# A Framework for Generating Radial and Surface-Oriented Regularized Stokeslets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Stokeslet and Regularized Stokeslet

#### 2.2. A Regularization Approach Utilizing the Vector Potential of the Stokeslet

#### 2.3. Radially Symmetric Regularizations

#### 2.3.1. Formulation of Smoothing Factors

**Theorem**

**1.**

- 1.
- The regularized velocity field scales as ${\mathit{G}}_{\u03f5}\left(\mathit{r}\right)={\u03f5}^{-1}{\mathit{G}}_{1}(\mathit{r}/\u03f5)$ and blob function scales according to (7).
- 2.
- The regularized velocity field ${\mathit{G}}_{\u03f5}$ is bounded and the blob function ${\varphi}_{\u03f5}$ is also bounded.
- 3.
- ${\varphi}_{\u03f5}$ integrates to unity over ${\mathbb{R}}^{3}$, as required by (6).

**Proof.**

#### 2.3.2. Error Analysis

#### 2.4. Surface-Oriented Regularizations

**Theorem**

**2.**

**Proof.**

#### 2.5. Test Case: A Translating Sphere

## 3. Results

#### 3.1. Radial Regularizations

#### 3.1.1. Example Smoothing Factors

#### 3.1.2. Forward Problem

#### 3.1.3. Inverse Problem

#### 3.2. A Surface-Oriented Regularization

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MRS | Method of Regularized Stokeslets |

IB | Immersed Boundary |

## Appendix A. Moments of Radial Blob Functions

## Appendix B. Expressions and Computational Cost of Regularized Stokeslets

**Table A1.**Functions ${h}_{1}$ and ${h}_{3}$ that are used in (18) to generate regularized Stokeslets by differentiating the radial smoothing factors given in Table 1. We have set $\u03f5=1$ for simplicity, but note that these functions obey the scaling properties ${h}_{m}(\u03f5;r)={\u03f5}^{-m}{h}_{m}(1,r/\u03f5)$ for $m=1,3$. The last column gives the time needed to evaluate both ${h}_{1}$ and ${h}_{3}$ together, relative to the alg2 regularization.

Label | ${\mathit{h}}_{1}(1;\mathit{r})$ | ${\mathit{h}}_{3}(1;\mathit{r})$ | Time |
---|---|---|---|

alg2 | $\frac{{r}^{2}+2}{{\left({r}^{2}+1\right)}^{3/2}}$ | $\frac{1}{{\left({r}^{2}+1\right)}^{3/2}}$ | $1.00$ |

alg4 | $\frac{2{r}^{4}+5{r}^{2}+6}{2{\left({r}^{2}+1\right)}^{5/2}}$ | $\frac{2{r}^{2}+5}{2{\left({r}^{2}+1\right)}^{5/2}}$ | $1.06$ |

tanh | $\frac{{tanh}^{2}r}{r}+{sech}^{2}r$ | $\frac{tanhr}{{r}^{2}}-\frac{{sech}^{2}r}{{r}^{3}}$ | $2.07$ |

erf | $\frac{erfr}{r}+\frac{2{e}^{-{r}^{2}}}{\sqrt{\pi}}$ | $\frac{erfr}{{r}^{3}}-\frac{2{e}^{-{r}^{2}}}{\sqrt{\pi}{r}^{2}}$ | $5.26$ |

alg2-c | $\frac{{r}^{4}+2{r}^{2}+4}{{\left({r}^{2}+1\right)}^{5/2}}$ | $\frac{{r}^{2}+4}{{\left({r}^{2}+1\right)}^{5/2}}$ | $1.06$ |

alg4-c | $\frac{2{r}^{6}+7{r}^{4}+2{r}^{2}+12}{2{\left({r}^{2}+1\right)}^{7/2}}$ | $\frac{2{r}^{4}+7{r}^{2}+20}{2{\left({r}^{2}+1\right)}^{7/2}}$ | $2.04$ |

