2.2 Varying body stiffness

To understand the role of body elasticity, we use our simulations to calculate the Stokes-normalized swimming speed while varying $B$ and $De$ for a fixed period ( $T=1$ ). We use a stroke defined by (2.1) with $A_{t}=5,$ and $A_{h}=2.$ This gives a high-amplitude stroke as in Teran et al. (Reference Teran, Fauci and Shelley2010) and Thomases & Guy (Reference Thomases and Guy2014). In figure 1(a) we plot normalized swimming speed as a function of $De$ for three characteristic stiffness values of $B=0.1,1.0,10.0$ , which we refer to as very soft, moderately soft and stiff, respectively. For very soft swimmers we see a monotonic boost in swimming speed, with a greater than 50 % boost for high $De.$

Figure 1.

(a) Swimming speed (normalized by Newtonian swimming speed) as a function of $De$ for different bending stiffness $B$ . Here we fix the period, $T=1$ . (b–d) Normalized swimming speed as a function of $De.$ Adaptations from the literature: (b) from Espinosa-Garcia et al. (Reference Espinosa-Garcia, Lauga, Zenit, Diego and Jolla2013), a physical model of a swimmer with a flexible tail. (c) From Teran et al. (Reference Teran, Fauci and Shelley2010) and Thomases & Guy (Reference Thomases and Guy2014), two different numerical simulations for a soft stroke with a large-amplitude tail. (d) From Thomases & Guy (Reference Thomases and Guy2014), numerical simulations for a stiff stroke with a large-amplitude tail.

This response is similar to what was reported in Espinosa-Garcia et al. (Reference Espinosa-Garcia, Lauga, Zenit, Diego and Jolla2013) using a physical model of a swimmer with a flexible tail (figure 1 b). For moderately soft swimmers, we see a non-monotonic speed-up, including a smaller speed boost over the Newtonian speed, followed by a slow-down at larger $De.$ This type of non-monotonic speed-up was first reported in Teran et al. (Reference Teran, Fauci and Shelley2010) and again in Thomases & Guy (Reference Thomases and Guy2014) for a soft stroke with high amplitude (figure 1 c). Finally, for stiff swimmers we see non-monotonic behaviour but no boost over the Newtonian speed, again followed by a slow-down at larger $De.$ This type of slow-down was reported in Thomases & Guy (Reference Thomases and Guy2014) for a stiff kicker (figure 1 d).

Figure 2.

Dimensional swimming speed in a Newtonian fluid for stiff, $B=10.0,$ and moderately soft, $B=1.0,$ swimmers over a range of frequencies $T^{-1}.$ Inset figures show shapes of swimmer over a period at the highest computed frequency for both soft and stiff swimmers.

In contrast to stiff, or rigid, swimmers, the dynamics of flexible swimmers involves an additional time scale. In a viscous fluid, rigid swimmers move with a velocity proportional to the beat frequency (the only time scale in the problem). The problem of a rigid swimmer in a viscoelastic fluid has two time scales, the beat frequency and the relaxation time, whose ratio is the dimensionless relaxation time $De.$ The swimming speed of soft swimmers depends nonlinearly on the frequency because the shape changes with the frequency. Figure 2 shows the swimming speed in a Newtonian fluid for both a stiff swimmer, $B=10.0$ , and a moderately soft swimmer, $B=1.0$ , over a range of beating frequencies. We see a linear response in the case of a stiff swimmer and a nonlinear response for the moderately soft swimmer. Inset swimmer shapes show how the shape changes as a function of the stiffness in the high frequency case.

To illustrate the significance of multiple time scales for flexible swimmers in viscoelastic fluids we compute the Stokes-normalized swimming speed as a function of $De$ varied two ways: by varying the relaxation time for a fixed period and by varying the period for a fixed relaxation time. Both simulations are performed with the same bending stiffness, $B=1.0$ , where passive body dynamics is significant. Results are shown in figure 3(a) for a swimmer with the same stroke from figure 1, and the two curves show remarkable qualitative differences. For a rigid swimmer these would give equivalent results. Thus this third time scale, arising from body flexibility, needs to be explicitly included in any discussion of swimming in elastic fluids. A more complete picture of how the swimming speed depends on both the relaxation time and period when the body is soft is shown in figure 3(b). Contours of constant $De=1-5$ are overlaid in black and the effect of body stiffness is clearly evident as the swimming speed varies significantly along any of the contours. The dashed lines denote the locations of the data in figure 3(a).

