Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe boundary integral formulation of Stokes flows includes slender-body theory
238
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 ribbons 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.
Hydrodynamic interactions (HIs) are important in biophysics research because they influence both the collective and the individual behaviour of microorganisms and self-propelled particles. For instance, HIs at the micro-swimmer level determine the attraction or repulsion between individuals, and hence their collective behaviour. Meanwhile, HIs between swimming appendages (e.g. cilia and flagella) influence the emergence of swimming gaits, synchronised bundles and metachronal waves. In this study, we address the issue of HIs between slender filaments separated by a distance larger than their contour length (d>L) by means of asymptotic calculations and numerical simulations. We first derive analytical expressions for the extended resistance matrix of two arbitrarily-shaped rigid filaments as a series expansion in inverse powers of d/L>1. The coefficients in our asymptotic series expansion are then evaluated using two well-established methods for slender filaments, resistive-force theory (RFT) and slender-body theory (SBT), and our asymptotic theory is verified using numerical simulations based on SBT for the case of two parallel helices. The theory captures the qualitative features of the interactions in the regime d/L>1, which opens the path to a deeper physical understanding of hydrodynamically governed phenomena such as inter-filament synchronisation and multiflagellar propulsion. To demonstrate the usefulness of our results, we next apply our theory to the case of two helices rotating side-by-side, where we quantify the dependence of all forces and torques on the distance and phase difference between them. Using our understanding of pairwise HIs, we then provide physical intuition for the case of a circular array of rotating helices. Our theoretical results will be useful for the study of HIs between bacterial flagella, nodal cilia, and slender microswimmers.
The Stokes equation describes the motion of fluids when inertial forces are negligible compared to viscous forces. In this article, we explore the consequence of parity-violating and non-dissipative (i.e. odd) viscosities on Stokes flows. Parity violating viscosities are coefficients of the viscosity tensor that are not invariant under mirror reflections of space, while odd viscosities are those which do not contribute to dissipation of mechanical energy. These viscosities can occur in systems ranging from synthetic and biological active fluids to magnetised and rotating fluids, in which parity (i.e. mirror reflection) symmetry is broken. We first systematically enumerate all possible parity-violating viscosities compatible with cylindrical symmetry, highlighting their connection to potential microscopic realizations. Then, using a combination of analytical and numerical methods, we analyze the effects of the parity violating viscosities on the Stokeslet solution, flow past a sphere or a bubble, and many-particle sedimentation. In all the cases we analyze parity violating viscosities give rise to an azimuthal flow even when the driving force is along the axis of cylindrical symmetry. For a few sedimenting particles, the azimuthal flow bends the trajectories compared to a traditional Stokes flow. For a cloud of particles, the azimuthal flow impedes the transformation into a torus and the subsequent breakup into smaller parts that would otherwise occur. The presence of azimuthal flows in cylindrically symmetric systems (sphere, bubble, cloud of particles) can serve as a probe for parity-violating viscosities in experimental systems.
We consider the creeping flow of a Newtonian fluid in a hemispherical region. In a domain with spherical, or nearly spherical, geometry, the solution of Stokes equation can be expressed as a series of spherical harmonics. However, the original Lamb solution is not complete when the flow is restricted to a semi-infinite space. The general solution in hemispherical geometry is then constructed explicitly. As an application, we discuss the solutions of Marangoni flows due to a local source at the liquid-air interface.
Slender-body approximations have been successfully used to explain many phenomena in low-Reynolds number fluid mechanics. These approximations typically use a line of singularity solutions to represent the flow. These singularities can be difficult to implement numerically because they diverge at their origin. Hence people have regularized these singularities to overcome this issue. This regularization blurs the force over a small blob therefore removing the divergent behaviour. However it is unclear how best to regularize the singularities to minimize errors. In this paper we investigate if a line of regularized Stokeslets can describe the flow around a slender body. This is achieved by comparing the asymptotic behaviour of the flow from the line of regularized Stokeslets with the results from slender-body theory. We find that the flow far from the body can be captured if the regularization parameter is proportional to the radius of the slender body. This is consistent with what is assumed in numerical simulations and provides a choice for the proportionality constant. However more stringent requirements must be placed on the regularization blob to capture the near field flow outside a slender body. This inability to replicate the local behaviour indicates that many regularizations cannot satisfy the non-slip boundary conditions on the bodies surface to leading order, with one of the most commonly used regularizations showing an angular dependency of velocity along any cross section. This problem can be overcome with compactly supported blobs { and we construct one such example blob which could be effectively used to simulate the flow around a slender body
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
