Real model for micro swimmer and the study of the relationship between the swimming speed, pitch angle, and rotation rate for the flagellum

This study focuses on the fluid mechanics of a microswimmer and explores the relationship between speed, pitch angle, and rotation rate for the flagellar during bacterial swimming. Based on the simulation using MATLAB, it is concluded that when the pitch angle of the flagellar helix is in the range of 0 to 90 degrees, the value of swimming speed increases firstly and decreases. When the angle reaches 46.83 degrees, the speed reaches the maximum point. The radius of the body of the microswimmer is determined by the Buckingham Pi theory. After calculating by using the equations in the related paper and measuring by the real model, we derive that the relationship between swimming speed and the rotation rate for the flagellar filament should be proportional at the low rotation rate so that it can be obtained to optimize the artificial micro swimming device with higher swimming efficiency.


Introduction
In previous studies, there were many factors affecting the swimming speed of bacteria.One of the factors is the rotation rate for the flagellar filament.Through the experiment by using laser dark-field microscopy, cells with faster-spinning flagella did not always swim faster [1].They claimed a model which considers the torque characteristics of the flagellar motor to explain this phenomenon.They concluded that swimming speeds all tend to saturate at high flagellar rotation rates.However, there is no real model to verify the result of the experiment.
Another factor is the pitch angle of the flagellar helix.Through the simulation of different helical geometries and handedness of typical bacterial flagella, they concluded that the speed of the rotating helix depends on the pitch angle of the helix [2].However, the conclusion of the relationship between swimming speed and the pitch angle of the flagellar helix is not very clear.
The range of related work is to identify the calculation method for the speed, force, moment, and their relationships with liquid viscosity when the bacteria swim under a low Reynolds number state.Here the author highlighted that the relationship between the swimming speed and pitch angle for the flagellar filament had not been previously investigated in the literature.Furthermore, we use the real model to test the relationship between the swimming speed and rotation rate for the flagellar filament.Previous research taken by E.M. Purcell in 1977 has established the way how bacteria swim and the calculation of the correlation coefficient [3].In his paper, he used the hypothesis method and matrix calculation to infer that the bacteria swim in a spiral way under a low Reynolds number state and found that the matrix must have diagonal elements.Through the principle of diffusion and related experiments, E.M. Purcell explained the relationship between the swimming speed and diffusion speed in an analogy way, which is when the diffusion speed of nutrient molecules is greater than the swimming speed of bacteria, the bacteria will stop swimming.
According to Purcell's research, when the bacteria are swimming, the tendency of flagella is spiral in the 3D dimension and must be in a circular motion in the 2D dimension.The significant point of this paper is to solve the problem of the flagella trajectory.At the same time, it provides the numerical values and calculation formulas for the speed, force, and torque parameters calculation during bacteria swimming, which is in the state of a low Reynolds number.
In Magariyama and Kudo's paper, in order to calculate the swimming speed of the bacteria in the polymer solution, they designed an extreme experiment case to prove that the polymer network does not affect the tangential motion of the microscopic slender body.However, it does affect its normal motion [4].They plotted the relationship between the swimming speed and viscosity and found the peak position of swimming speed in the case of different viscosities.For this paper, the key point is to calculate the speed, force, and torque of the bacteria swimming by using the mathematical expression of the modified resistance.They get the result that the polymer network only affects the normal movement of microscopic objects.
Researchers have become increasingly interested in mechanisms and some problems of bacterial swimming motility under a low Reynolds number state.Lauga and Powers used multiple mathematical models and applied physical principles to explain the question of flow singularities and bacteria swimming at a low Reynolds number [5].The significant part of this paper is to explain the basic hydrodynamics of swimming microorganisms, such as the calculation for flow singularities, the physical movement of microorganisms at a low Reynolds number, and the mechanism of microbial swimming.Through these theories, they proposed and optimized three models of the artificial micro-device [5].
The main purposes of the study are to design the microswimmer and explore the relationship between swimming speed and the pitch angle of the flagellar helix during bacterial swimming.Furthermore, we use the designed real model to test the relationship between swimming speed and the rotation rate for the flagellar filament.The significance of the topic is to use the relationship between the swimming speed of bacteria and related variables to promote the development of micro-robotics better.
The following are the contributions: • It describes the relationship between the velocity and pitch angle for the flagellar filament during bacterial swimming.
• It derives the pitch angle of the flagellar helix corresponding to the maximum swimming speed and the value of the maximum swimming speed.
• It proposes a real model with the size of the fluid mechanics part by using a 3D printer.
• It derives and tests the relationship between the velocity of the fluid mechanics part.The velocity is proportional to the rotation rate for flagellar filament at the low rotation rate.
• It describes the average velocity and rotation rate for the flagellar filament for the real model in water and laundry detergent.
The remainder of the paper is laid out as follows.Section 2 discusses the designed velocity for the microswimmer and the relationship between velocity and the pitch angle of the flagellar helix.Section 3 describes the size of the real swimmer model and the real model.Section 4 describes the result and test for the relationship between the velocity of the real swimmer model and the rotation rate for the flagellar filament, and Section 5 shows the conclusions.

