Cell motility in viscous fluids is ubiquitous and affects many biological processes, including reproduction, infection and the marine life ecosystem. Here we review the biophysical and mechanical principles of locomotion at the small scales relevant to cell swimming, tens of micrometers and below. At this scale, inertia is unimportant and the Reynolds number is small. Our emphasis is on the simple physical picture and fundamental flow physics phenomena in this regime. We first give a brief overview of the mechanisms for swimming motility, and of the basic properties of flows at low Reynolds number, paying special attention to aspects most relevant for swimming such as resistance matrices for solid bodies, flow singularities and kinematic requirements for net translation. Then we review classical theoretical work on cell motility, in particular early calculations of swimming kinematics with prescribed stroke and the application of resistive force theory and slender-body theory to flagellar locomotion. After examining the physical means by which flagella are actuated, we outline areas of active research, including hydrodynamic interactions, biological locomotion in complex fluids, the design of small-scale artificial swimmers and the optimization of locomotion strategies. (Some figures in this article are in colour only in the electronic version) This article was invited by Christoph Schmidt.

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The hydrodynamics of swimming microorganisms

▪ Abstract Filament-stretching rheometers are devices for measuring the extensional viscosity of moderately viscous non-Newtonian fluids such as polymer solutions. In these devices, a cylindrical liquid bridge is initially formed between two circular end-plates. The plates are then moved apart in a prescribed manner such that the fluid sample is subjected to a strong extensional deformation. Asymptotic analysis and numerical computation show that the resulting kinematics closely approximate those of an ideal homogeneous uniaxial elongation. The evolution in the tensile stress (measured mechanically) and the molecular conformation (measured optically) can be followed as functions of the rate of stretching and the total strain imposed. The resulting rheological measurements are a sensitive discriminant of molecularly based constitutive equations proposed for complex fluids. The dynamical response of the elongating filament is also coupled to the extensional rheology of the polymeric test fluid, and this can...

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Filament-stretching rheometry of complex fluids

The purpose of this paper is twofold: from one side we provide a general survey to the viscoelastic models constructed via fractional calculus and from the other side we intend to analyze the basic fractional models as far as their creep, relaxation and viscosity properties are considered. The basic models are those that generalize via derivatives of fractional order the classical mechanical models characterized by two, three and four parameters, that we refer to as Kelvin–Voigt, Maxwell, Zener, anti–Zener and Burgers. For each fractional model we provide plots of the creep compliance, relaxation modulus and effective viscosity in non dimensional form in terms of a suitable time scale for different values of the order of fractional derivative. We also discuss the role of the order of fractional derivative in modifying the properties of the classical models.

Creep, relaxation and viscosity properties for basic fractional models in rheology

Individual polymers at thermal equilibrium were exposed to an elongational flow producing a high strain rate, and their dynamics were recorded with video fluorescence microscopy. The flow was turned on suddenly so that the entire evolution of molecular conformation could be observed without initial perturbations. The rate of stretching of individual molecules is highly variable and depends on the molecular conformation that develops during stretching. This variability is due to a dependence of the dynamics on the initial, random equilibrium conformation of the polymer coil. The increasing appearance at high strain rates of slowly unraveling hairpin folds is an example of nonergodic dynamics, which can occur when a statistical mechanical system is subjected to nonadiabatic, or “sudden,” external forces.

https://science.sciencemag.org/content/sci/281/5381/1335.full.pdf

Response of Flexible Polymers to a Sudden Elongational Flow

Highly extensible Escherichia coli DNA molecules in planar extensional flow were visualized in dilute solution by fluorescence microscopy. For a narrow range of flow strengths, the molecules were found in either a coiled or highly extended conformation, depending on the deformation history of the polymer. This conformation hysteresis persists for many polymer relaxation times and is due to conformation-dependent hydrodynamic forces. Polymer conformational free-energy landscapes were calculated from computer simulations and show two free-energy minima for flow strengths near the coil-stretch transition. Hysteresis cycles may directly influence bulk-solution stresses and the development of stress-strain relations for dilute polymer flows.

https://science.sciencemag.org/content/sci/301/5639/1515.full.pdf

Observation of polymer conformation hysteresis in extensional flow.

The origins, uses and evaluation of constitutive equations for the stress tensor of polymeric liquids are discussed.

Constitutive equations for polymeric liquids

In this paper we examine molecular stretching in the inception of uniaxial elongational flow of dilute polymer solutions. The polymer molecules are modeled as bead–spring chains with finitely extensible nonlinear elastic springs, and we use the Peterlin approximation. This work is distinguished from earlier work because we model the macromolecules with chains instead of dumbbells, and we examine the time dependence of three average quantities describing the chain conformation in unsteady flows: root‐mean‐square end‐to‐end distance, root‐mean‐square extensions of the individual links, and mean moment of inertia about the axis of elongation. We observe a gradual transition from the coiled equilibrium state of the chain to the stretched state after the inception of strong uniaxial elongational flow, and we describe the nature of this transition which takes place in roughly four stages: I equilibrium coil; II deformed coil; III spring stretched (‘‘locally unraveled’’); and IV unfolded chain. Inclusion of hydrodynamic interaction changes the macromolecular response quantitatively but not qualitatively.

On coil–stretch transitions in dilute polymer solutions

In this paper we modify the kinetic theory for suspensions of elastic dumbbells by introducing anisotropic hydrodynamic drag (expressed by a tensorial Stokes's law) and anisotropic Brownian motion (resulting from the introduction of a tensor into the Maxwellian velocity distribution). We show how our general formulation leads to Giesekus's 1982 constitutive equation, to the encapsulated dumbbell model of Bird and DeAguiar, and to several other constitutive equations. It is also shown how the Giesekus postulate for the form of the hydrodynamic drag tensor can be used to re‐interpret hydrodynamic interaction. Finally this development emphasizes the necessity for modifying the stress tensor expression when reptation is introduced into molecular theories.

Anisotropic Effects in Dumbbell Kinetic Theory

A comparison is given between the rheological behavior of dilute solutions of linear polymers and ring-shaped polymers. Both types of molecules are modeled as freely jointed bead-spring structures, with Hookean springs. It is shown that the ring closure results in lowering the viscosity by a factor of two regardless of the number of beads, N . The first normal stress coefficient is lowered by a factor between 5 (for N = 3) and 8 (for N → ∞). The form of the constitutive equation for rings is the same as that for chains, but the spectrum of relaxation times is altered. Formal expressions are obtained for the ring configurational distribution function and the single link distribution function for an arbitrary time-dependent homogeneous flow.

John M. Wiest

Papers

Constitutive equations for polymeric liquids

On coil–stretch transitions in dilute polymer solutions

Anisotropic Effects in Dumbbell Kinetic Theory

Effect of ring closure on rheological behavior

Non‐Newtonian Liquids