Figure 3.

(a) Normalized swimming speed as a function of $De.$ Curves generated by varying only relaxation time (for fixed period $T=1$ ) or stroke period (for fixed relaxation time $\unicode[STIX]{x1D706}=0.5$ ). Body stiffness is fixed: $B=1.0$ . (b) Normalized swimming speed as a function of both $\unicode[STIX]{x1D706}$ and $T,$ with contours overlaid for constant $De$ values. Dashed lines correspond to the locations of the data in (a).

3.1 Linear theory: Newtonian fluids

We begin by considering small-amplitude displacements of a finite-length thin elastic rod in a Newtonian fluid driven by prescribed curvature, $\unicode[STIX]{x1D705}_{0}(s,t)$ , (equivalently, prescribed moments) along the body with free ends. The theoretical analysis is similar in two and three dimensions, however, we will make note of the differences when we compare with the numerical simulations in two dimensions. We proceed with the analysis in three dimensions for simplicity. The shape of the rod is determined by the balance between elastic forces and viscous drag. The vertical displacement, $y(s,t)$ satisfies

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}_{\bot }y_{t}=-{\mathcal{B}}(y_{ss}-\unicode[STIX]{x1D705}_{0})_{ss}, & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle y_{ss}-\unicode[STIX]{x1D705}_{0}=0,(y_{ss}-\unicode[STIX]{x1D705}_{0})_{s}=0,\quad \text{at }s=0,L. & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D701}_{\bot }$ is the perpendicular drag coefficient and ${\mathcal{B}}$ is the bending stiffness of the rod. Note that the use of new notation ${\mathcal{B}},$ is intended to distinguish this (dimensional) bending stiffness we use in the linear theory from our previously defined (non-dimensional) bending stiffness, $B$ which we use in our numerical simulations.

Non-dimensionalizing (3.1) using the body length $L$ and the period of the driving force $T$ results in the dimensionless parameter we call the body response time:

(3.3) $$\begin{eqnarray}G=\frac{T}{{\mathcal{B}}^{-1}\unicode[STIX]{x1D701}_{\bot }L^{4}}=\frac{\text{period of motion}}{\text{elasto-hydrodynamic beam relaxation time}}.\end{eqnarray}$$

We note that $G^{-1}$ could be called a body relaxation time. The same non-dimensional group has appeared previously, but has been interpreted differently. In Shelley & Ueda (Reference Shelley and Ueda2000) a quantity similar to $G$ was considered an ‘effective viscosity’ of growing elastic filaments. The Sperm number ( $Sp=G^{-1/4}$ ) is the ratio of the body length to the viscous decay length (Wiggins & Goldstein Reference Wiggins and Goldstein1998; Fu et al. Reference Fu, Powers and Wolgemuth2007). This interpretation is natural when considering filaments driven at one end rather than along the body as we do here.

We change variables from displacement to curvature deviation

(3.4) $$\begin{eqnarray}c(s,t)=\unicode[STIX]{x1D705}(s,t)-\unicode[STIX]{x1D705}_{0}(s,t)\end{eqnarray}$$

to facilitate comparing with large-amplitude simulations. For small displacements $y_{ss}(s,t)\approx \unicode[STIX]{x1D705}(s,t)$ and (3.1) (non-dimensionalized) becomes

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle c_{t}=-Gc_{ssss}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D705}_{0}}{\unicode[STIX]{x2202}t}, & \displaystyle\end{eqnarray}$$

(3.6a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle c=0,\quad c_{s}=0\quad \text{at }s=0,1. & \displaystyle\end{eqnarray}$$

For a given $\unicode[STIX]{x1D705}_{0},$ we use an orthogonal function expansion to solve the non-dimensional equations for $c(s,t)$ . We let the driving curvature be given as

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{0}(s,t)=\mathop{\sum }_{k=1}^{\infty }\unicode[STIX]{x1D6FC}_{k}^{\infty }\text{e}^{2\unicode[STIX]{x03C0}\text{i}\unicode[STIX]{x1D707}_{k}t}\unicode[STIX]{x1D6F9}_{k}(s),\end{eqnarray}$$

and solve the eigenvalue problem,

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D707}\unicode[STIX]{x1D6F9}(s)=-\unicode[STIX]{x1D6F9}_{ssss}, & \displaystyle\end{eqnarray}$$