tanh-c | $\begin{array}{cc}& (2ln2{sech}^{2}r+1)\frac{tanhr}{r}\hfill \\ & \phantom{\rule{1.em}{0ex}}+(6ln2{sech}^{2}r-4ln2+1){sech}^{2}r\hfill \end{array}$ | $\begin{array}{cc}& (2ln2{sech}^{2}r+1)\frac{tanhr}{{r}^{3}}\hfill \\ & \phantom{\rule{1.em}{0ex}}-(6ln2{sech}^{2}r-4ln2+1)\frac{{sech}^{2}r}{{r}^{2}}\hfill \end{array}$ | $2.56$ |

erf-c | $\frac{erfr}{r}-\frac{(4{r}^{2}-6){e}^{-{r}^{2}}}{\sqrt{\pi}}$ | $\frac{erfr}{{r}^{3}}+\frac{(4{r}^{2}-2){e}^{-{r}^{2}}}{\sqrt{\pi}{r}^{2}}$ | $5.31$ |

## Appendix C. Matrix Condition Numbers

**Figure A1.**Skeel’s condition number for the inverse problem of a translating sphere described in Section 2.5 using (

**a**) uncorrected radial, (

**b**) corrected radial, and (

**c**) surface-oriented regularizations. For the radial (

**a**,

**b**) and surface-oriented (

**c**) regularizations, $n=4096$ and $n=1024$ discretization points are used, respectively. The shaded regions indicate the values of $\u03f5$ where the error in the drag is approximately minimized.

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**Figure 1.**Behavior of the radially symmetric smoothing factors ${s}_{1}$ (

**a**,

**b**) and blob functions ${\varphi}_{1}$ (

**c**,

**d**) given in Table 1, plotted as a function of $r=\parallel \mathit{r}\parallel $. All smoothing factors have been normalized by substituting $r\to r\sqrt[3]{{\varphi}_{1}\left(0\right)}$ in the arguments to ${s}_{1}$, and the blob functions correspond to the normalized smoothing factors. No boundary corrections are utilized in (

**a**,

**c**) whereas (

**b**,

**d**) incorporates the boundary-velocity correction given by (25).

**Figure 2.**Error (sup-norm) in the computed velocity field on the sphere boundary versus the regularization parameter $\u03f5$ for the forward problem of a unit sphere translating at unit velocity $\mathit{U}$ in a fluid with unit viscosity. Here, the surface traction is prescribed as the constant vector $\mathit{q}=3\mathit{U}/2$. The error is given as the sup-norm of $\mathit{u}\left(\mathit{x}\right)-\mathit{U}$ for $\mathit{x}$ in the set of discretization points on the sphere surface. The regularizations listed in the legend are those derived from the radial smoothing factors given by Table 1. Smoothing factors utilized in (

**a**) leave the boundary velocity uncorrected whereas in (

**b**), corrections according to (25) are included. In (

**b**), a log-log scale is used to show the power-law dependence (dashed lines) of the error on $\u03f5$ in the discretization-error- and regularization-error-dominated regimes.

**Figure 3.**Similar to Figure 2, but for the corresponding inverse problem. Here, the error is the relative error with respect to the analytical result for the drag on the sphere, ${D}_{\mathrm{err}}=D/6\pi -1$, where D is the drag. Results for uncorrected and corrected smoothing factors are shown in (

**a**) and (

**b**), respectively. In (

**b**), a log-log scale is used, and the absolute value of ${D}_{\mathrm{err}}$ is plotted. We use `▼’s to indicate under-prediction (${D}_{\mathrm{err}}<0$) and `▲’s to indicate drag over-prediction (${D}_{\mathrm{err}}>0$). The dashed lines indicate the power-law scaling of ${D}_{\mathrm{err}}$ in the regularization-error-dominated regime.

**Figure 4.**Error in the computed drag versus number of discretization points n using the erf-c regularization. The absolute value of the error is shown, but the direction of the triangle indicates the sign, following the same convention as that in Figure 3.