Velocity and pitch angle of the flagellar helix
To design a fluid mechanics part, the first thing is to solve for the velocity of the microswimmer.In this design, the microswimmer is a combination of a spherical head and a helical tail.Figure 1 presents the front view of the design model.In the simulation, the radius of the spherical head of the fluid mechanics part is 1 × 10 ି ݉ , because the fluid mechanics part is assumed as a sphere.The viscosity of water is 0.89 × 10 ିଷ ܰ • ܵ • ݉ ିଶ , which is assumed in an environment of 25 degrees (Korson et al., n.d.).The value for apparent viscosities in the normal and tangential directions is the same as the viscosity of water at 25 degrees.To determine the drag coefficients of the cell body, Equations ( 1) to (4) are used [4].Table 1 shows the values of the pitch of the flagellar helix, rotation rates of the flagellar filament, and the drag coefficients of the cell body.

‫ߠ݊ܽݐ‬ = ଶగ
(1) where ߠ is the helical angle of the flagellar helix, which is equal to 45 degrees   5) to (7) are used to calculate the drag coefficients of the flagellar filament.Table 2 presents the values of drag coefficients of the flagellar filament.
where ߙ , ߚ , and ߛ are the drag coefficients of flagellar filament, ‫ܮ‬ is the length of flagellar helix [m], which is 10 × 10 ି ݉. ݀ is half of the diameter of the flagellar filament [m], which is equal to 5 × 10 ିଽ ݉.  8) to (10) are used to calculate the motion, drag force, and torque acting on a cell body and the drag force and torque acting on a flagellar filament.Table 3 shows the value of the speed of the fluid mechanics part.
where ‫ܨ‬ and ‫ܨ‬ are the hydrodynamic forces acting on the cell body and flagellar filament, ‫ݒ‬ is the speed of the fluid mechanics part [m/s].
Table 3. Value of the speed of the fluid mechanics part.
(/) Value 1.7999 × 10 ିସ In the processing of the simulation, Figure 2 shows the relationship between the swimming speed and pitch angle of the flagellar helix.From Figure 2, it presents that when the pitch angle of the flagellar helix reaches 46.63 degrees, the swimming speed of the fluid mechanics part will get the maximum point which is equal to1.80285 × 10 ିସ ‫.ݏ/݉‬Moreover, as the pitch angle increases, the trend of swimming speed shows an upward trend from 0 to 46.63 degrees, and it can be observed that the trend drops from 46.83 to 90 degrees.The rate of change of swimming speed decreases first and then increases with the pitch angle.

Determine the size of the fluid mechanics part and real model
The Buckingham Pi theory is used to build and test the fluid mechanics part on the cm size with a similar Re number [6].Equation ( 11) is used to list all the variables in the problem.Table 4 presents each variable in terms of basic dimensions.Equation (12) shows the dimension function.Equation ( 13) is used to determine the relationship between the characteristic's length in cm size and Re number.
Where ܶ is the torque of the motor, ܽ is the characteristic's length.‫ݒ‬ is the swimming speed for the fluid mechanics part, and ߩ is the fluid density.ߟ is the fluid viscosity.Table 4.Each variable in terms of basic dimensions.

Variables
Unit Where ‫ܥ‬ is a constant.We need to design the size of the fluid mechanics part with a similar Re number.In that case, Equation ( 13) is used to plot the function related to the characteristic's length and Re number.For this experiment, we want to approach the Re number equal to 0.1 or lower so that the laundry detergent is used as the viscous agent to simulate the relationship between the Re number and the size of the real model.The property for the laundry detergent-related fluid density is0.9 × 10 ିଷ ݇݃/ܿ݉ ଷ .The assumption for building the model is that ܶ is the motor torque at a rotation rate of 0, which is 1.5 × 10 ିଵ଼ ܰ • ݉.The swimming speed is assumed as 1.80285 × 10 ିସ ‫,ݏ/݉‬ which is the maximum point we obtain.Figure 3 presents the relationship between the Re number and the size of the real swimmer model.To build the design model, we use the data about the characteristic's length, helical angle of the flagellar helix, the pitch angle of the flagellar helix, and the radius of the flagellar to conduct the 3D modeling.Figure 4 presents the simulation model in SolidWorks.We use the 3D printer to print the real model.Figure 5 shows the real model for the fluid mechanic part.