(3.9a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F9}=0,\quad \unicode[STIX]{x1D6F9}_{s}=0\quad \text{at }s=0,1, & \displaystyle\end{eqnarray}$$

for eigenvalues $\unicode[STIX]{x1D707}_{k}$ and eigenfunctions $\unicode[STIX]{x1D6F9}_{k}(s).$ The expansion coefficients of the realized curvature, $\unicode[STIX]{x1D705}$ , are then

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{k}=\unicode[STIX]{x1D6FC}_{k}^{\infty }\left(1-\left(1-\frac{G\unicode[STIX]{x1D707}_{k}}{2\unicode[STIX]{x03C0}\text{i}}\right)^{-1}\right).\end{eqnarray}$$

From this solution we can see that as the rod is stiffened ( $G\rightarrow \infty$ ), the resultant curvature tends to the prescribed curvature, $\unicode[STIX]{x1D6FC}_{k}\rightarrow \unicode[STIX]{x1D6FC}_{k}^{\infty }.$ We also see that for softer rods, i.e. smaller values of the body response time $G$ , the amplitude of the curvature decreases and there is a phase lag relative to the prescribed shape.

As mentioned above, we use intrinsic coordinates and curvature deviations, to allow us to consider large prescribed curvatures. However we note that (3.5) lacks terms coming from geometric nonlinearities and inextensibility that may not be small when the prescribed curvature is large (Goldstein & Langer Reference Goldstein and Langer1995; Camalet & Jülicher Reference Camalet and Jülicher2000). In §§ 3.3 and 4.2 we compare our simulations to theoretical analysis using (3.5), and in appendix A we show that the influence of the additional terms is in fact small for the amplitudes we consider.

3.2 Linear theory: viscoelastic fluids

We can modify the linear theory for elastic rods to include fluid elasticity. This is similar to what was done in Fulford, Katz & Powell (Reference Fulford, Katz and Powell1998) and Fu et al. (Reference Fu, Wolgemuth and Powers2008). In Fulford et al. (Reference Fulford, Katz and Powell1998) modifications to linear rod theory to include linear viscoelastic fluid effects were presented, and the authors concluded that while fluid elasticity does not change swimming speed, it reduces total work and thus can boost efficiency. However, it was pointed out in Lauga (Reference Lauga2007a ) that it is essential to use a nonlinear elasticity model in these types of calculations because the swimming speed itself is second order in amplitude, where the nonlinear effects are relevant. We note that with these higher-order terms Lauga (Reference Lauga2007a ) shows that swimming speed is always hindered by fluid elasticity for the case considered – infinite-length low-amplitude swimmer with sinusoidal undulations. In Fu et al. (Reference Fu, Wolgemuth and Powers2008) the authors analysed shape changes induced by fluid elasticity in a linearly elastic fluid. Unlike swimming speed, shape changes due to fluid elasticity come in to the asymptotic expansion at first order in amplitude, and hence it is reasonable to use a linearly elastic fluid to look at shape changes. Fu et al. (Reference Fu, Wolgemuth and Powers2008) did not make conclusions about how these shape changes affect swimming speed. Here we perform a similar analysis as in Fu et al. (Reference Fu, Wolgemuth and Powers2008), but by applying the analysis to deviations in curvature we are able to study shape changes in low- and high-amplitude finite-length flexible rods. In § 4 we discuss how these shape changes affect swimming speed.

As in (3.1) we can write a force balance relation between the force on a fluid and from the beam as

(3.11) $$\begin{eqnarray}f_{fluid}-{\mathcal{B}}(y_{ss}-\unicode[STIX]{x1D705}_{0})_{ss}=0,\end{eqnarray}$$

where the $f_{fluid}$ represents the normal force on the rod from the viscoelastic fluid. If we define the fluid force to be based on the total deviatoric stress $\unicode[STIX]{x1D749}=\dot{\unicode[STIX]{x1D738}}+\unicode[STIX]{x1D749}_{\boldsymbol{p}}$ then (upon linearization) using (2.5):

