me Silas Alben

Associate Professor and AIM Director
Department of Mathematics
4858 East Hall
530 Church Street
University of Michigan
Ann Arbor, MI  48109-1043

last name at

phone: 734-647-5518

Math 501: AIM Student Seminar Course

My research addresses problems from biology (especially biomechanics) and engineering which can be studied with the tools of applied mathematics and continuum mechanics. My work consists of modeling, theoretical analysis and development of numerical methods, with the general goal of obtaining new physical insight into these problems.


Improved convection cooling in steady channel flows
S. Alben,
Phys. Rev. Fluids (2), 104501--1-26, 2017
Convective cooling of the heated walls of a 2D channel is a benchmark problem with recent applications. We find steady flows that maximize the heat removed from fixed-temperature walls for a given rate of energy used to drive the flow, Pe2. These flows have a boundary layer structure, and heat transfer proportional to
Pe2/5, an improvement over the Pe1/3 scaling of Poiseuille flow.

Optimal convection cooling flows in general 2D geometries S. Alben, J. Fluid Mech. (814) 484-509, 2017 We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with a hot interior and cold exterior.

Stability and scalability of piezoelectric flags X.Wang, S. Alben, C. Li, and Y.L. Young. Phys. Fluids, (28) 023601-1-19, 2016
We investigate the effect of piezoelectric material on the flutter speed, vibration mode and frequency, and energy harvesting power and efficiency of a flexible flag. We develop a fully coupled fluid-solid-electric model by combining the inviscid vortex sheet model with a linear electro-mechanical coupling model. A consistent optimal resistance is found that maximizes the flutter speed and the energy harvesting power. For a resonant RL circuit, by tuning the inductance to match the circuit frequency to the flag’s vibration frequency, the flutter speed can be greatly decreased, and a larger averaged power and efficiency are obtained. We find that the resistance only circuit is more effective when the flag is placed in a lighter fluid (e.g., air), while the RL circuit is able to reduce the flutter speed when the flag is placed in a heavier fluid (e.g., water).

The dynamics of vortex streets in channels X.Wang and S. Alben,
Phys. Fluids, (27) 073603-1-15, 2015

We study the dynamics of regular and reverse von Karman vortex streets in channels. Reverse streets maintain their structure while regular vortex streets undergo inversion. We find a transition to asymmetry when the vortices are strong, viscosity is low, or the street is compressed horizontally or extended vertically.

Flag flutter in inviscid channel flow S. Alben, Phys. Fluids, (27) 033603, 2015
We determine when a straight flag in a channel-bounded inviscid flow is unstable to flapping motions. Channel confinement destabilizes heavier flags, but has little influence on lighter flags' stability. The analytical stability boundary for an infinite flag shows similar results. As the channel walls approach the flag, its flapping amplitude decreases roughly in proportion to the near-wall distance. Meanwhile, its dominant flapping frequency and wave number increase in a nearly stepwise fashion. I.e. at certain critical values of channel width, the flag jumps to a higher flapping mode.

Passive Vibration Control of Flexible Hydrofoils using Piezoelectric Material
C. Li, E.J. Chae, Y.L. Young, X. Wang, and S. Alben. Fourth International Symposium on Marine Propulsors, 2015.
This work explores the use of a piezoelectric material for passive vibration control of a cantilevered, flexible hydrofoil. The integrated fluid-solid-electrical response is modeled by coupling a 2-D viscous unsteady Reynolds Averaged Navier Stokes (uRANS) solver for the fluid with a two-degrees-of-freedom (2-DOF) solid model, which is coupled with the circuit equation. The results show that activation of the PZT circuit increases the vibration frequencies, as the electromechanical coupling force adds an additional contribution to the system stiffness. The changes in system stiffness and response frequency depend on the resistance. A well-tuned resistor in parallel with a PZT circuit can be used to delay flutter and control flow-induced vibrations.

Bending of bilayers with general initial shapes S. Alben,  Adv. Comput. Math., 2014 
We present a simple discrete formulation of bilayer bending and use it to compute equilibrium configurations of actuated bilayers with general initial shapes. We identify typical bending behaviors: overall bending directions along longest and shortest dimensions, inward bending at corners, curvature intensification near boundaries, and conical bending and partitioned bending zones in some cases.

