CHAPTER 1

Fluid–Structure Interactions and Flagellar Actuation

Henry C. Fu

Department of Mechanical Engineering, University of Nevada, Reno, North Virginia 89557, USA

1.1 Introduction 3
1.2 Hydrodynamics of slender filaments 4
1.3 Elastic forces in slender filaments 7
1.3.1 Straight undeformed filament 7
1.3.2 Helical undeformed filament 8
1.4 Swimming velocity of bacterium with helical flagellum 9
1.5 Fluid–structure interactions in bacterial flagella 10
1.6 Flagella in viscoelastic fluids 12
1.6.1 First-order solutions for a cylinder with prescribed
beating pattern 15
1.6.2 Forces on a cylinder with prescribed beating pattern 16
1.6.3 Velocity of a cylinder with prescribed beating pattern 17
1.7 Fluid–structure interaction in eukaryotic flagella 20
1.8 Probing dynein coordination using models of spontaneous flagellar beating 25
References 27

1.1 Introduction

In this chapter, we address the interaction between hydrodynamic forces and flagellar shapes for swimming bacteria and sperm. Although the filamentous propulsive structures used by prokaryotic bacteria and eukaryotic sperm are distinct, they are both called "flagella." A bacterial flagellum is a passive filament with diameter of 10–20 nm and about 10-µm long, actuated by a rotary motor at its base. The sperm flagellum has larger dimensions, with diameter of about 200 nm and length of 30–300µm, and has a complicated internal structure that allows actuation along the length of the flagellum.

Despite these differences, the treatment of fluid–structure interactions in both bacterial and sperm flagella is unified by the fact that both are slender filaments, so that hydrodynamic and elastic forces in both cases can be described using the same techniques. Due to the small size of the flagella, the hydrodynamics fall into the low Reynolds number regime. The elastic forces can be described using the theory appropriate for thin rods.

Estimating the Reynolds number of bacterial flagella using the flagellar length ≈ 10µm and swimming speed 100µm/s results in Re = 10^-3. For eukaryotic flagella, estimating the Reynolds number using the flagellar length of ≈ 50µm and swimming speed ≈ 50µm/s results in Re = 2.5 × 10^-3. For such filaments, the hydrodynamic forces can be efficiently described using slender body theory or its lowest order approximation, resistive force theory. Such small Reynolds numbers indicate that viscous forces dominate over inertial forces in flagellar hydrodynamics. One consequence of is that the dynamics are in the strongly overdamped regime; hence, the dynamics simplify to the condition of force balance. In equations, this means that the governing equations are not the Navier–Stokes equation,

ρ[D/Dt]v = ρg + [nabla] · σ, (1.1)

but rather the Stokes flow equation,

0 = ρg + [nabla] · σ. (1.2)

Physically, this corresponds to the observation that the viscous stresses in the low-Reynolds number limit are much bigger than the inertial stresses, and therefore, the viscous stresses must nearly cancel each other out to satisfy momentum conservation. The condition of force balance will be used throughout this chapter to calculate flagellar dynamics.

The chapter is organized as follows: in Sections 1.2 and 1.3, we introduce the descriptions of hydrodynamic and elastic forces appropriate for thin filaments, which will be used in the rest of the chapter. Then in Sections 1.4–1.5, we focus on bacterial flagella and discuss modifications to hydrodynamic interactions in Section 1.6, and in Sections 1.7 and 1.8, we focus on eukaryotic flagella. In both, we examine how fluid–structure interactions affect the shapes of the flagella during swimming motion and ultimately swimming properties. In addition to fluid–structure interactions in Newtonian fluids such as water, we also address how swimming shapes and properties are affected by swimming in viscoelastic fluids such as mucus. Symbols are listed in Table 1.1.

1.2 Hydrodynamics of slender filaments

To compute fluid–structure interactions, we must calculate the viscous force exerted on the filament due to movement through the fluid. In many cases, it is sufficient to use resistive force theory, the lowest order approximation to slender body theory. Thus, we first describe the resistive force theory, then describe refinements to it utilizing the full slender body theory.

