Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows

Abstract

The present study furthcr explores the fundamental singular solutions for Stokes flow that can be useful £or constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associatedwith a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, strcsslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.These fundamental singularities are employed here to construct exact solutions to a number of cxterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary ilow. The different primary flows considered here include the uniform strcam, shear flows, parabolic profiles and extensional flows (hyperbolic profiles), while the body shapcs cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for tlhe construction of a solution in each specific case, their distribution densities and the rangc of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, arc discussed.

Jet propulsion without inertia

“…We consider the same prolate spheroid as in the previous example, and compute the swimming velocity U = Ux when two jets (with fluxes ±q) are placed at x = ±bx. The resistance problem has a tractable solution and representation using the singularity methods described by Chwang and Wu [46]. In that work it was shown that a prolate spheroid translating along its minor axis with velocityŨ =Ũx generates flow and pressure fields given by…”

Unlocking the secrets of multi-flagellated propulsion: drawing insights fromTritrichomonas foetus

“…The cell body was modelled as a prolate ellipsoid, with a major axis (a), minor axis (b) and eccentricity (e), where e ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 À ðb 2 /a 2 Þ: p Using this approximation, the viscous drag and torque on the cell body can be calculated from the following equations [10,78]: and m is the viscosity of water. Equations (4.1a,b) govern the viscous drag of the body, whereas equation (4.1c) shows the relationship between the torque and rotation of a prolate ellipsoid around its minor axis.…”

“…Equations (4.1a,b) govern the viscous drag of the body, whereas equation (4.1c) shows the relationship between the torque and rotation of a prolate ellipsoid around its minor axis. Equations (4.1d,e) are the linear drag coefficients of a prolate ellipsoid, and equation (4.1f ) is the drag coefficient of rotation of a prolate ellipsoid around its minor axis [78]. The viscous drag (F G ) and net moment (M G ) on the body must be balanced by the fluid forces of the flagella (F f and M Gf ), as shown below: Considering that T. foetus has four flagella, the fluid forces and net moment generated from the flagella can be expanded to the To accurately derive the flagellar forces in the laboratory frame, the absolute velocity (laboratory frame of reference) was calculated from the following equation:…”

Propulsion Efficiency of a Dynamic Self-Assembled Helical Ribbon

“…They generate eddies, thereby efficiently producing inertial thrust. Micron sized small swimmers lack these possibilities and must use nonreciprocal less efficient conformational dynamics [3][4][5][6]. Swimming strategies of low-Reynoldsnumber swimmers vary.…”