# Simulation of Convection–Diffusion Transport in a Laminar Flow Past a Row of Parallel Absorbing Fibers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flow Field in a System of Parallel Fibers at Re > 1

**u**= 0 is set at the fiber surface 4, while the condition of undisturbed flow

**u**= {1,0} is set at the entrance of the simulation cell 1; the condition of vanishing viscous stresses (zero gradients) is applied at the outlet boundary 3; and at the boundaries labelled 2, the conditions of symmetry for the velocity components are used. The dimensionless fiber drag force per unit length was found as the surface integral of the projection of the local total stress on the flow direction:

**n**is the outer normal vector to the surface, $d\mathsf{\Sigma}$ is the surface element, and ${S}_{g}$ is the fiber surface area. The fiber drag force is related with the dimensionless pressure drop across the fiber row as shown by $\Delta p=Fa/2h$.

## 3. Convection–Diffusion Mass Transfer in a System of Absorbing Fibers

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix

Coefficients in Equation (A1) | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{5}$ |
---|---|---|---|---|---|

A | 2.1217 | 289.9297 | 26.9531 | 39.913 | −300.7297 |

B | −1.1934 | −215.0368 | −19.077 | −31.4594 | 222.6869 |

C | 0.2835 | 61.5072 | 5.3081 | 9.5571 | − 63.5653 |

D | −0.0312 | −7.4111 | −0.635 | −1.1996 | 7.654 |

E | 0.0013 | 0.3224 | 0.0277 | 0.5381 | −0.3330 |

Coefficientsin Equation (A1) | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{5}$ | ${a}_{6}$ | ${a}_{7}$ |
---|---|---|---|---|---|---|---|

A | −16,813.6833 | −8371.0393 | −737.937 | 10.4997 | 1.348 | −24.9981 | 16,818.5536 |

B | 5090.8957 | 2496.3936 | 264.9326 | −39.2739 | 5.7935 | 31.2227 | −5096.2083 |

C | 176.1791 | 107.063 | −13.1498 | 18.2408 | −3.2293 | −11.6302 | −174.2677 |

D | 42,356.9489 | 21,175.9228 | 1767.2858 | 57.3744 | 1.0235 | 1.5894 | −42,357.2119 |

E | −406.1734 | −202.9051 | −17.1196 | −0.4041 | −0.0559 | −0.1074 | 406.1907 |

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**Figure 1.**A scheme of the computational cell: flow past a row of parallel fibers, where the Reynolds number (Re) = 20 and $a/h$ = 0.5; the flow direction is shown by the arrow. The streamline ordinates are: 10

^{−4}(a), 0.2 (b), 0.5 (c), 1 (d), and 1.5 (e); 1—inlet, 2—symmetry boundaries, 3—outlet, 4—fiber surface.

**Figure 2.**The dimensionless fiber drag force (F) in a row of fibers vs. the Reynolds number for different distances between the fiber axes with $a/h$ computed with the step 0.05 starting from $a/h$ = 0.1 (curve 1) to 0.9 (curve 2); curve 3 is determined by the regression formula for an isolated fiber, Equation 7 [32]; and curve 4 is determined by the Lamb formula for an isolated fiber [33].

**Figure 3.**The fiber drag force in a row of fibers vs. the Reynolds number; in the case of low Reynolds numbers and sparse rows: $a/h$ = 0.1 (1), 0.01 (2), 0.001 (3); 4 is given by the regression formula for an isolated cylinder, Equation (7) [32]; 5 is given by the Lamb formula for an isolated cylinder [33]; horizontal asymptotic lines (⋅⋅⋅⋅⋅)—values for the Stokes flow limit at Re ≪ 1.

**Figure 4.**The fiber drag force in a row of fibers vs. the Reynolds number: in the case of intermediate Reynolds numbers, the $a/h$ values are marked on the curves.

**Figure 5.**The flow and concentration C fields past a row of fibers at different values of the Reynolds number, where Sc = 1000, $a/h$ = 0.5, and the streamline inlet ordinates are 10

^{−4}, 0.2, 0.5, 1, and 1.5.

**Figure 6.**Fiber retention efficiencies per unit length in a row of fibers with different interfiber distances: (

**a**) $a/h$ = 0.2 and (

**b**) 0.5, vs. Reynolds number, found from the combined numerical solution of the Navier–Stokes and convection–diffusion equations (curves 1): Schmidt numbers are marked on the curves. Curves 2 were plotted by Equation (10) with allowance for $F=F\left(\mathrm{Re}\right)$; curves 3 were plotted by Equation (10) with $F$ = ${F}_{0}$ for the Stokes flow approximation at Re ≪ 1; curves 4 were plotted by the asymptotic formula (Equation (12)) for Pe ≪ 1.

**Figure 7.**Fiber retention efficiencies per unit length in a dense row of fibers with $a/h$ = 0.8 vs. Reynolds number, found from the combined numerical solution of the Navier–Stokes and convection–diffusion equations (curves 1): Schmidt numbers are marked on the curves; curves 2 were plotted by Equation (10) with $F$ = ${F}_{0}$ for the Stokes flow at Re ≪ 1; curves 3 were plotted by the asymptotic formula (Equation (12)) for Pe ≪ 1.

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**MDPI and ACS Style**

Kirsch, V.A.; Bildyukevich, A.V.; Bazhenov, S.D.
Simulation of Convection–Diffusion Transport in a Laminar Flow Past a Row of Parallel Absorbing Fibers. *Fibers* **2018**, *6*, 90.
https://doi.org/10.3390/fib6040090

**AMA Style**

Kirsch VA, Bildyukevich AV, Bazhenov SD.
Simulation of Convection–Diffusion Transport in a Laminar Flow Past a Row of Parallel Absorbing Fibers. *Fibers*. 2018; 6(4):90.
https://doi.org/10.3390/fib6040090

**Chicago/Turabian Style**

Kirsch, Vasily A., Alexandr V. Bildyukevich, and Stepan D. Bazhenov.
2018. "Simulation of Convection–Diffusion Transport in a Laminar Flow Past a Row of Parallel Absorbing Fibers" *Fibers* 6, no. 4: 90.
https://doi.org/10.3390/fib6040090