Viscous propulsion in active transversely-isotropic media: Corrigendum

We report a corrigendum to the paper 'Viscous propulsion in active transversely-isotropic media' [J. Fluid Mech. 812, 501-524, 2017 / arxiv 1608.01451].

On viscous propulsion in active transversely isotropic media 409 for fibre-reinforced media, similar in nature to the glycofilament structure of cervical mucus, through the transversely isotropic constitutive equations of Ericksen (1960); we now detail a correction and further analysis, which in particular shows the importance of fibre orientation for both passive and active fluid cases.
Our previous paper (Cupples et al. 2017) consisted of calculating the mean swimming velocity and energy dissipation of an infinite waving sheet in a transversely isotropic fluid in two dimensions, extending the classical Taylor's swimming sheet model to include anisotropic effects and active rheology. A surprising conclusion was that fibre orientation only affected swimming velocity in the active case. However a recent study by Shi & Powers (2017) investigated microscopic propulsion in nematic liquid crystals and found that in a common limit (passive, zero elasticity, zero shear viscosity and small extensional viscosity) the models disagreed, with their study finding an angle dependence in swimming speed. Here we find that this discrepancy is due to missed terms in the solution of the governing equation in Cupples et al. (2017). These terms are relevant to both the passive and active cases, and qualitatively change the conclusions.
The analysis involves a perturbation expansion in the small parameter ε = k * b * , where b * is the amplitude and k * the wavenumber. The leading-order solution at O(ε) is unchanged from Cupples et al. (2017), and we here discuss a correction to the O(ε 2 ) solution which determines the swimming velocity. First the passive transversely isotropic fluid case is discussed (µ 1 = 0) in § 2, which is shown to be consistent with Shi & Powers (2017) in a common limit; the mean swimming velocity is recalculated and presented for a wide range of anisotropic extensional and shear viscosities in § 2.2. After this, a solution to the active case is considered in § 3, where a spatially averaged swimming velocity is calculated and discussed.

