# On the Relationship between Aquatic Plant Stem Characteristics and Drag Force: Is a Modeling Application Possible?

^{1}

^{2}

^{*}

## Abstract

**:**

^{b}, and two approaches based on a regression analysis were applied to determine the parameters of the model—a and b. In the first method, the parameters were identified separately for each day of measurement, while in the second method, the coefficient b was calculated for all data from all days as a unified number for individual plants. The results suggest that coefficient b may provide information about the proportion of changes in drag forces depending on plant stiffness. The values of this coefficient were associated with the shape of the stem cross-section. The more circular the cross-section, the closer the value of the parameter was to 1. The parameter values were 1.60 for E. canadensis, 1.98 for P. pectinatus, and 2.46 for P. crispus. Moreover, this value also depended on the density of the cross-section structure. Most of the results showed that with an increase in stem diameter, the ratio between the drag and bending forces decreased, which led to fewer differences between these two forces. The model application may be introduced in many laboratory measurements of flow–biota interactions as well as in aquatic plant management applications. The implementation of these results in control methods for hydrophytes may help in mitigating floods caused by increases to a river channel’s resistance due to the occurrence of plants.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Assumptions

_{D}and the bending force of this plant F

_{B}. The description of this ratio was proposed by Nikora [1] in the following form:

_{D}is the drag coefficient, U is the reference velocity, A is the reference plant area, R is the radius of curvature at a point where the bending force is defined, λ is the distance from the bed to the point where the resultant fluid force acts, and EI is the flexural rigidity. Referring to Figure 1, most of these parameters depend on the material characteristics of the plant, which have not yet been investigated in relation to the biomechanical traits of plants. Thus, our calculations focused on the stem diameter, which is easily measured and could be used to determine the phenological development stage of a plant.

- The flexural rigidity EI, which has a constant modulus of elasticity E [MPa] for solid materials with a circular cross-sectional shape, is expressed as [53]$$EI=E\xb7{I}_{C}=\frac{E\xb73.14\xb7{d}^{4}}{64}\cong 0.05\xb7E\xb7{d}^{4},$$
_{C}is the second moment of area for the circular cross-section and is defined as ${I}_{C}=\frac{\pi {d}^{4}}{64}$, [mm^{4}], and d is the stem diameter [mm]. - When the cross-section of the stem plant has a circular shape (the ideal situation), the second moment of area I depends on the plant diameter d to the fourth power, i.e., I ~ d
^{4}. This condition is true even in the case when the shape scales with the increase in diameter—that is, when the cross-sections are similar (in the sense of the similarity of the figures). On the other hand, stems of aquatic plants do not have uniform cross-sections or perfect circular shapes (Figure 3). To simplify the biomechanical measurements, the cross-sectional area of the stem is usually compared to a specific shape with a solid structure [35,36]. However, a plant stem is not a solid material. The internal structure of a plant is more complex and it is not uniform for the plant’s entire lifecycle. In addition, a cross-sectional area does not change proportionally with the increase in diameter. Hence, the assumption of diameter to the fourth power as described by Equation (3) that is used for solid materials is not correct for biological systems. Therefore, in our nonideal situation, the second moment of area is approximately I ~ F(d) d^{4}, where F(d) is a function of the shape, which, in turn, is a function of time d(t), as hydrophytes change their dimensions throughout the growing season. In addition, we assume here that the shapes of cross-sections of plants with the same diameters are (roughly) the same. - In the ideal situation, flexural modulus E does not depend on the plant diameter. E is the property of the material from which the plant was “built”. However, our situation is not ideal, and when the plant grows, the material from which it was built changes. Thus, again, E ~ H(d, t).

_{m}and the stem diameter d has the following form:

#### 2.2. Data

^{−1}[36]. The Wilga River is a lowland river, and the bed of the river is mostly covered with medium and coarse sand [36,54]. The channel bed at the sampling site, which is in a natural unregulated part of the river, is strongly vegetated by aquatic plants, containing small stones and gravel [36].

