A note on the stability of slip channel flows

We consider the influence of slip boundary conditions on the modal and non-modal stability of pressure-driven channel flows. In accordance with previous results by Gersting (1974) (Phys. Fluids, 17) but in contradiction with the recent investigation of Chu (2004) (C.R. Mecanique, 332), we show that slip increases significantly the value of the critical Reynolds number for linear instability. The non-modal stability analysis however reveals that the slip has a very weak influence on the maximum transient energy growth of perturbations at subcritical Reynolds numbers. Slip boundary conditions are therefore not likely to have a significant effect on the transition to turbulence in channel flows

Low-Reynolds number swimming in gels

Many microorganisms swim through gels, materials with nonzero zero-frequency elastic shear modulus, such as mucus. Biological gels are typically heterogeneous, containing both a structural scaffold (network) and a fluid solvent. We analyze the swimming of an infinite sheet undergoing transverse traveling wave deformations in the "two-fluid" model of a gel, which treats the network and solvent as two coupled elastic and viscous continuum phases. We show that geometric nonlinearities must be incorporated to obtain physically meaningful results. We identify a transition between regimes where the network deforms to follow solvent flows and where the network is stationary. Swimming speeds can be enhanced relative to Newtonian fluids when the network is stationary. Compressibility effects can also enhance swimming velocities. Finally, microscopic details of sheet-network interactions influence the boundary conditions between the sheet and network. The nature of these boundary conditions significantly impacts swimming speeds.Comment: 6 pages, 5 figures, submitted to EP

Enhanced active swimming in viscoelastic fluids

Swimming microorganisms often self-propel in fluids with complex rheology. While past theoretical work indicates that fluid viscoelasticity should hinder their locomotion, recent experiments on waving swimmers suggest a possible non-Newtonian enhancement of locomotion. We suggest a physical mechanism, based on fluid-structure interaction, leading to swimming in a viscoelastic fluid at a higher speed than in a Newtonian one. Using Taylor's two-dimensional swimming sheet model, we solve for the shape of an active swimmer as a balance between the external fluid stresses, the internal driving moments, and the passive elastic resistance. We show that this dynamic balance leads to a generic transition from hindered rigid swimming to enhanced flexible locomotion. The results are physically interpreted as due to a viscoelastic suction increasing the swimming amplitude in a non-Newtonian fluid and overcoming viscoelastic damping.We thank R. E. Goldstein and T. J. Pedley for useful discussions. This work was funded in part by the European Union (CIG grant to EL).This is the author accepted manuscript. The final version is available from IOP Science via : http://dx.doi.org/10.1209/0295-5075/108/3400

Micro-Tug-of-War: A Selective Control Mechanism for Magnetic Swimmers

One of the aspirations for artificial microswimmers is their application in noninvasive medicine. For any practical use, adequate mechanisms enabling control of multiple artificial swimmers will be of paramount importance. Here we theoretically propose a multihelical, freely jointed motor as a selective control mechanism. We show that the nonlinear step-out behavior of a magnetized helix driven by a rotating magnetic field can be exploited when used in conjunction with other helices to obtain a velocity profile that is non-negligible only within a chosen interval of operating frequencies. Specifically, the force balance between the competing opposite-handed helices is tuned to give no net motion at low frequencies (tug-of-war), while in the middle-frequency range, the magnitude and, potentially, the sign of the swimming velocity can be adjusted by varying the driving frequency. We illustrate this idea on a two-helix system and demonstrate how to generalize to $\textit{N}$ helices, both numerically and theoretically. We then explain how to solve the inverse problem and design an artificial swimmer with an arbitrarily complex velocity vs frequency relationship. We finish by discussing potential experimental implementation.This work is funded in part by the European Union through a Marie Curie CIG Grant (E. L.) and by the Engineering and Physical Sciences Research Council (P. K.).This is the author accepted manuscript. The final version is available from the American Physical Society via http://dx.doi.org/10.1103/PhysRevApplied.5.06401

Analytical solutions to slender-ribbon theory

The low-Reynolds number hydrodynamics of slender ribbons is accurately captured by slender-ribbon theory, an asymptotic solution to the Stokes equation which assumes that the three length scales characterising the ribbons are well separated. We show in this paper that the force distribution across the width of an isolated ribbon located in a infinite fluid can be determined analytically, irrespective of the ribbon's shape. This, in turn, reduces the surface integrals in the slender-ribbon theory equations to a line integral analogous to the one arising in slender-body theory to determine the dynamics of filaments. This result is then used to derive analytical solutions to the motion of a rigid plate ellipsoid and a ribbon torus and to propose a ribbon resistive-force theory, thereby extending the resistive-force theory for slender filaments

The boundary integral formulation of Stokes flows includes slender-body theory

The incompressible Stokes equations can classically be recast in a boundary integral (BI) representation, which provides a general method to solve low-Reynolds number problems analytically and computationally. Alternatively, one can solve the Stokes equations by using an appropriate distribution of flow singularities of the right strength within the boundary, a method particularly useful to describe the dynamics of long slender objects for which the numerical implementation of the BI representation becomes cumbersome. While the BI approach is a mathematical consequence of the Stokes equations, the singularity method involves making judicious guesses that can only be justified a posteriori. In this paper we use matched asymptotic expansions to derive an algebraically accurate slender-body theory directly from the BI representation able to handle arbitrary surface velocities and surface tractions. This expansion procedure leads to sets of uncoupled linear equations and to a single one-dimensional integral equation identical to that derived by Keller and Rubinow (1976) and Johnson (1979) using the singularity method. Hence we show that it is a mathematical consequence of the BI approach that the leading-order flow around a slender body can be represented using a distribution of singularities along its centreline. Furthermore when derived from either the single-layer or double-layer modified BI representation, general slender solutions are only possible in certain types of flow, in accordance with the limitations of these representations