Equation formulation
The full system of equations is derived from the dimensionless Navier-Stokes equations, at zero Reynolds number, along with Ericksen's (1960) constitutive equation for a transversely isotropic fluid (equations (2.1)-(2.3) in Cupples et al. 2017). A streamfunction ψ, satisfying incompressibility, and an equation governing the perturbation to the fibre orientation θ around a uniform initial fibre angle φ (equation (2.5) in Cupples et al. 2017), complete the model. At O(ε 2 ) the system of partial differential equations is where F and G are known functions of the O(ε) solutions and are given in appendix A. This is stated in full in Cupples et al. (2017, equation (C 1) of appendix C). These functions involve terms proportional to cos 2 (x − t), sin 2 (x − t) and sin(x − t) cos(x − t) with coefficients in terms of the anisotropic parameters. In § 2 we take µ 1 = 0, which we refer to as the 'passive fluid' case, and solve the resulting system to determine the mean swimming velocity. The steps in this calculation are elucidated in more detail in order to highlight how to correct the solution. In § 3 we reconsider the active case for non-zero µ 1 .
G. Cupples, R. J. Dyson and D. J. Smith
2.1. Corrected solution The first step we take is to note that x and t only appear together as x − t and so we make the substitution z = x − t; in what follows we will be precise regarding which variable we are averaging over as the active case is not t-periodic in general.
Equation (2.1) becomes 1 + µ 2 4 sin 2 2φ + µ 3 ∇ 4 Ψ 1 + µ 2 2 cos 4φ ∂ 4 Ψ 1 ∂z 2 ∂y 2 + sin 4φ 2 On viscous propulsion in active transversely isotropic media 411 where Ψ 1 (z, y) = ψ 1 (x − t, y) and F z is (2.9) and the boundary conditions, equations (2.3) and (2.4), are The periodic nature of the swimming sheet means Ψ 1 is also periodic in z and so, upon taking the z-average of the system (2.8)-(2.11), the z derivatives disappear and the system becomes where · z ≡ (1/2π) π −π · dz. At this stage in Cupples et al. (2017) an incorrect ansatz was assumed which neglected inhomogeneous terms. Hence we alter this ansatz to correctly determine the first-order streamfunction Ψ 1 and thus the swimming velocity. Consider a complementary solution to the homogeneous problem and a particular integral satisfying the inhomogeneous portion; i.e.
( 2.15) For the homogeneous problem we have hence assume Ψ z P (y) takes the form where P (k) are constants to be determined. Substituting this form into (2.17) and rearranging for constants P (k) , we find (2.20) The boundary conditions are used to determine the constants in (2.20); for the velocity to be bounded we require A 3 = A 2 = 0 and for the z-averaged problem the boundary condition (2.3) becomes and A 0 can be set to zero without loss of generality. Hence the full solution is Due to the periodicity of the problem, the time and z-averages are identical, i.e. U z = U t . In the far field the mean swimming velocity is thus In addition to µ 1 = 0, the anisotropic shear viscosity µ 3 is set to zero and µ 2 = 0.05. It is immediately seen that the inclusion of the extra term has altered the mean swimming velocity and introduced a dependence on the initial orientation angle φ. Aside from the minimum values of the mean swimming velocity, the corrected solution agrees well with the work from Shi & Powers; this small difference is due to the 1/(1 + µ 2 sin 2 2φ/4 + µ 3 ) multiplying the second term in (2.24), as can be seen in the magnified view in figure 1.
Next we consider a larger range of µ 2 and µ 3 and compare the mean swimming velocity. First consider small µ 2 and µ 3 (figure 2). Increasing µ 2 dominates the impact of the initial orientation angle on the mean swimming velocity, and the anisotropic shear viscosity works to collapse the results back towards the Newtonian value; when both parameters are zero we return to the Newtonian solution as expected. For very small µ 2 (dashed lines which are not seen) the variation from the Newtonian solution is very small.
Finally we investigate the impact when both µ 2 and µ 3 may take on large values. Here we have separated the results into two cases: when µ 3 = 0 (figure 3a) and when µ 3 = 900 (figure 3b). When µ 3 = 0, the mean swimming velocity takes on large values near φ = 0 and φ = π; these sharp peaks are consistent with the results in Cupples et al. (2017) occurring when one parameter was much larger than the others. Away from these regions, the mean swimming velocity takes on values similar to those presented in figures 1 and 2. When both parameters are large (figure 3b), the mean swimming velocity reduces in comparison to figure 3(a). The shape of the φ-U t curve is similar as the anisotropic parameters are varied, only the magnitude changes.  π 0 ƒ π/4 π/2 3π/4 π 0 FIGURE 3. Corrected mean swimming velocity for large µ 2 . (a) µ 3 = 0 and (b) µ 3 = 900. Four choices for µ 2 are compared: µ 2 = 0 (solid lines), µ 2 = 100 (dashed lines), µ 2 = 500 (dot-dashed lines) and µ 2 = 900 (dotted lines). Panel (a) contains a magnified view of the middle section of the results.

Mean swimming velocity in active media
Next consider active transversely isotropic media, where µ 1 = 0. The equations governing the flow and orientation are given by (1.1) and (1.2) respectively. Due to the time derivatives that force the evolution of orientation (equation (1.2)) we can no longer seek a solution depending on z = x − t and instead look at an x-average of the coupled system.

Corrected solution
Based on the geometry of the problem, ψ 1 and θ 1 will be periodic in x. Hence, an x-average is taken, 1 + µ 2 4 sin 2 2φ + µ 3 ∂ 4 ψ x 1 ∂y 4 + µ 1 cos 2φ where · x ≡ (1/2π) 2π 0 · dx. Equation (3.1) can be directly integrated twice with respect to y, and substituted into (3.2) to give where B 0 (t) and B 1 (t) are functions of time to be determined. To simplify the following calculations, the functions are written in the form B 0 (t) =Ḟ 0 (t) exp(µ 1 Γ t) and B 1 (t) =Ḟ 1 (t) exp(µ 1 Γ t), where the dot notation represents a time derivative, F 0 (t) and F 1 (t) are functions of time to be determined and Γ = cos 2φ sin 2 φ 1 + µ 2 4 sin 2 2φ + µ 3 . (3.5) Then, equation (3.4) can be solved via an integrating factor to give and c(y) is a function to be determined. The full solution is detailed in appendix B. Since the fibres have initial orientation φ, the initial condition for the angle is θ x 1 (x, y, 0) = 0 and so where F 0 j = F j (0). The solution is thus