In resistive force theory, the force per unit length f on a slender body is determined by the local velocity v of the slender body relative to the macroscopic background flow v_background. At a position corresponding to the arc length s along the filament, the force per unit length is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

In this expression, [??] is the tangent vector along the filament, and ζ_[parallel] and ζ_{[perpendicular to]} are the drag coefficients for motion parallel and perpendicular to the filament arc length, respectively. The values of the drag coefficients depend on both the fluid viscosity and the geometry of the slender filament. For very slender filaments, ζ_{[perpendicular to]} is approximately twice as large as ζ_[parallel]. The anisotropy in drag coefficients turns out to be necessary for filaments to generate propulsion. There are many formulations for the drag coefficients (see for example, Ref. [1]), but for example, one set of concrete expressions for the drag coefficients is

ζ_{[perpendicular to]} = 8πη/ln((0.18l)²/a²) + 1 (1.4)

ζ_[parallel] = 4πη/ln((0.18l)²/a²) - 1 (1.5)

where η is viscosity, l is a length scale corresponding to the wavelength of flagellar undulations, and a is the radius of the flagellar filament. The variety of different expressions of drag coefficients in resistive force theory is an indication that the hydrodynamic force per unit length is not actually determined locally but instead depends on the details of the filament waveform far from the point under consideration.

Slender body theory provides a more accurate representation of the hydrodynamic forces on the filament, which takes into account the hydrodynamic interactions between sections of the filament that are far removed from each other. In this formulation, the flows are generated from a line density of singular solutions to the Stokes flow equations (1.1)–(1.3), so the flow v(r) at position r is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where -f(s) is the force exerted by the flagellum on the fluid (equal but opposite to the hydrodynamic force exerted on the flagellum), and the operator δk_j - [??]_k[??]_j projects onto the space normal to the filament. S_ij and D_ik are the Stokeslet and doublet solutions to the Stokes equations, respectively, given by

S_ij(r) = 1/8πμ(δ_ij/r + r_ir_j/r³) (1.7)

D_ij(r) = 1/4π(δ_ij/r³ - 3 r_ir_j/r⁵. (1.8)

Thus, Eq. (1.6) represents a line of Stokeslets of strength density -f(s) and source dipoles of strength f_{[perpendicular to]}a²/4μ. This particular combination of Stokeslet and source dipoles is chosen so that the no-slip condition can be satisfied at the surface of the filament: at a radius of a from the centerline, the velocity is constant (to leading order in a/L, where L is some length scale over which the filament direction varies).

In Eq. (1.6), the force distribution f(s) is unknown, but due to the condition of force balance that holds in the overdamped limit of Stokes flow, f(s) is equal and opposite to the force per unit length exerted on a segment of flagellum by elastic forces resulting from filament deformations, which we turn to next.

1.3 Elastic forces in slender filaments

This chapter will focus on the continuous deformations of flagella. As mentioned in the Introduction, the discussion is restricted to small deformations from the rest configuration of the flagellum, which is typically helical for bacterial flagella and straight for sperm flagella. Throughout this chapter, we do not account for the large-scale deformations associated with polymorphic transitions in bacterial flagella, the treatment of which requires extensions of linear elasticity theory.

The flagellar configuration can be specified by the position r(s) which the centerline of the flagellum takes as a function of the arc length s or distance along the centerline. For a slender rod, standard elasticity theory relates the resultant force F(s) and moment M(s) at cross sections of the rod to the external force per unit length f and external moment per unit length m acting on the filament,

0 = [partial derivative]F/[partial derivative]s + f (1.9)

0 = [partial derivative]M/[partial derivative]s + [??] × F + m, (1.10)

where s is the arc length and [??] = [partial derivative]_sr is the tangent along the filament. Thus, knowledge of the external forces and moments f and m, for example, due to hydrodynamic forces, can be used to find the resultant elastic force and moment on the filament cross section.

Once those resultant quantities are known, we use them to determine the deformed configuration of the filament. In this chapter, we consider two cases: (1) when the undeformed state of the filament is straight (as in the case of sperm flagella) and (2) when the undeformed state of the filament is helical (as in the case of bacterial flagella).

1.3.1 Straight undeformed filament

For sperm flagella, we will consider filaments that are straight in their undeformed states and that have deformations restricted to a plane and no torsion. The latter restriction is a simplifying approximation; in reality, most sperm have beating patterns that are not completely planar. In that case, the force can be derived from the energy functional

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)