Hydrodynamics maneuver of a single helical flagellum swimming robot at low-Reynolds condition

Abstract

Helical swimming robots with a capable propulsion system at low-Reynolds numbers have been proposed for many applications. Although linear propulsion characteristics of swimming robots with a single helical flagellum have been extensively studied, the characteristics of maneuverability have not been completely investigated yet. This study presents a new method for the maneuverability of the helical swimming robot with a single helical flagellum. This mechanism is based on the change in the angle between the helical and body axes. This study shows that a change in the aforementioned angle can enable the swimming robot to have turning maneuvers in clockwise or counterclockwise directions. Moreover, the swimming robot will move in a straight line if the helical and body axes are parallel. To investigate this new method and predict the robot’s behavior at various inclination angles, a hydrodynamics model is used. To validate the hydrodynamics model, an experimental prototype of a macro-size swimming robot with specific inclination angles is fabricated. The results indicate that the helical swimming robot swims on circular trajectories through specific inclination angles between the helical flagellum and the body axis. Moreover, the radius of curvature decreases by increasing the inclination angle. Results of the validated hydrodynamics model indicate that the turning velocity has approximately a constant value at different inclination angles depending on the rotational frequency and geometrical parameters of the swimming robot. Finally, the effects of geometrical parameters of the body and the helical flagellum on the radius of curvature and turning velocity are investigated through the proposed hydrodynamics model. The verified results indicate that the hydrodynamics model provides a viable alternative model to predict the behavior of a helical swimming robot at various inclination angles within a range of design variables. This new method can be introduced as a mechanism for maneuverability of the helical swimming robots with a single helical flagellum and will be able to control the parameters in this type of swimmers for the implementation of predefined missions.

Appendix A

1.1 Hydrodynamics parameters

In this appendix, we present the value of the hydrodynamics parameters of the helical flagellum and body, which are represented by ${H}_{ij}^{\alpha \beta }$ and${B}_{ij}^{\alpha \beta }$, respectively:

$$H_{{xx}}^{{FU}}=H_{{yy}}^{{FU}}={b_1},\;\;~H_{{xx}}^{{F\Omega }}=H_{{yy}}^{{F\Omega }}={b_2},$$

(18a)

$$~B_{{xx}}^{{FU}}={a_{1x}},~~~B_{{yy}}^{{FU}}=~{a_{1y}},$$

(18b)

$$~B_{{xx}}^{{M\Omega }}=B_{{yy}}^{{M\Omega }}=~B_{{zz}}^{{M\Omega }}={a_2},$$

(18c)

$$H_{{xx}}^{{MU}}=H_{{yy}}^{{MU}}={c_1},\;\;~H_{{xx}}^{{M\Omega }}=~H_{{yy}}^{{M\Omega }}={c_2}~,$$

(18d)

$$H_{{zy}}^{{MU}}={b_1} \cdot ~P{G_x},\;\;~H_{{zy}}^{{M\Omega }}={b_2} \cdot ~P{G_x}~,$$

(18e)

$${a_1}=\left( {{a_{1x}}{{\sin }^2}\alpha +{a_{1y}}{{\cos }^2}\alpha } \right),$$

(19a)

$${a_{1x}}=\frac{{16\pi \mu ({L_{{\text{body}}}}/2){e^3}}}{{\left[ {\left( {1+{e^2}} \right)E - 2e} \right]}}~,~\;\;{a_{1y}}=\frac{{32\pi \mu ({L_{{\text{body}}}}/2){e^3}}}{{\left[ {\left( {3{e^2} - 1} \right)E+2e} \right]}},$$

(19b)

$$E=Ln\frac{{1+e}}{{1 - e}},$$

(19c)

$$e=\sqrt {{{({L_{{\text{body}}}}/2)}^2} - {{({D_{{\text{body}}}}/2)}^2}} /({L_{{\text{body}}}}/2),$$

(19d)

$${a_2}=\left( {({a_{2x}}+{{(P{G_x})}^2} \cdot {a_{1x}}){\text{si}}{{\text{n}}^2}\alpha ~~~+\;({a_{2y}} \cdot {\text{co}}{{\text{s}}^2}\alpha )} \right),$$

(19e)

$${a_{2x}}=\frac{{32\pi \mu ({L_{{\text{body}}}}/2){{({D_{{\text{body}}}}/2)}^2}{e^3}}}{{3\left[ {2e - \left( {1 - {e^2}} \right)E} \right]}},$$

(19f)

$${a_{2y}}=\frac{{32\pi \mu ({L_{{\text{body}}}}/2){{({D_{{\text{body}}}}/2)}^2}{e^3}\left( {2 - {e^2}} \right)}}{{\left( {3({e^2} - 1} \right)\left[ {\left( {1+{e^2}} \right)E - 2e} \right]}}~,$$

(19g)

$$~{b_2}={h_{{\text{helix}}}}n\lambda ~\left( {{\xi _n} - {\xi _t}} \right){\text{sin}}\beta ,$$

(20a)

$${b_1}=n\lambda ~{\text{cos}}\beta \left( {{\xi _n}~{\text{ta}}{{\text{n}}^2}\beta +{\xi _t}} \right),$$

(20b)

$${c_1}={h_{{\text{helix}}}}n\lambda ~\left( {{\xi _n} - {\xi _t}} \right){\text{sin}}\beta ,$$

(20c)

$${c_2}={h_{{\text{helix}}}}^{2}n\lambda ~{\text{cos}}\beta \left( {{\xi _t}~{\text{ta}}{{\text{n}}^2}\beta +{\xi _n}} \right),$$

(20d)

$${m_{{\text{tail}}}}=4\mu \pi n\lambda {d^2}{\text{cos}}\beta .$$

(20e)