equations

## 3. Results and Examples

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Sketch of a catalytic microswimmer with a nonuniform release of ions; (

**b**) Inner and outer regions for fields of concentration and electric potential.

**Figure 2.**${\Phi}_{s}$ as a function of $\theta $ for swimmers of the first (

**a**) and second (

**b**) type calculated using $\mathrm{Da}=0.5$ (dashed curve), 1 (solid curve) $1.5$ (dash-dotted curve).

**Figure 3.**An apparent slip velocity as a function of $\theta $ for swimmers of the first (

**a**) and second (

**b**) type calculated using ${\varphi}_{s}=-0.5$ and $\mathrm{Da}=0.5$ (dashed curve), 1 (solid curve) $1.5$ (dash-dotted curve).

**Figure 4.**A propulsion velocity of swimmers of the first type (

**a**) calculated using ${\varphi}_{s}=0.5$, 0, and $-0.5$ (solid curves from top to bottom) and second type (

**b**) computed with the same surface potentials (solid curves from bottom to top). Dashed lines correspond to calculations within the linear theory.

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**MDPI and ACS Style**

Asmolov, E.S.; Nizkaya, T.V.; Vinogradova, O.I. Accurate Solutions to Non-Linear PDEs Underlying a Propulsion of Catalytic Microswimmers. *Mathematics* **2022**, *10*, 1503.
https://doi.org/10.3390/math10091503

**AMA Style**

Asmolov ES, Nizkaya TV, Vinogradova OI. Accurate Solutions to Non-Linear PDEs Underlying a Propulsion of Catalytic Microswimmers. *Mathematics*. 2022; 10(9):1503.
https://doi.org/10.3390/math10091503

**Chicago/Turabian Style**

Asmolov, Evgeny S., Tatiana V. Nizkaya, and Olga I. Vinogradova. 2022. "Accurate Solutions to Non-Linear PDEs Underlying a Propulsion of Catalytic Microswimmers" *Mathematics* 10, no. 9: 1503.
https://doi.org/10.3390/math10091503