Variational properties of steady fall in Stokes flow

1972 ◽  
Vol 52 (2) ◽  
pp. 321-344 ◽  
Author(s):  
H. F. Weinberger
Keyword(s):  
Stokes Flow ◽  
Variational Principles ◽  
Vertical Axis ◽  
The Body ◽  
Centre Of Mass ◽  
Slender Body Theory ◽  
First Approximation ◽  
Slender Bodies ◽  
Settling Speed ◽  
Body Theory

It is shown that for a given body and a given orientationgthere is always a position of the centre of mass which produces a stable falling motion in a very viscous fluid withgvertical and, in general, with a spin about the vertical axis. The corresponding terminal settling speed is bounded by means of several variational principles.Relations between the terminal speeds for falls with different downward directions and between the terminal speed and the geometry of the body are deduced. In particular, it is proved that for a large class of slender bodies the first approximation to the drag obtained from the slender-body theory of Burgers (1938) is correct. It follows that the ratio of the terminal speeds for falls with the long axis vertical and horizontal is near two.

Calculation of Subsonic Normal Force and Centre of Pressure Position of Bodies of Revolution Using a Slender Body Theory – Boundary Layer Method

1980 ◽  
Vol 31 (1) ◽  
pp. 1-25
Author(s):  
K.D. Thomson
Keyword(s):  
Boundary Layer ◽  
Velocity Distribution ◽  
Normal Force ◽  
The Body ◽  
Slender Body ◽  
Centre Of Pressure ◽  
Slender Body Theory ◽  
Bodies Of Revolution ◽  
Slender Bodies ◽  
Body Theory

SummaryThe aim of this paper is to present a method for predicting the aerodynamic characteristics of slender bodies of revolution at small incidence, under flow conditions such that the boundary layer is turbulent. Firstly a panel method based on slender body theory is developed and used to calculate the surface velocity distribution on the body at zero incidence. Secondly this velocity distribution is used in conjunction with an existing boundary layer estimation method to calculate the growth of boundary layer displacement thickness which is added to the body to produce the effective aerodynamic profile. Finally, recourse is again made to slender body theory to calculate the normal force curve slope and centre of pressure position of the effective aerodynamic profile. Comparisons made between predictions and experiment for a number of slender bodies extending from highly boattailed configurations to ogive-cylinders, and covering a large range of boundary layer growth rates, indicate that the method is useful for missile design purposes.

A Slender-Body Theory in Acoustics

1995 ◽  
Author(s):  
Michel Tran Van Nhieu
Keyword(s):  
Sound Field ◽  
Practical Method ◽  
The Body ◽  
Slender Body ◽  
Slender Body Theory ◽  
New Approach ◽  
Singular Perturbation Methods ◽  
Slender Bodies ◽  
Computation Load ◽  
Body Theory

Abstract A new approach has been proposed to calculate the sound pressure field radiated or scattered from slender bodies. The theoretical analysis is based upon mathematical singular perturbation methods leading to the so-called slender-body approximation. The theory provides a practical method to compute the sound field as it allows a great simplification in the geometrical representation of the body surface and a reduced computation load. This paper summarizes the main results obtained with this approximation at the present state of investigation.

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.

Slender-body theory for slow viscous flow

1976 ◽  
Vol 75 (4) ◽  
pp. 705-714 ◽  
Author(s):  
Joseph B. Keller ◽  
Sol I. Rubinow
Keyword(s):  
Integral Equation ◽  
Incompressible Fluid ◽  
Viscous Incompressible Fluid ◽  
Unit Length ◽  
The Body ◽  
Slender Body ◽  
Slow Flow ◽  
Slender Body Theory ◽  
Slow Viscous Flow ◽  
Body Theory

Slow flow of a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions. The main result is an integral equation for the force per unit length exerted on the body by the fluid. The novelty is that the body is permitted to twist and dilate in addition to undergoing the translating, bending and stretching, which have been considered by others. The method of derivation is relatively simple, and the resulting integral equation does not involve the limiting processes which occur in the previous work.

Motion of slender bodies in unsteady Stokes flow

2011 ◽  
Vol 688 ◽  
pp. 66-87 ◽  
Author(s):  
Efrath Barta
Keyword(s):  
Stokes Flow ◽  
Slenderness Ratio ◽  
Creeping Flow ◽  
Slender Body Theory ◽  
Wide Range ◽  
Unsteady Stokes Flow ◽  
Slender Bodies ◽  
Set Up ◽  
Micro Organisms ◽  
Parameter Values

AbstractThe flow regime in the vicinity of oscillatory slender bodies, either an isolated one or a row of many bodies, immersed in viscous fluid (i.e. under creeping flow conditions) is studied. Applying the slender-body theory by distributing proper singularities on the bodies’ major axes yields reasonably accurate and easily computed solutions. The effect of the oscillations is revealed by comparisons with known Stokes flow solutions and is found to be most significant for motion along the normal direction. Streamline patterns associated with motion of a single body are characterized by formation and evolution of eddies. The motion of adjacent bodies results, with a reduction or an increase of the drag force exerted by each body depending on the direction of motion and the specific geometrical set-up. This dependence is demonstrated by parametric results for frequency of oscillations, number of bodies, their slenderness ratio and the spacing between them. Our method, being valid for a wide range of parameter values and for densely packed arrays of rods, enables simulation of realistic flapping of bristled wings of some tiny insects and of locomotion of flagella and ciliated micro-organisms, and might serve as an efficient tool in the design of minuscule vehicles. Its potency is demonstrated by a solution for the flapping of thrips.

scholarly journals Dropping slender-body theory into the mud

2019 ◽  
Vol 862 ◽  
pp. 1-4
Author(s):  
S. E. Spagnolie
Keyword(s):  
Fluid Flow ◽  
Aspect Ratio ◽  
Slender Body ◽  
Viscoplastic Fluid ◽  
Slender Body Theory ◽  
Viscous Fluid Flow ◽  
Physical Systems ◽  
Surrounding Environment ◽  
Slender Bodies ◽  
Body Theory

The equations describing classical viscous fluid flow are notoriously challenging to solve, even approximately, when the flow is host to one or many immersed bodies. When an immersed body is slender, the smallness of its aspect ratio can sometimes be used as a basis for a ‘slender-body theory’ describing its interaction with the surrounding environment. If the fluid is complex, however, such theories are generally invalid and efforts to understand the dynamics of immersed bodies are almost entirely numerical in nature. In a valiant effort, Hewitt & Balmforth (J. Fluid Mech., vol. 856, 2018, pp. 870–897) have unearthed a theory to describe the motion of slender bodies in a viscoplastic fluid, ‘fluids’ such as mud or toothpaste which can be coaxed to flow, but only with a sufficiently large amount of forcing. Mathematical theories for some tremendously complicated physical systems may now be within reach.

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.

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.

Line source distributions and slender-body theory

1963 ◽  
Vol 17 (2) ◽  
pp. 285-304 ◽  
Author(s):  
John P. Moran
Keyword(s):  
Potential Flow ◽  
Line Source ◽  
The Body ◽  
Successive Approximations ◽  
External Disturbances ◽  
Slender Body Theory ◽  
Bodies Of Revolution ◽  
Plane Flow ◽  
Body Theory

A systematic procedure is presented for the determination of uniformly valid successive approximations to the axisymmetric incompressible potential flow about elongated bodies of revolution meeting certain shape requirements. The presence of external disturbances moving with respect to the body under study is admitted. The accuracy of the procedure and its extension beyond the scope of the present study—e.g. to problems in plane flow - are discussed.

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