6/1/2023 0 Comments Slender bodyIf this condition is not satisfied, the no-slip boundary condition on the surface of the body cannot be satisfied. Although the range over which these equations apply can change from blob to blob, these equations must be true when ρ = ρ 0. These conditions can be viewed as an integral transform into ρ, which depends on the tails of the blobs. These conditions, therefore, specify the types of regularizations that can be used in numerical simulations to asymptotically capture the flow near a slender boy. These conditions are necessary and sufficient for the line regularized Stokeslets to agree with classical SBT to O(a) near the boundary as the integrals of E i represent the difference between the two representations. The above conditions provide a set of restrictions on the types of blobs, f ( v ), that can replicate the near field flow around a slender body. Where ζ = ξ / ρ = ( t − s ) / ( a ρ ), and the equality needs to hold for ρ ≳ ρ 0. The former is often called resistive force theory and provides useful analytical estimates, while the latter has been shown to provide highly accurate results in numerous situations. Broadly, SBTs have been divided into two types: those that expand the system in powers of 1 / ln ( a ) and those that expand the system in powers of a. These theories seek to expand the governing equations in terms of the large aspect ratio 1 / a to produce simplified models for the hydrodynamics. Hence, asymptotic methods, called slender-body theories (SBTs), have been developed to overcome these issues. This large aspect ratio causes boundary element methods to typically require high surface resolutions. However, this slender-body hydrodynamics can be hard to simulate thanks to the very large aspect ratio 1 / a (major/minor lengths) of the body. The slow viscous hydrodynamics of long thin bodies is important in many systems. This singularity representation diverges on the centreline of the body, and it can become difficult to implement numerically. Many of these theories place the singularity solutions of Stokes equations along the centreline of the body. Algebraically accurate SBTs have proven to be very accurate and provide exact results for prolate spheroids, while logarithmically accurate SBTs are useful for analytical estimations. These theories are called slender-body theories (SBTs), and they expand the system in powers of a or 1 / ln a. In the small a limit, asymptotic theories have been developed for the flow around slender bodies. Let a denote the ratio of the radius of a slender body to its length. These bodies typically have arc length much larger than their radius and can be difficult to resolve numerically. Slender bodies immersed in fluids are frequently studied in biology, polymer mechanics and colloids. This problem can be overcome with compactly supported blobs, and we construct one such example blob, which can be effectively used to simulate the flow around a slender body. This inability to replicate the local behaviour indicates that many regularizations cannot satisfy the no-slip boundary conditions on the body’s surface to leading order, with one of the most commonly used blobs showing an angular dependency of velocity along any cross section. ![]() However, more stringent requirements must be placed on the regularization blob to capture the near field flow outside a slender body. This is consistent with what is assumed in numerical simulations and provides a choice for the proportionality constant. 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 achieved by comparing the asymptotic behaviour of the flow from the line of regularized Stokeslets with the results from slender-body theory. ![]() In this paper, we investigate if a line of regularized Stokeslets can describe the flow around a slender body. However, it is unclear how best to regularize the singularities to minimize errors. This regularization blurs the force over a small blob and thereby removing divergent behaviour. Hence, people have regularized these singularities to overcome this issue. These singularities can be difficult to implement numerically because they diverge at their origin. These approximations typically use a line of singularity solutions to represent flow. Slender-body approximations have been successfully used to explain many phenomena in low-Reynolds number fluid mechanics.
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