Regularized Stokeslets Lines Suitable for Slender Bodies in Viscous Flow

Abstract

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 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 and thereby removing 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 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. 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.

Slender body theories for rotating filaments

“…(inner) f (s) is defined in (34). The velocity (37) has a form similar to that of SBT, but with different bounds on the integral. Unlike SBT, the integal (37) is not a finite part integral, but a nearly singular integral that makes sense for each term separately.…”

“…To compare with SBT, we observe that the integrand in (37) is O(ǫ) when R ≤ 2â, and so we can add the excluded part back in without changing the asymptotic accuracy. This gives a velocity of the exact same form as SBT,…”

“…Later extensions, including the force coupling method [22], method of regularized Stokeslets [7,8], and Rotne-Prager-Yamakawa (RPY) tensor [33,35], focused on choosing the regularized delta function such that the Stokes equations are solvable analytically. The accuracy of using regularized singularities to approximate slender translating filaments has been studied numerically [5] and analytically [9,24,29,37]. The conclusion of these studies is that regularization methods can effectively match slender body theory in the limit ǫ → 0, with a judicious choice of regularization radius â.…”

Slender body theories for rotating filaments

“…Recently, the accuracy of using regularized singularities to approximate slender translating filaments has come under scrutiny both from a numerical (Bringley & Peskin 2008) and analytical (Cortez & Nicholas 2012; Maxian, Mogilner & Donev 2021; Ohm 2021; Zhao & Koens 2021) perspective. Generally speaking, the recent analysis has affirmed (Cortez & Nicholas 2012; Maxian et al.…”

“…Generally speaking, the recent analysis has affirmed (Cortez & Nicholas 2012; Maxian et al. 2021; Ohm 2021) that regularization methods can effectively match the SBT centreline and far-field fluid velocity with a judicious choice of regularization radius , while others (Ohm 2021) have shown that the global flow field near the slender body cannot be matched unless very specific conditions on the regularization function are met (Zhao & Koens 2021, § 4). While the latter restriction is somewhat disappointing, it has little effect in practice: since regularized singularities can match the SBT filament centreline velocity, the only impact of the flow near the filament being incorrect is in computing disturbance flows on other nearly touching filaments.…”