#### 2.3. Regression Calculations

_{1}, …, a

_{n}and b values in the following function based on the whole dataset:

## 3. Results

#### 3.1. Regression Analysis

#### 3.1.1. Case 1

#### 3.1.2. Case 2

#### 3.1.3. Cases Comparison

#### 3.2. Data Analysis

_{D}/F

_{B}relationship may be presented in the following form:

_{D}, and U are independent of the plant diameter d. In our case, EI = a · d

^{b}, which finally gives

_{D}and the bending force F

_{B}. For the cases when the coefficient b ≥ 1, this ratio decreases and has positive limits equal to 0. On the other hand, in the case when the coefficient b ∈ 〈0, 1), the ratio increases, and it is proportional to the root function of the plant diameter. For b = 1, a state of equilibrium between these two forces is expected. Specifically, for case 2 (Table 1), the values of coefficient b were equal to 1.60, 1.98, and 2.46 for each sample. Substituting a specific value for this coefficient, we are able to estimate the proportion of changes in drag forces depending on the plant stiffness. More detailed descriptions of the obtained results are presented in the discussion section.

## 4. Discussion

_{D}and F

_{B}for the same value of the coefficient b (assuming that b > 1) decreases; thus, the difference between the drag and bending forces is proportionally lower. On the other hand, when the coefficient b < 1, the situation is reversed (see Equation (8)). In the case 2 example (Table 1), the results for each investigated plant species showed that the increase in diameter caused a proportionally lower influence in drag. E. canadensis was characterized by the highest effect of the diameter on the proportion of forces, whereas for P. crispus, the dimensions led to a downtrend in the ratio of forces. Interesting results were obtained for P. crispus in case 1, where the coefficient b was calculated for each period. Biomechanical measurements of this hydrophyte were carried out in 2016 and 2017 [36]. During the first season (2016), most of the modeled coefficients b varied between 1 and 10 with two exceptions, which were less than but close to 1 (Table 1). Thus, the majority of the periods were characterized by a downtrend in the ratio of forces with the plant expansion, whereas the second season (2017) showed that the coefficient b was approximately 0.3 (Table 1), which meant that the ratio between F

_{D}and F

_{B}increased proportionally to the increase in the plant diameter. Moreover, we cannot assume that the influences that the parameters have on the ratio are hidden in the assumed variable X (Equation (8)). However, a lower value of the ratio may suggest that the plant belongs to the “bending” plants category, and the plants in this category are stiffer and cause higher resistance [1].

## 5. Conclusions

- (1)
- The relationship between flexural rigidity of aquatic plant stem and drag has the following form: EI = ad
^{b}. - (2)
- Our work showed that two approaches may be used for estimating plant stiffness based on plant morphology in a detailed (case 1) or general way (case 2), which is needed to obtain drag forces (Equation (1)).
- (3)
- With a constant coefficient b, the increase in the diameter of the plant stem may cause monotonous changes in the ratio of the drag and bending forces.
- (4)
- The model may be applied in many laboratory measurements of flow–biota interactions as well as in widely understood aquatic plant management.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The main forces acting upon a plant stem: the drag force F

_{d}, the buoyancy force F

_{b}, the gravity force F

_{g}, and the resultant force F

_{r}.

**Figure 3.**Examples of cross-sections of aquatic plants: P. pectinatus L. (

**a**), P. crispus L. (

**b**), E. canadensis Michx. (

**c**). (

**a**,

**b**) were modified from Łoboda et al. [36].

**Figure 5.**Model matching for E. canadensis (

**a**), P. pectinatus (

**b**) and P. crispus (

**c**). The red line represents case 1, and the blue line represents case 2.

E. canadensis | |||||||||

Date of Collection | Mean Stem Diameter * | Mean Flexural Modulus * | Case 1 | Case 2 | MSE Relative Diff. | ||||

[mm] | [MPa] | a | b | MSE | a | b | MSE | [%] | |

2016-06-09 | 1.07 | 62.54 | 3.47 | 2.20 | 6.51 | 3.67 | 6.58 | 1.08 | |

2016-06-24 | 1.10 | 95.32 | 3.36 | 1.93 | 8.19 | 5.59 | 8.23 | 0.49 | |

2016-08-16 | 1.18 | 141.20 | 10.32 | 1.14 | 7.64 | 9.47 | 1.60 | 7.97 | 4.32 |

2016-09-13 | 1.37 | 60.07 | 5.49 | 2.03 | 75.86 | 6.36 | 76.09 | 0.30 | |

2016-10-04 | 1.24 | 32.31 | 2.22 | 2.31 | 2.27 | 2.63 | 2.33 | 2.64 | |

P. pectinatus | |||||||||

Date of Collection | Mean stem Diameter ** | Mean Flexural Modulus ** | Case 1 | Case 2 | MSE Relative Diff. | ||||

