Analytical solutions to slender-ribbon theory

Koens, Lyndon; Lauga, Eric

doi:10.1103/physrevfluids.2.084101

Cited by 32 publications

(6 citation statements)

References 41 publications

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“…The total drift displacement in the direction, given in (3.50), depends on the time required for the filament to reorientate parallel to the gravitational field, which depends on , and . This result is similar to the sedimentation of a ribbon torus (Koens & Lauga 2017), which also reorientates in a direction parallel to the gravitational field. For the case of the ribbon, the total drift in the direction depends on the angle between the plane of the ribbon and its centreline normal vector, whereas here the total drift of the filament depends on the imposed viscosity gradient .…”

Section: Straight Filament In a Linear Viscosity Gradientmentioning

confidence: 80%

Resistive-force theory of slender bodies in viscosity gradients

Kamal

Lauga

2023

J. Fluid Mech.

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In many natural settings, spatial variations in a fluid's viscosity arise due to changes in its local physico-chemical environment. We consider the low-Reynolds-number dynamics of slender bodies in fluids with a linearly varying viscosity. Assuming the spatial change in viscosity to be small compared to the spatial deviation of the body but large relative to the body's aspect ratio, we derive the modifications to resistive-force theory in a fluid due to a constant viscosity gradient. At leading order in the body slenderness, the results are identical to the classical theory in a constant-viscosity fluid but with the viscosity taking everywhere its local, non-constant value. At next order in slenderness, non-local terms arise due to the non-zero viscosity gradient. We use our results to predict the motion of straight and toroidal filaments settling under the action of gravity. We show that viscosity gradients induce rigid-body rotation of the filaments at a rate proportional to the components of the gradients along the filaments. This result contrasts with constant-viscosity fluids where the filaments do not rotate. We demonstrate further that if the viscosity gradient acts in the direction opposite to the gravitational field, then the filaments rotate towards a stable orientation, whose value depends on the ratio between the viscosity gradients parallel and perpendicular to the gravitational field; otherwise, the filaments align in the direction of the gravitational field. Our work shows that viscosity gradients can exert new forces on slender bodies, which could, for example, be used to control their orientation and drift.

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Section: Straight Filament In a Linear Viscosity Gradientmentioning

confidence: 80%

Resistive-force theory of slender bodies in viscosity gradients

Kamal

Lauga

2023

J. Fluid Mech.

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“…(3.18) for different i and j up to O( 2 ). substitution χ = γ sinh(φ) and then expanded in to find the leading-order contribution (Koens & Lauga 2017). A list of the expansions relevant to this derivation is found in Table 1.…”

Section: Totalmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

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

2018

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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 & 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. † Email: e.lauga@damtp.cam.ac.uk arXiv:1806.04192v1 [physics.flu-dyn] 11 Jun 2018 † The integration factor is obtained by recognising that ∂sS(s, θ) =t(s) + ∂s[ρ(s)êρ(s, θ)], ∂ θ S(s, θ) = ρ(s)∂ θêρ (s, θ), andt × ∂ θêρ = −êρ and performing a Taylor expansion in .

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“…Since the cylinder has a large aspect ratio, L a, we can exploit the framework of slender-body theory (SBT) [17][18][19][20][21][22][23][24][25] to find the leading-order value (in a/d) of the reaction flow as a flow due to a distribution of point forces f set along the axis of the cylinder, as…”

Section: Setup and General Frameworkmentioning

confidence: 99%

“…The literature on the response of slender filaments to external flows is substantial. The majority of the past theoretical results relies on the well-known framework of slender-body theory (SBT) [17][18][19][20][21][22][23][24][25]. Built on the idea of approximating the flow around a slender filament by a distribution of hydrodynamic singularities along its centreline (primarily point forces and source dipoles), this framework can be thought of as an asymptotic approximation of the boundary integral formulation of the Stokes flow [26,27].…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamic interactions between a point force and a slender filament

Tanasijević¹,

Lauga²

2021

Preprint

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The Green's function of the incompressible Stokes equations, the stokeslet, represents the singular flow due to a point force. Its classical value in an unbounded fluid has been extended near surfaces of various shapes, including flat walls and spheres, and in most cases the presence of a surface leads to an advection flow induced at the location of the point force. In this paper, motivated by the biological transport of cargo along polymeric filaments inside eukaryotic cells, we investigate the reaction flow at the location of the point force due to a rigid slender filament located at a separation distance intermediate between the filament radius and its length (i.e. we compute the advection of the point force induced by the presence of the filament). An asymptotic analysis of the problem reveals that the leading-order approximation for the force distribution along the axis of the filament takes a form analogous to resistive-force theory but with drag coefficients that depend logarithmically on the distance between the point force and the filament. A comparison of our theoretical prediction with boundary element computations show good agreement. We finally briefly extend the model to the case of curved filaments.

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Analytical solutions to slender-ribbon theory

Cited by 32 publications

References 41 publications

Resistive-force theory of slender bodies in viscosity gradients

Resistive-force theory of slender bodies in viscosity gradients

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

Hydrodynamic interactions between a point force and a slender filament

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