Slender-ribbon theory

Abstract

Ribbons are long narrow strips possessing three distinct material length scales (thickness, width, and length) which allow them to produce unique shapes unobtainable by wires or filaments. For example when a ribbon has half a twist and is bent into a circle it produces a Möbius strip. Significant effort has gone into determining the structural shapes of ribbons but less is know about their behavior in viscous fluids. In this paper we determine, asymptotically, the leading-order hydrodynamic behavior of a slender ribbon in Stokes flows. The derivation, reminiscent of slender-body theory for filaments, assumes that the length of the ribbon is much larger than its width, which itself is much larger than its thickness. The final result is an integral equation for the force density on a mathematical ruled surface, termed the ribbon plane, located inside the ribbon. A numerical implementation of our derivation shows good agreement with the known hydrodynamics of long flat ellipsoids, and successfully captures the swimming behavior of artificial microscopic swimmers recently explored experimentally. We also study the asymptotic behavior of a ribbon bent into a helix, that of a twisted ellipsoid, and we investigate how accurately the hydrodynamics of a ribbon can be effectively captured by that of a slender filament. Our asymptotic results provide the fundamental framework necessary to predict the behavior of slender ribbons at low Reynolds numbers in a variety of biological and engineering problems.a)

Microscale flow dynamics of ribbons and sheets

“…The mesh used for the RiBEM calculations, generated using simple custom routines, is shown in figure 4. This mesh is closer to the experimental system than those used in previous studies 21,33 , since it does not exhibit the curved edges, which are not present in the experiment, found from the ribbon parameterisation based upon Eqs. (11) and (12).…”

“…The rotational coefficient R c is overestimated significantly by all three models. Koens and Lauga 21 hypothesised that the curved edge of the SRT/Keaveny parameterisation might be responsible for this discrepancy. However, since this approximation is not present in our RiBEM model, the source of this discrepancy remains unclear.…”

“…24 These ABFs comprise a magnetic head rigidly attached to a helical ribbon tail, and swim when driven by an external rotating magnetic field. The swimming behaviour has in turn been used to experimentally determine the resistance coefficients of such a swimmer, 24 which have also been analysed theoretically using SRT 21 and BEM. 33 The flexibility of RiBEM allows us generate a realistic computational mesh of the experimental system (Fig.…”

“…[17][18][19][20] Recently, slender-ribbon theory (SRT) was developed to accurately explore the microscale hydrodynamics of slender ribbons. 21,22 This method expanded on the work of Johnson 16 and so captures the non-local interactions of the shape and is accurate to order w/ . The slenderness condition in SRT is contingent on the assumption that all three internal length scales of the ribbon, the length, , width, w, and thickness, h, are separated w h (Fig.…”

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

“…In the limit of very large aspect ratios, a prolate ellipsoid becomes a long slender body with a straight centreline. For such slender shapes, numerical implementations of the BI integrals tend to require a high resolution to resolve both length scales of the body, and so many singularity representation methods, called slender-body theories (SBTs), have been developed to overcome this difficulty (Cox 1970;Batchelor 1970;Clarke 1972;Lighthill 1976;Keller & Rubinow 1976;Johnson 1979;Sellier 1999;Götz 2000;Koens & Lauga 2016, 2017. Early SBTs used a line of stokeslets to represent the rigid body motion of the object, and typically expanded the system in orders of 1/ ln(r f / ) where 2 is the total length of the slender body and r f its maximum radius (Cox 1970;Batchelor 1970).…”