(3.12) $$\begin{eqnarray}De\;{\dot{f}}_{fluid}+f_{fluid}=(1+\unicode[STIX]{x1D709})f_{vis}+De\;{\dot{f}}_{vis},\end{eqnarray}$$

where $f_{vis}$ is the viscous drag force. Note that given the form of the system in (2.3)–(2.5), we have assumed a total viscosity of $1+\unicode[STIX]{x1D709}$ . The swimmer motion is time periodic so we take the Fourier transform in time of (3.12) to solve for viscoelastic modifications to the fluid drag. This yields, $\hat{f}_{fluid}=\left((1+\unicode[STIX]{x1D709}+2\unicode[STIX]{x03C0}\text{i}De)/(1+2\unicode[STIX]{x03C0}\text{i}De)\right)\hat{f}_{vis}.$

As in the viscous theory, we can solve for modifications to the curvature from body stiffness and use the modifications to the fluid drag to account for the fluid elasticity:

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{k}=\unicode[STIX]{x1D6FC}_{k}^{\infty }\left(1-\left(1-\frac{G\unicode[STIX]{x1D707}_{k}}{\unicode[STIX]{x1D701}_{ve}2\unicode[STIX]{x03C0}\text{i}}\right)^{-1}\right),\quad \text{with }\unicode[STIX]{x1D701}_{ve}=\frac{1+\unicode[STIX]{x1D709}+2\unicode[STIX]{x03C0}\text{i}De}{1+2\unicode[STIX]{x03C0}\text{i}De}.\end{eqnarray}$$

The coefficients in (3.13) give an analytical expression for the modifications to the rod shapes relative to the prescribed shapes as fluid and body elasticity are varied.

3.3 Elastic shape changes: theory and numerical comparison

The analysis in the previous sections made use of resistive force theory which relates the drag force and velocity on a long thin cylindrical object. More generally, for small amplitudes the vertical displacement satisfies

(3.14) $$\begin{eqnarray}y_{t}=MFy,\end{eqnarray}$$

where $M$ is the mobility operator and $F$ is the linearized bending force operator. Resistive force theory makes the approximation $M\approx 1/\unicode[STIX]{x1D701}_{\bot }.$ Our analysis of shapes (3.13) contains the quantity $G\unicode[STIX]{x1D707}_{k},$ where $G$ depends on the drag coefficient, $\unicode[STIX]{x1D701}_{\bot },$ (3.3). To use the more general linear theory in our analysis one can identify $\unicode[STIX]{x1D707}_{k}/\unicode[STIX]{x1D701}_{\bot }=\unicode[STIX]{x1D707}_{k}^{MF}$ , where $\unicode[STIX]{x1D707}_{k}^{MF}$ denotes the $k\text{th}$ eigenvalue of the operator $MF.$ We relate $G\unicode[STIX]{x1D707}_{k}$ to the dimensionless bending stiffness, $B$ used in our simulations, through

(3.15) $$\begin{eqnarray}G\unicode[STIX]{x1D707}_{k}=\frac{T{\mathcal{B}}}{L^{4}\unicode[STIX]{x1D701}_{\bot }}\unicode[STIX]{x1D707}_{k}=\left(\frac{T{\mathcal{B}}}{L^{4}}\right)\left(\frac{\unicode[STIX]{x1D707}_{k}}{\unicode[STIX]{x1D701}_{\bot }}\right)=B\unicode[STIX]{x1D707}_{k}^{MF}.\end{eqnarray}$$

To compare the linear analysis with our two-dimensional simulations we numerically approximate (3.14). For small deviations to the vertical displacement, $M$ is the integral operator which is the convolution of the vertical force with the fundamental solution to Stokes equations. We approximate $M$ using the method of regularized Stokeslets (Cortez Reference Cortez2001), which is a numerical method based on a regularized Greens function for the Stokes equations. We can also numerically approximate $\unicode[STIX]{x1D707}_{k}^{F},$ the $k\text{th}$ eigenvalue of the bending force operator $F,$ using a second-order finite difference, and we find that with point spacing $\unicode[STIX]{x0394}s=0.002,$ the eigenvalues of $F$ are within 1 % of the eigenvalues of the continuous operator. We give the eigenvalues for the first four non-trivial modes in table 1. Note that to compute $\unicode[STIX]{x1D707}_{k}^{MF}$ we assume a viscosity of one. Also in table 1, we give the first four (mode dependent) drag coefficients computed as $\unicode[STIX]{x1D701}_{k}=\unicode[STIX]{x1D707}_{k}^{F}/\unicode[STIX]{x1D707}_{k}^{MF}$ .

Table 1.

