scholarly journals Remarks on Regularized Stokeslets in Slender Body Theory

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 283
Author(s):  
Laurel Ohm
Keyword(s):  
Stokes Flow ◽  
Regularization Parameter ◽  
Slender Body ◽  
Slender Body Theory ◽  
Classical Counterpart ◽  
Regularized Stokeslets ◽  
The Difference ◽  
Body Theory ◽  
Force Data ◽  
Well Posed

We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius ϵ>0. Denoting the regularization parameter by δ, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive L∞ bounds for the difference between regularized SBT and its classical counterpart in terms of δ, ϵ, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to log(δ/ϵ); in particular, δ=ϵ is necessary to avoid an O(1) discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to log(1+δ2/ϵ2), and any choice of δ∝ϵ will result in an O(1) discrepancy as ϵ→0. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed slender body PDE which classical SBT is known to approximate. Numerics verify this O(1) discrepancy but also indicate that the difference may have little impact in practice.

scholarly journals Regularized Stokeslets Lines Suitable for Slender Bodies in Viscous Flow

Fluids ◽  
2021 ◽  
Vol 6 (9) ◽  
pp. 335
Author(s):  
Boan Zhao ◽  
Lyndon Koens
Keyword(s):  
Regularization Parameter ◽  
Near Field ◽  
The Body ◽  
Slender Body ◽  
Slender Body Theory ◽  
Slip Boundary ◽  
Slender Bodies ◽  
Angular Dependency ◽  
Regularized Stokeslets ◽  
Body Theory

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.

scholarly journals Metachronal Swimming with Rigid Arms near Boundaries

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 24 ◽  
Author(s):  
Rintaro Hayashi ◽  
Daisuke Takagi
Keyword(s):  
Mathematical Model ◽  
Stokes Flow ◽  
Experimental System ◽  
Phase Delay ◽  
Slender Body ◽  
Inertial Effects ◽  
Slender Body Theory ◽  
Measured Displacement ◽  
Body Theory ◽  
Over Time

Various organisms such as crustaceans use their appendages for locomotion. If they are close to a confining boundary then viscous as opposed to inertial effects can play a central role in governing the dynamics. To study the minimal ingredients needed for swimming without inertia, we built an experimental system featuring a robot equipped with a pair of rigid slender arms with negligible inertia. Our results show that directing the arms to oscillate about the same time-averaged orientation produces no net displacement of the robot each cycle, regardless of any phase delay between the oscillating arms. The robot is able to swim if the arms oscillate asynchronously around distinct orientations. The measured displacement over time matches well with a mathematical model based on slender-body theory for Stokes flow. Near a confining boundary, the robot with no net displacement every cycle showed similar behavior, while the swimming robot increased in speed closer to the boundary.

A note on the S-transform and slender body theory in Stokes flow

2010 ◽  
Vol 75 (3) ◽  
pp. 343-355 ◽  
Author(s):  
J. R. Blake ◽  
E. O. Tuck ◽  
P. W. Wakeley
Keyword(s):  
Stokes Flow ◽  
Slender Body ◽  
Slender Body Theory ◽  
S Transform ◽  
Body Theory

A Wave Resistance Formulation for Slender Bodies With Fine Bow Shapes

2010 ◽  
Author(s):  
Brandon M. Taravella ◽  
William S. Vorus ◽  
Daniel R. Givan
Keyword(s):  
Near Field ◽  
Surface Profile ◽  
Wave Resistance ◽  
Slender Body ◽  
Slender Body Theory ◽  
Rates Of Change ◽  
Order Of Magnitude ◽  
The Difference ◽  
Flow Variables ◽  
Body Theory

Ogilvie (1972) investigated a Green’s function method for calculating the wave profile of slender ships with fine bows. He recognized that near a slender ship’s bow, rates of change of flow variables axially should be greater than those typically assumed in slender body theory. Ogilvie’s (1972) result is still a slender body theory in that the rates of change in the near field are different transversely (a half-order) than axially; however, the difference in order of magnitude between them is less than in the usual slender body theory. Typical of slender body theory, this formulation results in a downstream stepping solution (along the ship’s length) in which downstream effects are not reflected upstream. Ogilvie (1972), however, only developed a solution for wedge-shaped bodies. In this paper, a general solution for arbitrary slender ships has been developed based on Ogilvie’s (1972) formulation. The results for wave resistance have been calculated and are compared to previously published model test data and calculated results. The free surface profile has also been calculated.

Stokes flow about a sphere attached to a slender body

1974 ◽  
Vol 64 (4) ◽  
pp. 817-826 ◽  
Author(s):  
N. J. De Mestre ◽  
D. F. Katz
Keyword(s):  
Stokes Flow ◽  
Unit Length ◽  
Interactive Effect ◽  
Slender Body ◽  
Slip Condition ◽  
Image Point ◽  
Image System ◽  
Slender Body Theory ◽  
Spherical Head ◽  
Body Theory

Stokes flow is analysed for a combination body, consisting of a sphere attached to a slender body, translating along its axis in an infinite and otherwise un-disturbed fluid. The cross-section of the after-body, or tail, is circular; the radius, while not necessarily constant, is small compared with the radius of the spherical head. The tail is represented by a distribution of Stokeslets of strength per unit length F(z), located and directed along its axis. The interactive effect of head-tail attachment is manifested by the presence of image singularities located within the sphere. The image system for a single tail Stokeslet must be such that the no-slip condition is satisfied on the surface of the sphere. It is shown that this system consists of a Stokeslet, a Stokes doublet (stresslet only) and a source doublet located a t the image point. The strength F(z) is obtained by applying the no-slip condition to the combination body. The solution follows the lines of traditional slender-body theory, an expansion being performed in ascending powers of the reciprocal of the logarithm of the aspect ratio. The integral force parameters and F(z) are obtained to second order. The interactive effect is assessed, and the results are discussed in the context of a sedimenting micro-organism, such as a spermatozoon. The drag on the combination body is shown to be less by around 10% than the sum of the drags on an isolated sphere and tail. This drag, for a sperm-shaped body, is divided approximately equally between head and tail.

Note on the swimming of slender fish

1960 ◽  
Vol 9 (2) ◽  
pp. 305-317 ◽  
Author(s):  
M. J. Lighthill
Keyword(s):  
Swimming Speed ◽  
Critical Value ◽  
Slender Body ◽  
Vortex Wake ◽  
Frictional Drag ◽  
Slender Body Theory ◽  
Propulsive Efficiency ◽  
Deformable Bodies ◽  
Work Done ◽  
Body Theory

The paper seeks to determine what transverse oscillatory movements a slender fish can make which will give it a high Froude propulsive efficiency, $\frac{\hbox{(forward velocity)} \times \hbox{(thrust available to overcome frictional drag)}} {\hbox {(work done to produce both thrust and vortex wake)}}.$ The recommended procedure is for the fish to pass a wave down its body at a speed of around $\frac {5} {4}$ of the desired swimming speed, the amplitude increasing from zero over the front portion to a maximum at the tail, whose span should exceed a certain critical value, and the waveform including both a positive and a negative phase so that angular recoil is minimized. The Appendix gives a review of slender-body theory for deformable bodies.

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