Slender Body Interactions for Low Reynolds Numbers—Part  II:  Body-Body Interactions

Barta, E; Liron, N.

doi:10.1137/0148077

Cited by 12 publications

(15 citation statements)

References 12 publications

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“…A slip condition may be an appropriate boundary condition on the 'surface' of the microvilli. In the case of muco-ciliary transport in the liquid lining of the lung, the inclusion of porous medium effects to simulate epithelial permeability may assist with bringing theoretical modelling into closer correspondence with biological understanding (see, for example Matsui et al, 1998Matsui et al, , 2000Smith et al, 2007). r Closely approaching cilia: Bartha and Liron (1988b) used the asymptotic techniques of Johnson (1980) to consider the interaction of two slender bodies either in an unbounded fluid, or near a wall. They found asymptotically accurate second-kind integral equations for the force distributions and drag on each body, subject to their separation being of the order of the cilium length.…”

Section: Future Developmentsmentioning

confidence: 99%

Discrete Cilia Modelling with Singularity Distributions: Application to the Embryonic Node and the Airway Surface Liquid

2007

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We discuss in detail techniques for modelling flows due to finite and infinite arrays of beating cilia. An efficient technique, based on concepts from previous 'singularity models' is described, that is accurate in both near and far-fields. Cilia are modelled as curved slender ellipsoidal bodies by distributing Stokeslet and potential source dipole singularities along their centrelines, leading to an integral equation that can be solved using a simple and efficient discretisation. The computed velocity on the cilium surface is found to compare favourably with the boundary condition. We then present results for two topics of current interest in biology. 1) We present the first theoretical results showing the mechanism by which rotating embryonic nodal cilia produce a leftward flow by a 'posterior tilt,' and track particle motion in an array of three simulated nodal cilia. We find that, contrary to recent suggestions, there is no continuous layer of negative fluid transport close to the ciliated boundary. The mean leftward particle transport is found to be just over 1 mum/s, within experimentally measured ranges. We also discuss the accuracy of models that represent the action of cilia by steady rotlet arrays, in particular, confirming the importance of image systems in the boundary in establishing the far-field fluid transport. Future modelling may lead to understanding of the mechanisms by which morphogen gradients or mechanosensing cilia convert a directional flow to asymmetric gene expression. 2) We develop a more complex and detailed model of flow patterns in the periciliary layer of the airway surface liquid. Our results confirm that shear flow of the mucous layer drives a significant volume of periciliary liquid in the direction of mucus transport even during the recovery stroke of the cilia. Finally, we discuss the advantages and disadvantages of the singularity technique and outline future theoretical and experimental developments required to apply this technique to various other biological problems, particularly in the reproductive system.

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Section: Future Developmentsmentioning

confidence: 99%

Discrete Cilia Modelling with Singularity Distributions: Application to the Embryonic Node and the Airway Surface Liquid

2007

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“…long computation time and problems of convergence. While we do not need the capability to deal with centreline curvature, we adopt the Johnson (1980) and Barta & Liron (1988) axial singularity distribution model. The advantages of this approach are its rigorous use of the various types of singularities, the uniform validity of the solution over the whole surfaces of the bodies and its computational simplicity.…”

Section: Introductionmentioning

confidence: 99%

Creeping flow around a finite row of slender bodies in close proximity

Barta

Weihs

2006

J. Fluid Mech.

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The flow through and around a finite row of parallel slender bodies in close proximity moving in a viscous incompressible fluid is studied. The motion occurs under creeping flow ($\hbox{\it Re}\,{\ll}\,1$) conditions. This row is a model of a comb-wing configuration found in insects of the Thrips family and being developed for use for flying vehicles of mm size, operating in the creeping flow regime. We show here that such wings utilize viscous effects to carry along enough fluid to approximate continuous surfaces. The comb is described as a row of rod-like ellipsoids of slenderness ratio smaller than 0.01 at distances apart of order 10 times the minor axis and the flow field is computed by distributing singularities along the major axes of the ellipsoids. Results for the drag on the individual rods, as well as for the full row are presented. It is shown that above a certain number of rods, dependent on the geometric parameters of the comb, the row acts very much like a continuous surface, with over 95% of the flow moving around, and not through the comb. This allows a potential saving of tens of percents in wing weight. Parametric results for number of rods, rod density (ratio of inter-rod distance to rod length) and slenderness ratio are presented demonstrating the dependence of the flow field on the configuration. It is found that 50–80 rods are required to approach the asymptotic limit of large number of rods, for various combinations of rod parameters with inter-rod distances of order of the cross-section diameter.

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“…In §4.1, we solve explicitly problem (3), showing that, far enough from the sphere of radius L, u is a good approximation of the solution v of problem (2). Moreover, u takes at the origin the same value taken by v on the spherical surface ∂ Σ , and it turns out that, to obtain also u = U at infinity, we must take the drag force exerted by the point-like spherical particle as…”

Section: Dimensional Reduction and Hyperviscositymentioning

confidence: 99%

“…The solution to the Stokes equation (2) in the entirety of R 3 is divergent at the origin; the solution of (3) (with L > 0) is bounded and continuous in 0, making explicit the reason why the hyperviscous equation (1) is considered a regularization of the classical one, namely (1) with L = 0. It is important to notice that, in the limit L → 0, the problem (3) becomes ill-posed.…”

Section: Dimensional Reduction and Hyperviscositymentioning

confidence: 99%

“…Despite the possibility of approximating with good accuracy many quantities of physical interest, it is difficult to apply classical slender-body theory to particles of arbitrary shape and to situations featuring the presence of many particles, which are clearly important for applications. For this reason, modifications of the theory have been proposed by de Mestre [8], Johnson [16], Barta and Liron [2], and Barta [1], and also somewhat different theories have been developed by Lighthill [19,20] and Keller and Rubinow [17]. In a recent paper, Cortez and Nicholas [6] presented a new theory in which the concentrated force densities are replaced by forces which are localized in a small three-dimensional region lying within the slender body.…”

Section: Introductionmentioning

confidence: 99%

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Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization

Giusteri

Fried

2014

Meccanica

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A new slender-body theory for viscous flow, based on the concepts of dimensional reduction and hyperviscous regularization, is presented. The geometry of flat, elongated, or point-like rigid bodies immersed in a viscous fluid is approximated by lower-dimensional objects, and a hyperviscous term is added to the flow equation. The hyperviscosity is given by the product of the ordinary viscosity with the square of a length that is shown to play the role of effective thickness of any lower-dimensional object. Explicit solutions of simple problems illustrate how the proposed method is able to represent with good approximation both the velocity field and the drag forces generated by rigid motions of the immersed bodies, in analogy with classical slender-body theories. This approach has the potential to open up the way to more effective computational techniques, since the complexity of the geometry can be significantly reduced. This, however, is achieved at the expense of involving higherorder derivatives of the velocity field. Importantly, both the dimensional reduction and the hyperviscous regularization, combined with suitable numerical schemes, can be used also in situations where inertia is not negligible.

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Slender Body Interactions for Low Reynolds Numbers—Part II: Body-Body Interactions

Cited by 12 publications

References 12 publications

Discrete Cilia Modelling with Singularity Distributions: Application to the Embryonic Node and the Airway Surface Liquid

Discrete Cilia Modelling with Singularity Distributions: Application to the Embryonic Node and the Airway Surface Liquid

Creeping flow around a finite row of slender bodies in close proximity

Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization

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