Eigenvalues of the discretized operators $F$ and $MF$ using $\unicode[STIX]{x0394}s=0.002,$ and the effective drag coefficient $\unicode[STIX]{x1D701}_{k}=\unicode[STIX]{x1D707}_{k}^{F}/\unicode[STIX]{x1D707}_{k}^{MF}$ for the first four non-trivial modes.

In order to compare the predicted shape changes given by (3.13) with our numerical simulations we prescribe a curvature of the form

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{0}(s,t)=A\sin (2\unicode[STIX]{x03C0}t),\end{eqnarray}$$

in our model equations (2.3)–(2.1). The prescribed standing wave of constant curvature corresponds to a motion through circular arcs with peak curvature $A$ . By symmetry, this motion does not result in any horizontal translation of the body. We refer to these non-translating ‘swimmers’ as flexors. We consider both low- and high-amplitude curvatures, $A=0.5$ and $A=4.0.$ The shapes are shown inset in figure 4(a).

Figure 4.

(a) Normalized amplitude of the first mode, $|\unicode[STIX]{x1D6FC}_{1}|/|\unicode[STIX]{x1D6FC}_{1}^{\infty }|,$ from (3.13), for the flexor at low $A=0.5$ and high $A=4.0$ amplitude over a range of bending stiffness ${\mathcal{B}}.$ Linear theory is shown in solid lines and simulation data are indicated by markers. (b,c) Data from (a) renormalized by Newtonian ( $De=0$ ) data, as a function of both ${\mathcal{B}}$ and $De.$

In figure 4 we plot the theoretical predictions from (3.13) (solid lines) along with values computed from numerical simulations; low amplitude ( $A=0.5$ ) are indicated by hollow markers, and high amplitude ( $A=4.0$ ) are indicated with filled markers. In figure 4(a) we plot the normalized amplitude of the first mode ( $|\unicode[STIX]{x1D6FC}_{1}|/|\unicode[STIX]{x1D6FC}_{1}^{\infty }|$ ) to see how the amplitude deviates from the prescribed amplitude as a function of bending stiffness $B$ . We see that generically the amplitude of the flexor decreases as the flexor is softened for fixed $De$ . For sufficiently soft flexors ( $B\lesssim 1$ ) viscoelasticity increases the amplitude monotonically with $De$ , but for stiffer swimmers the amplitude changes non-monotonically with fluid elasticity.

In figure 4(b,c) we renormalize the data by the amplitude in a viscous fluid to see the effects of viscoelasticity more clearly. Again we see that fluid elasticity can increase the amplitude significantly for a soft flexor, but that effect is lost as the flexor is stiffened. When we plot the amplitude as a function of $De$ for the very soft, moderately soft and stiff cases we see again that three qualitatively different regimes emerge. For very soft flexors the amplitude is monotonically increased by elasticity, for moderately soft flexors the response is non-monotonic, and can decrease or increase the amplitude, and for stiff flexors there is little change in the amplitude due to fluid elasticity. It is notable that the linear theory does such a good job predicting shape changes for low and high amplitude and for low and high Deborah number. In appendix A we derive the theory for both the limit of small amplitude and the limit of high stiffness. We see in figure 4(a,b) that the largest differences are for moderate stiffnesses at high amplitude. We note that we are showing results only for the first mode. For higher modes the trends are similar but the transition from stiff to soft behaviour occurs at lower values of $B$ because the eigenvalues $\unicode[STIX]{x1D707}_{k}$ increase with $k$ .

4.1 Swimming speed: two-mode swimmer

To keep the analysis simple we define a gait whose swimming speed in a viscous fluid we can compute analytically. We define a ‘two-mode swimmer’ given by the curvature:

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D705}(s,t)=A_{1}\cos (2\unicode[STIX]{x03C0}t/T+\unicode[STIX]{x1D719}_{1})\unicode[STIX]{x1D6F9}_{1}(s)+A_{2}\cos (2\unicode[STIX]{x03C0}t/T+\unicode[STIX]{x1D719}_{2})\unicode[STIX]{x1D6F9}_{2}(s),\end{eqnarray}$$

where the $\unicode[STIX]{x1D6F9}_{i}(s)$ for $i=1,2,$ are the first and second bending modes. The modulation of a single mode results in a standing wave and will not translate in a Newtonian fluid. We use a sum of the first two modes with a phase difference to generate a non-reciprocal motion. Shapes of the first, second and sum of the first and second modes are plotted in figure 5 for both low and high amplitudes.

Figure 5.