[mm] | [MPa] | a | b | MSE | a | b | MSE | [%] | |

2016-05-14 | 1.30 | 94.51 | 7.93 | 1.10 | 49.70 | 5.41 | 55.99 | 12.66 | |

2016-06-09 | 0.97 | 86.59 | 2.93 | 2.66 | 6.39 | 3.45 | 6.75 | 5.63 | |

2016-06-24 | 1.37 | 90.18 | 11.16 | 0.24 | 62.14 | 5.26 | 81.85 | 31.72 | |

2016-08-16 | 1.56 | 94.09 | 7.26 | 2.17 | 212.51 | 8.20 | 213.32 | 0.38 | |

2016-09-13 | 1.68 | 36.66 | 1.74 | 3.31 | 49.88 | 4.44 | 54.80 | 9.86 | |

2016-10-04 | 1.26 | 55.08 | 4.26 | 0.98 | 30.49 | 2.91 | 1.98 | 32.43 | 6.36 |

2017-06-14 | 1.22 | 168.50 | 16.32 | 0.07 | 26.49 | 10.35 | 43.03 | 62.44 | |

2017-07-12 | 0.84 | 252.97 | 6.55 | 4.32 | 14.71 | 9.70 | 31.96 | 117.27 | |

2017-08-08 | 1.21 | 109.72 | 8.89 | 0.28 | 8.04 | 5.75 | 13.19 | 64.05 | |

2017-10-31 | 1.24 | 117.56 | 6.77 | 2.02 | 18.15 | 6.90 | 18.17 | 0.11 | |

2017-11-21 | 1.17 | 174.56 | 9.18 | 2.69 | 43.26 | 11.49 | 50.22 | 16.09 | |

P. crispus | |||||||||

Date of Collection | Mean Stem Diameter ** | Mean Flexural Modulus ** | Case 1 | Case 2 | MSE Relative Diff. | ||||

[mm] | [MPa] | a | b | MSE | a | b | MSE | [%] | |

2016-05-14 | 2.10 | 33.54 | 13.28 | 0.89 | 261.59 | 3.57 | 299.24 | 14.37 | |

2016-06-09 | 2.43 | 19.78 | 0.12 | 6.29 | 395.53 | 4.16 | 451.57 | 14.18 | |

2016-06-24 | 1.87 | 51.59 | 14.99 | 0.84 | 194.40 | 4.53 | 247.34 | 27.28 | |

2016-08-16 | 1.97 | 105.21 | 20.86 | 1.65 | 706.28 | 10.82 | 781.83 | 10.69 | |

2016-09-13 | 2.23 | 36.38 | 6.23 | 2.45 | 1329.52 | 6.17 | 2.46 | 1329.53 | 0 |

2016-10-04 | 2.21 | 43.17 | 0.01 | 9.90 | 349.31 | 7.06 | 1062.55 | 204.18 | |

2016-11-04 | 1.84 | 55.22 | 2.17 | 4.10 | 275.05 | 7.09 | 323.51 | 17.64 | |

2016-12-06 | 1.70 | 52.71 | 8.64 | 1.35 | 239.71 | 4.57 | 248.21 | 3.55 | |

2017-08-08 | 1.90 | 38.50 | 12.93 | 0.29 | 51.47 | 2.56 | 81.82 | 59.14 | |

2017-11-10 | 1.65 | 77.41 | 20.40 | 0.27 | 54.41 | 5.72 | 114.66 | 110.85 |

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Łoboda, A.M.; Karpiński, M.; Bialik, R.J.
On the Relationship between Aquatic Plant Stem Characteristics and Drag Force: Is a Modeling Application Possible? *Water* **2018**, *10*, 540.
https://doi.org/10.3390/w10050540

**AMA Style**

Łoboda AM, Karpiński M, Bialik RJ.
On the Relationship between Aquatic Plant Stem Characteristics and Drag Force: Is a Modeling Application Possible? *Water*. 2018; 10(5):540.
https://doi.org/10.3390/w10050540

**Chicago/Turabian Style**

Łoboda, Anna Maria, Mikołaj Karpiński, and Robert Józef Bialik.
2018. "On the Relationship between Aquatic Plant Stem Characteristics and Drag Force: Is a Modeling Application Possible?" *Water* 10, no. 5: 540.
https://doi.org/10.3390/w10050540