Relation between Velocity and Rotation rate
After building the real model, we want to find the relationship between the velocity of the fluid mechanics part and the rotation rate for the flagellar filament.The velocity of the fluid mechanics part and rotation rate of the flagellar filament are obtained from Equations ( 14) to (16).For calculating the ߙ and ߛ , we assume that they follow the traditional RFT following Equations ( 17) and (18).
Where ߙ ܽ݊݀ ߚ are the drag coefficients of the cell body.ߙ , ߚ , and ߛ are the drag coefficients of the flagellar filament.L is the length of the flagellar filament.2d is the diameter of the flagellar helix.p is the pitch of the flagellar helix.r is the radius of the flagella helix.ߤ ே * are the apparent viscosities in the normal direction.ߤ is the viscosity.Due to the cell body of the model being a sphere, cell width should equal cell length.
Due to ߙ , ߙ and ߛ are the drag coefficients for cell body and flagellar filament.In some cases, they should be constants.In that case, the relationship between swimming speed and the rotation rate for flagellar filament should be proportional.
After analyzing the relationship between the velocity and rotation rate, we conduct the experiments in water using the real model.Table 5 presents the data for the experiments in water and laundry detergent, respectively.Figure 7 shows the relationship between the swimming speed and rotation rate in water and laundry.7, when the rotation rate for flagellar filament increases, the swimming speed increases at the low rotation rate.When the rotation rate reaches 10.5 rads/s, the swimming speed remains at nearly 4.1 cm/s.This experiment proves that the relationship between the swimming speed and the rotation rate for the flagellar filament is proportional at a low rotation rate.The swimming speed will remain at the same values at a high rotation rate because of apparent saturation.We repeat the experiment in laundry detergent by using the same process.Table 6 shows the data for the experiments in laundry detergent.Figure 8 presents the relationship between the swimming speed and rotation rate in laundry detergent.8, we can find that the trend of swimming speed increases as rotation increases.There are two reasons why the swimming speed increases slowly at a low rotation rate.The first reason is that the dynamic viscosity of laundry detergent is quite higher than water.The second reason is that the suds from the laundry detergent prevent the swimmer from advancing in the water at a low rotation rate.

Conclusions
With the in-depth study, we can recognize the useful propulsion efficiency of bacteria swimming motility at a low Reynolds number.Moreover, the peak position of swimming speed at different viscosities can be obtained through simulation.Using the real model, we find that the relationship between the swimming speed and rotation rate for flagellar filament is proportional to the low rotation rate.This relationship can be obtained to design and optimize the high-efficiency micro swimming device in the future.
The goal of the simulation began with finding the relationship between the swimming speed and pitch angle of the flagellar helix.The shows that when the pitch angle increases, the swimming speed will increase firstly and then decrease.The rate of change of the swimming speed will drop and then rise while the pitch angle increases.We determine the size of the real model and use the model to measure the velocity and rotation cycle.The relationship between the swimming speed and rotation rate for the flagellar filament is obtained, which is proportional to the low rotation rate.At the high rotation rate for the flagellar filament, the swimming speed remains the same in water because of apparent saturation.
There are two areas where this experiment could be improved.The first aspect is to improve the measurement technology of the model, using more advanced instruments to increase the speed of the measurement model in water and laundry detergent to improve the accuracy of the experimental results.The other is to increase the randomness of the experiment to reduce the bias caused by the accidental error.Repeated experiments can be performed many times to increase the randomness and reduce the sample bias.

Figure 1 .
Figure 1.Front view of the design model.
[°]. ‫‬ is the pitch of the flagellar helix [m] and ‫ݎ‬ is the radius of the flagellar, which is 1 × 10 ି [m].߱ is the rotation rate of the flagellar filament [rps], and ݂ is the rotation frequency of the flagellar filament[Hz], which is equal to 200 Hz.ߙ and ߚ are the drag coefficients of flagellar filament, respectively.ߤ ே * and ߤ ் * are the apparent viscosities in the normal and tangential directions, which is equal to the viscosity of water at 25 degrees, 0.89 × 10 ିଷ ܰ • ‫ݏ‬ • ݉ ିଶ .ܽ and ܾ are the half of the cell width and length, which is approximately the radius of the cell, 1 × 10 ି ݉.

Figure 2 .
Figure 2. Relationship between the swimming speed and pitch angle of the flagellar helix.From Figure2, it presents that when the pitch angle of the flagellar helix reaches 46.63 degrees, the swimming speed of the fluid mechanics part will get the maximum point which is equal to1.80285 × 10 ିସ ‫.ݏ/݉‬Moreover, as the pitch angle increases, the trend of swimming speed shows an upward trend from 0 to 46.63 degrees, and it can be observed that the trend drops from 46.83 to 90 degrees.The rate of change of swimming speed decreases first and then increases with the pitch angle.

Figure 3 .
Figure 3. Relationship between the Re number and the size of the real swimmer model.As shown in Figure 3, the suitable point for the size of the real swimmer model is when the Re number is 0.1.The size of the fluid mechanics part of the characteristic's length is 0.00172442 m, which is 1.72442 cm.

Figure 5 .Figure 6 .
Figure 5.The real model for the microswimmer.We installed a gear starter inside the real model to simulate the dynamics of bacterial swimming.Figure6shows the gear starter inside the real model.

Table 1 .
Values of the pitch of the flagellar helix, rotation rates of flagellar filament, and drag coefficients of the cell body.

Table 2 .
Values of drag coefficients of the flagellar filament.To determine the speed of the fluid mechanics part, Equations (

Table 5 .
The data for the experiments in water.Relationship between swimming speed and rotation rate in water.As shown in Figure

Table 6 .
Data for the experiments in the laundry detergent.Relationship between the swimming speed and rotation rate in laundry detergent.From Figure

Table 7 .
Table 7 presents the average velocity and rotation rate in water and laundry detergent.Average velocity and rotation rate in water and laundry detergent.