**Figure 5.**Force density profiles of the surface-oriented regularized Stokeslets, derived from (31), with $\u03f5=1$. The orientation is along the z-axis. (

**a**,

**b**) plot the cross section of ${\varphi}_{1}^{a}$ through the $xy$- and $xz$- planes, respectively, and (

**c**,

**d**) similarly plot ${\varphi}_{1}^{c}$. A detailed profile of ${\varphi}_{1}^{a}$ along the x- (or y-) axis is shown in (

**e**), as well as parallel to the z- axis for $\rho =0$ and $\rho =1$ in (

**f**). Similar plots of ${\varphi}_{1}^{c}$ are shown in (

**g**,

**h**). Values sampled along particular lines in (

**e**–

**h**) are indicated by dashed lines in (

**a**–

**d**), where there is a correspondence between columns of figures. For example, the blue and orange dashed lines in (

**b**) respectively correspond to the blue and orange force density profiles in (

**f**).

**Figure 6.**Error of surface oriented regularizations versus the radial alg2 regularization. The error in the drag is shown in (

**a**) and the error in the boundary condition or “leak” at points on the surface of the sphere but in between the discretization points is shown in (

**b**).

**Table 1.**Radially symmetric smoothing factors ${s}_{1}$ in use in (17) and their corresponding blob functions ${\varphi}_{1}$. Here, $r=\parallel \mathit{r}\parallel =\parallel \mathit{x}-\mathit{y}\parallel $ for a point force at $\mathit{y}$. The “label” corresponds to that used in figure captions for the particular regularized Stokeslets and blobs that correspond to these smoothing factors. Smoothing factors are given with $\u03f5=1$; recall that ${s}_{\u03f5}={s}_{1}(r/\u03f5)$. The correction is the term added to the original smoothing factor (in the second column) to satisfy (25) at discretization points on the boundary. Formulas for ${h}_{1}$ and ${h}_{3}$ from (18) are given separately in Appendix B, Table A1.

Label | ${\mathit{s}}_{1}$ | Correction Term | ${\mathit{\varphi}}_{1}$ | Correction Term |
---|---|---|---|---|

alg2 | $\frac{r}{\sqrt{{r}^{2}+1}}$ | $\frac{r}{{({r}^{2}+1)}^{3/2}}$ | $\frac{15}{8\pi {({r}^{2}+1)}^{7/2}}$ | $-\frac{15(4{r}^{2}-3)}{8\pi {({r}^{2}+1)}^{9/2}}$ |

alg4 | $\frac{r\left(2{r}^{2}+3\right)}{2{\left({r}^{2}+1\right)}^{3/2}}$ | $\frac{3r}{{({r}^{2}+1)}^{5/2}}$ | $\frac{15(5-2{r}^{2})}{16\pi {({r}^{2}+1)}^{9/2}}$ | $\frac{15(8{r}^{4}-40{r}^{2}+15)}{16\pi {({r}^{2}+1)}^{11/2}}$ |

tanh | $tanh\left(r\right)$ | $2ln2tanhr{sech}^{2}r$ | $\frac{(r+4tanhr-3r{tanh}^{2}r){sech}^{2}r}{4\pi r}$ | $\mathrm{See}\phantom{\rule{4.pt}{0ex}}(*)\phantom{\rule{4.pt}{0ex}}\mathrm{below}.$ |

erf | $erf\left(r\right)$ | $\frac{2r{e}^{-{r}^{2}}}{\sqrt{\pi}}$ | $\frac{(5-2{r}^{2}){e}^{-{r}^{2}}}{2{\pi}^{3/2}}$ | $\frac{\left(4{r}^{4}-20{r}^{2}+15\right){e}^{-{r}^{2}}}{2{\pi}^{3/2}}$ |

$\frac{ln2}{\pi r}(-15r{sech}^{2}r{tanh}^{2}r+2r-12{tanh}^{3}r+8tanhr){sech}^{2}r.$ | (*) |

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Chisholm, N.G.; Olson, S.D. A Framework for Generating Radial and Surface-Oriented Regularized Stokeslets. *Fluids* **2022**, *7*, 351.
https://doi.org/10.3390/fluids7110351

**AMA Style**

Chisholm NG, Olson SD. A Framework for Generating Radial and Surface-Oriented Regularized Stokeslets. *Fluids*. 2022; 7(11):351.
https://doi.org/10.3390/fluids7110351

**Chicago/Turabian Style**

Chisholm, Nicholas G., and Sarah D. Olson. 2022. "A Framework for Generating Radial and Surface-Oriented Regularized Stokeslets" *Fluids* 7, no. 11: 351.
https://doi.org/10.3390/fluids7110351