Shapes for first, second and sum of first two modes for ‘two-mode’ swimmer defined by (4.1). Snapshots of one period for low- and high-amplitude strokes.

Using resistive force theory one can derive the (time-averaged) swimming speed for a given small-amplitude motion:

(4.2) $$\begin{eqnarray}\left\langle U\right\rangle =\left(\frac{\unicode[STIX]{x1D701}_{\bot }}{\unicode[STIX]{x1D701}_{\Vert }}-1\right)\frac{1}{LT}\int _{0}^{\text{T}}\int _{0}^{L}y_{s}y_{t}\,\text{d}s\,\text{d}t,\end{eqnarray}$$

where $U$ is the swimming speed, $y(s,t)$ is the vertical displacement of the swimmer and $\unicode[STIX]{x1D701}_{\bot }$ and $\unicode[STIX]{x1D701}_{\Vert }$ are the perpendicular and parallel drag coefficients, respectively, (Wiggins & Goldstein Reference Wiggins and Goldstein1998; Lauga Reference Lauga2007b ).

For small amplitudes, the shape of the swimmer (up to translation and rotation) is given by integrating equation (4.1) twice in space to compute the swimming speed via (4.2). The swimming speed (in a viscous fluid) for the two-mode swimmer is proportional to the product of the amplitudes and the sine of the phase difference:

(4.3) $$\begin{eqnarray}\left\langle U\right\rangle \propto A_{1}A_{2}\sin (\unicode[STIX]{x1D719}_{2}-\unicode[STIX]{x1D719}_{1}).\end{eqnarray}$$

With this expression and the theoretical prediction for shape changes, we define a stroke-induced swimming speed which is the swimming speed in a Newtonian fluid that depends on the shape changes due to fluid elasticity and body flexibility. Specifically, for a given $De$ and $G,$ we compute $\unicode[STIX]{x1D6FC}_{k}$ from (3.13) ( $A_{j}=|\unicode[STIX]{x1D6FC}_{j}|,$ and $\unicode[STIX]{x1D719}_{j}=\arg (\unicode[STIX]{x1D6FC}_{j})$ ) and the stroke-induced swimming speed from (4.3). We parametrize the shape changes due to changes in $De$ using a parameter we call the stroke Deborah number (Thomases & Guy Reference Thomases and Guy2014), $\text{Stroke}_{De}.$ In other words, $\text{Stroke}_{De}$ represents the value of $De$ used in (3.13) to compute the stroke-induced swimming speed via (4.3). Our analytical expression for the shape changes is based on a linearly elastic fluid, but because the nonlinear elastic effects and swimming speed are both second order in amplitude, we do not expect the stroke-induced swimming speed to capture the true viscoelastic swimming speed. An analytical expression for the swimming speed in a nonlinear viscoelastic fluid, as was computed in Lauga (Reference Lauga2007a ), is not tractable in the finite-length case, because translational invariance, which facilitates the calculation for infinite-length swimmers, is lost.

We plot the stroke-induced swimming speed for the two-mode swimmer over a range of $\text{Stroke}_{De},$ and as with the flexor, we see the emergence of three regimes dependent on the body stiffness, see figure 6(a). Shape changes boost the stroke-induced swimming speed if the swimmer is very soft, a smaller boost is obtained for the moderately soft swimmer and additionally there is a non-monotonic response to increasing elasticity including a regime where shape changes slow-down the swimmer, and finally if the swimmer is stiff there is a negligible effect.

4.2 Swimming speed: theory and numerical comparison

Figure 6.

(a) Theoretically predicted stroke-induced swimming speed (computed using (4.3)) normalized by the Newtonian stroke-induced swimming speed. (b) Stokes-normalized swimming speed in simulations with the two-mode swimmer. (c) Ratio of speed to stroke-induced speed for low- and high-amplitude strokes. (d) Ratio of high- to low-amplitude swimming speed. (Dashed lines are for graphical interpretation.)