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G. Cupples, R. J. Dyson and D. J. Smith The form for θ x 1 can now be substituted back into (3.3), 1 + µ 2 4 sin 2 2φ + µ 3 ∂ 2 ψ x 1 which can then be directly integrated with respect to y to obtain (3.11) To determine the functions of integration, reconsider the boundary condition 12) and note that since the velocity must remain bounded in the far field we require B 0 (t) + µ 1 Γ B 2 (t)e µ 1 Γ t = 0 and B 1 (t) + µ 1 Γ B 3 (t)e µ 1 Γ t = 0. It can be shown that this is equivalent to B 0 (t) = B 1 (t) = B 2 (t) = B 3 (t) = 0 (see appendix C). Since B 5 has no impact on the velocity it can, without loss of generality, be set to zero. The final function B 4 (t) is determined from (3.12) as (3.13) Hence, the solutions ψ where the swimming velocity is given at far field as (3.16) 3.2. Comments Equation (3.16) will be valid only when µ 1 Γ 0 or for very short time scales. The sign of µ 1 Γ is determined by µ 1 cos 2φ; for 'puller' type behaviour, where µ 1 is positive, the solution is valid only for π/4 φ 3π/4 and these exponential terms decay with time. This however leads to a steady-state swimming velocity U x = 0 and so the active properties of the fluid halt any propulsion. For 'pusher' type behaviour, where µ 1 is negative, this validity is for 0 φ π/4 and the same result for the swimming velocity is obtained.
Outside this region, the solution for θ x 1 and further the swimming velocity U x grow exponentially and hence will not be valid in the perturbation expansion currently considered. To fully understand microscopic propulsion in active transversely isotropic media, it will be necessary to consider a numerical solution to the full swimming problem.

Discussion
New results arising from correction of an error in Cupples et al. (2017) have been described, prompted by Shi & Powers (2017) who investigated propulsion in nematic liquid crystals and discovered a discrepancy between the two models in a common limit. The corrected swimming velocity was calculated for a passive fluid, from which it was found that the extra terms introduce a dependence of the mean swimming velocity on the initial orientation angle. By setting µ 2 to be small and µ 3 = 0 our corrected result agrees with Shi & Powers (2017) in the common limit.
The corrected swimming velocity was then compared for a range of µ 2 and µ 3 . The effects of the initial orientation angle on U t were increased by increasing the anisotropic extensional viscosity and larger anisotropic shear viscosities reduce the effect of the initial orientation angle. Further, when one parameter is large and the other small, rapid changes in the swimming velocity and a reversal in the swimming direction (i.e. negative swimming velocity) were seen; a result seen consistent with the mean rate of working found in Cupples et al. (2017).
Finally a solution for the swimming velocity in active media (µ 1 = 0) was sought. Periodicity in x was imposed for the streamfunction and evolution of orientation angle due to the problem geometry; this observation simplified the calculations required. The coupled equations were solved to determine the first-order evolution of orientation and the swimming velocity. The swimming velocity varied exponentially in time, with the sign of the exponent dependent on µ 1 and the initial orientation angle. Thus the expansion is valid only for very short time periods, or for specific µ 1 and initial orientation angles where the exponent is negative; in these cases the active properties appear to halt propulsion. Setting µ 1 = 0 returned the result for the passive case. A topic of significant interest for future work is to investigate a fully numerical solution to the swimming problem in active transversely isotropic media. We set B 0 (t) = F 0 (t) exp(µ 1 Γ t) and B 1 (t) = F 1 (t) exp(µ 1 Γ t), where Γ is given by (3.5), to simplify the following calculations.
Introduce an integrating factor such that for function c(y) determined via boundary condition (3.12) in § 3.1.