Figure 1

The wave form of a human sperm observed by Ishijima et al. [4] in a solution with very high viscosity $\left(400\mathrm{cP}\right)$. The sperm head, on the left side of the image, is held in place with a micropipette tip. The contour length of the flagellum is approximately $40\mu \mathrm{m}$.Reuse & Permissions

Figure 2

Sliding filament model. The dotted line is the centerline of the flagellum when it is straight; the circles divide it in quarters. The solid lines represent the bent centerline and ${\mathbf{r}}_{\pm }$; the dots divide each line in quarters as measured along the respective contours. All three curves have length $L$.Reuse & Permissions

Figure 3

(Color online) Shapes of beating patterns for passive filaments driven by prescribed angle at the left end $(s=0)$. A half cycle of the pattern is shown for viscous $(\mathrm{De}=0)$ and viscoelastic $(\mathrm{De}=100)$ cases, and sperm number 2, 7, and 20. Time sequence: solid (red), short dash (orange), long dash (green), long dash-short dash (blue), dash-dot (purple). The ratio of solvent viscosity to the total viscosity is ${\eta }_{\mathrm{s}}∕\eta ={\mathrm{De}}_2∕\mathrm{De}={10}^{-4}$. At the top of each plot, we print the (dimensionless) angular magnitude $ϵ$ required to produce motion with amplitude $0.1L$, and the (dimensionless) power dissipated by the motion.Reuse & Permissions

Figure 4

Time-averaged power dissipated by passive filaments driven by prescribed angle, as a function of $\mathrm{De}=\omega \lambda$ (dashed curve). Power is normalized to the power dissipated in the viscous case $\mathrm{De}=0$. The black curve corresponds to the power dissipated if the beating pattern does not change from the viscous case, but the filament is placed in a viscoelastic medium. In both cases, $\mathrm{Sp}=7$. The driving amplitude is chosen so that the maximum amplitude of the beating pattern is $L∕10$ in the viscous case. The ratio of solvent viscosity to the total viscosity is ${\eta }_{\mathrm{s}}∕\eta ={\mathrm{De}}_2∕\mathrm{De}={10}^{-4}$.Reuse & Permissions

Figure 5

(Color online) Shapes of beating patterns for filaments with internal sliding forces and fixed head position, with $\mathrm{Sp}=7$. A half cycle of the pattern is shown for viscous $(\mathrm{De}=0)$ and viscoelastic $(\mathrm{De}=100)$ cases, and for internal sliding forces given by Eq. (48) with $k∕L$ varying from 0 (uniform force) to $8\pi$, as indicated. Time sequence: solid (red), short dash (orange), long dash (green), long dash-short dash (blue), dash-dot (purple). At the top of each figure, we print the (dimensionless) magnitude ${\overline{f}}_{\mathrm{m}}$ required to produce motion with amplitude $0.1L$, and the (dimensionless) power dissipated by the motion. The ratio of solvent viscosity to the total viscosity is ${\eta }_{\mathrm{s}}∕\eta ={\mathrm{De}}_2∕\mathrm{De}={10}^{-4}$.Reuse & Permissions

Figure 6

(Color online) Shapes of beating patterns for hinged filaments with internal sliding forces, with varying $\mathrm{Sp}$. A half cycle of the pattern is shown for viscous $(\mathrm{De}=0)$ and viscoelastic $(\mathrm{De}=100)$ cases, and for internal sliding forces given by Eq. (48) with $k∕L=8\pi$. Time sequence: solid (red), short dash (orange), long dash (green), long dash-short dash (blue), dash-dot (purple). At the top of each figure, we print the (dimensionless) magnitude ${\overline{f}}_{\mathrm{m}}$ required to produce motion with amplitude $0.1L$, and the (dimensionless) power dissipated by the motion. The ratio of solvent viscosity to the total viscosity is ${\eta }_{\mathrm{s}}∕\eta ={\mathrm{De}}_2∕\mathrm{De}={10}^{-4}$.Reuse & Permissions

Figure 7

Time-averaged power dissipated by filaments with active internal sliding forces, as a function of $\mathrm{De}=\omega \lambda$. Power is normalized to the power dissipated in the viscous case $\mathrm{De}=0$, with force amplitude such that maximum filament displacement in the viscous case $\mathrm{De}=0$ is $0.1L$. Different curves correspond to sliding forces given by Eq. (48) with different values of $k$: dashes, $k=0$; long dashes, $k=\pi ∕L$; long dash-short dash, $k=3\pi ∕L$. The solid curve is the power dissipated by a filament with a beating shape corresponding to the viscous case $(\mathrm{De}=0)$. In all cases, $\mathrm{Sp}=7$. The ratio of solvent viscosity to the total viscosity is ${\eta }_{\mathrm{s}}∕\eta ={\mathrm{De}}_2∕\mathrm{De}={10}^{-4}$. (a) Fixed head with clamped attachment point. (b) Fixed head with hinged attachment point.Reuse & Permissions

Figure 8

(Color online) (a) Beating patterns of human sperm observed by Ishijima et al. [4] in (from left to right) $1\mathrm{cP}$ Hanks’ solution, $35\mathrm{cP}$ Hanks’ solution, $4000\mathrm{cP}$ Hanks’ solution, $4360\mathrm{cP}$ cervical mucus. The sperm heads (on the left side of images) are held in place with a micropipette tip. The contour lengths of the flagella are approximately $40\mu \mathrm{m}$. The thick (blue) line on the third panel indicates the maximum deflection angle $\theta \left(s\right)$. (b) Shapes of beating patterns for filaments with internal sliding forces, fixed head position, and clamped boundary conditions. A half cycle of the pattern is shown for internal sliding forces given by Eq. (51) with $k∕L=3\pi$. Time sequence: solid (red), short dash (orange), long dash (green), long dash-short dash (blue), dash-dot (purple). From left to right, the sperm numbers are 5.6, 13.6, 37.7, and 45, corresponding to viscosities of $1\mathrm{cP}$, $35\mathrm{cP}$, $4000\mathrm{cP}$, and $4360\mathrm{cP}$. In the first two plots, the medium is purely viscous, while in the third plot, $\mathrm{De}=0.5$ and ${\mathrm{De}}_2=0.5∕4000$. In the fourth plot, $\mathrm{De}=50$ and ${\mathrm{De}}_2=50∕4360$. At the top of each figure, we print the (dimensionless) magnitude ${\overline{f}}_{\mathrm{m}}$ required to produce motion with amplitude $0.1L$, and the length scale $\xi$.Reuse & Permissions

Figure 9

(Color online) Shape of beating pattern for filament with internal sliding forces, fixed head position, and clamped boundary conditions obtained from asymptotic analysis. A half cycle of the pattern is shown for internal sliding forces given by Eq. (51) with $k∕L=3\pi$. Time sequence: solid (red), short dash (orange), long dash (green), long dash-short dash (blue), dash-dot (purple). The sperm number is 37.7, corresponding to a viscosity of $4000\mathrm{cP}$, and $\mathrm{De}=0.5$, while ${\mathrm{De}}_2=0.5∕4000$. At the top of the figure, we print the (dimensionless) magnitude ${\overline{f}}_{\mathrm{m}}$ required to produce motion with amplitude $0.1L$, and the lengthscale $\xi$.Reuse & Permissions

Figure 10

(Color online) Infinite filament in a viscoelastic medium on a table rotating with angular speed $\Omega$. The coordinates $\{x,y\}$ rotate with the table; $\{x^{\prime },y^{\prime }\}$ are space fixed.Reuse & Permissions