We simulate a two-mode swimmer of both low and high amplitude by prescribing a curvature of the form given in (4.1) with $A_{1}=0.8A$ , $A_{2}=0.6A$ , $\unicode[STIX]{x1D719}_{2}-\unicode[STIX]{x1D719}_{1}=\unicode[STIX]{x03C0}/2$ , for $A=0.5$ (low) and $4.0$ (high). These values come from projections of the stroke used to generate figure 1. The Stokes-normalized swimming speeds for a very soft, moderately soft and stiff swimmer at both low (hollow markers) and high (filled markers) amplitude are plotted in figure 6(b). We note that the three regimes seen in figure 6(a) still emerge from these simulations, but for this two-mode swimmer the simulation swimming speeds are always slower than the stroke-induced swimming speeds. The ratio of swimming speed to stroke-induced swimming speed is shown in figure 6(c). This quantity can be interpreted as the effect of fluid elasticity that is not related to shape changes. It is notable that these curves collapse onto a single curve for the low-amplitude swimmers at all stiffnesses as well as the high-amplitude swimmer in the very soft regime. This additional elastic fluid effect on swimming speed is likely to be highly stroke dependent.

Figure 7.

Amplitudes $|\unicode[STIX]{x1D6FC}_{1}|,|\unicode[STIX]{x1D6FC}_{2}|$ and $\sin (\unicode[STIX]{x1D719}_{2}-\unicode[STIX]{x1D719}_{1})$ , for the two-mode swimmer, equation (4.1). Linear theory is shown in solid lines (dependence on ${\mathcal{B}},$ and $De$ coming from (3.13)) and simulation data are indicated by markers for high amplitude $A=4.0,$ and $De=0.5$ (a), $De=4.0$ (b).

The additional effects of fluid elasticity are fundamentally different for the large-amplitude, large $De$ regime. In figure 6(d) we plot the ratio of swimming speeds for the high-to-low amplitude strokes, and see that for $De>1$ (for sufficiently stiff swimmers) a significant difference in swimming speed arises. This difference is not related to shape changes because, like the flexor, the elasticity-induced shape changes predicted by the theory for the two-mode swimmer agree very well with the simulation results, for all $De,$ at low and high amplitudes; see figure 7. At low amplitude (not shown) the relative error between theoretical and numerical predictions is less than 1 %, and for high amplitude the error at low $De$ is at most 5 $\%$ and at high $De$ the error is at most 9 %. These results indicate that the theoretically predicted shape changes and their isolated effects on swimming speed can be well approximated by the analytical results for the amplitudes simulated and the range of $De$ considered. A mechanistic understanding is lacking to explain what causes the dramatic slow-downs of swimmers in the high-amplitude, high $De$ regime.

Figure 8.

(a) Average elastic–viscous stress ratio in a two-mode swimmer over a range of bending stiffness ${\mathcal{B}}$ for both low- and high-amplitude strokes and low and high $De.$ (b) Colour field of the magnitude of the elastic–viscous stress ratio around stiff swimmers ( ${\mathcal{B}}=10$ ) at low and high $De$ and low and high amplitudes. (c) Tail amplitudes over a range of bending stiffness ${\mathcal{B}}$ , measured as the maximum displacement at the tail over a period, for both low- and high-amplitude strokes and low and high $De.$

We conjecture that the slow-downs in the high-amplitude, high $De$ regime must be attributed in part to the large localized stresses that accumulate near the body (Teran et al. Reference Teran, Fauci and Shelley2010; Thomases & Guy Reference Thomases and Guy2014). To explore this conjecture, in figure 8(a) we plot the average elastic to viscous stress ratio, computed as the time average over one period of $\Vert \unicode[STIX]{x1D749}_{\boldsymbol{p}}\Vert /\Vert \unicode[STIX]{x1D735}\boldsymbol{u}\Vert$ where $\Vert \cdot \Vert$ is the Frobenius norm, over a range of body stiffness $B$ for both low- (hollow markers) and high-amplitude (filled markers) strokes for $De=0.5,4.0.$ There is a notable transition in the stress ratio in the high $De$ , high-amplitude swimmer as the body is stiffened, while this stress ratio is flat for both low-amplitude and low $De$ swimmers. Stiff swimmer shapes along with the elastic–viscous stress ratio are plotted on a log scale in figure 8(b). The low-amplitude strokes are surrounded by elastic stresses that are at least two orders of magnitude smaller than the high-amplitude strokes, but even at large amplitude the low $De$ swimmer still has relatively low stress near the body. Lastly, we plot the tail amplitude, as one measure of the swimmer stroke, in figure 8(c). We see that for sufficiently soft swimmers the ‘high-amplitude’ stroke has a lower amplitude, which explains why in figure 6 the very soft high-amplitude swimmer behaves like the low-amplitude swimmers. For the high-amplitude strokes it is only in the large-amplitude and large $De$ regime where significant stress accumulates near the swimmer.