Optimizing snake locomotion on an inclined plane
X.Wang, M.T. Osborne, and S. Alben,  Phys. Rev. E
, (89) 012717, 2014
We develop a model to study the locomotion of snakes on inclined planes. We determine numerically which snake motions are optimal for two retrograde traveling-wave body shapes, triangular and sinusoidal waves, across a wide range of frictional parameters and incline angles. In the regime of large transverse friction coefficients, we find power-law scalings for the optimal wave amplitudes and corresponding costs of locomotion. We give an asymptotic analysis to show that the optimal snake motions are traveling waves with amplitudes given by the same scaling laws found in the numerics.

Efficient kinematics for jet-propelled swimming 
S. Alben, L.A. Miller, and J. Peng,  J. Fluid Mech.
, (733) 100-133, 2013
We use computer simulations and an analytical model to study the relationship between kinematics and performance in jet-propelled jellyfish swimming.  In the simulations, two types of efficient kinematics are found: a bell radius which is nearly a linear function of time, and a "burst-and-coast" kinematics. The analytical model studies the contraction phase only, and finds that the efficiency-optimizing bell radius as a function of time transitions from nearly linear for small-to-moderate output power to exponentially decaying for large output power.

Optimizing snake locomotion in the plane  S. Alben, Proc. Roy. Soc. A,  469 (2156), 2013
We determine which planar snake motions are optimal for locomotory efficiency, across frictional parameter space. With large transverse friction, our numerics and analysis show that the optimal motion is a retrograde traveling wave with amplitude scaling as the -1/4 power of the transverse friction coefficient. At zero transverse friction, a triangular direct wave is optimal. Between these extremes, we find a variety of optimal motions including standing waves (or ratcheting motions).

Functional morphology of the fin rays of teleost fishes  B.E. Flammang, S. Alben, P.G.A. Madden, and G.V. Lauder, J. Morphology 2013   We examined the range of motion and curvatures of the pectoral fin rays of bluegill sunfish during steady swimming, turning maneuvers, and hovering behaviors and during a vortex perturbation impacting the fin during the fin beat. Our model showed that impacting vortices transferred little force to the fin rays due to their flexibility.  This flexibility may offer intrinsic damping of environmental fluid perturbations encountered by swimming fish.

Optimization of two- and three-link snake-like locomotion  F. Jing and S. Alben, Phys. Rev. E 87, 022711, 2013
We analyze the crawling of bodies with two and three links. The links are connected by hinge joints, and the hinge angles are actuated periodically in time. We determine the acutations which lead to the most efficient crawling motions, for different choices of coefficients of friction between the snake and the ground. High efficiency is obtained with a large backward coefficient of friction and a small transverse coefficient of friction, compared to the forward coefficient of friction. For the three-link case, the efficiency-maximizing paths are triangles in the parameter space of internal angles.

Using Computational and Mechanical Models to Study Animal Locomotion
L.A. Miller, D.I. Goldman, T.L. Hedrick, E.D. Tytell, Z.J. Wang, J. Yen, and S. Alben,
Integr. Comp. Biol. (2012) 52(5): 553-575
We review mathematical, computational, and experimental studies which compare different methods of locomotion in variable environments. A common theme of the work is applying physical principles to the understanding of living systems.

Passive Robotic Models of Propulsion by the Bodies and Caudal Fins of Fish
G.V. Lauder, B.E. Flammang, and S. Alben,
Integr. Comp. Biol. (2012) 52(5): 576-587
We discuss the use of simple robotic models of flexing fish bodies during self-propulsion. We report unexpected non-linear effects of changing the length and stiffness of the foil, and analyze the effect of changing the shape of the trailing edge on self-propelled swimming speed and kinematics. 

Interfacing Mathematics and Biology: A Discussion on Training, Research, Collaboration, and Funding
L.A. Miller and S. Alben,
Integr. Comp. Biol. (2012) 52(5):616-621
This article summarizes a workshop discussion at the 2012 Annual Meeting of the Society for Integrative and Comparative Biology on training and research at the intersection of the biological, physical, engineering, and mathematical sciences.

Dynamics of freely swimming flexible foils
S. Alben,
C. Witt, T.V. Baker, E.J. Anderson, and G.V. Lauder
Physics of Fluids 24, 051901, 2012
We use modeling and simulations, guided by initial experiments, to study thin foils which are oscillated at the leading edge and are free to move unidirectionally under the resulting fluid forces. We find resonant-like peaks in the swimming speed, which is proportional to foil length to the -1/3 power and foil rigidity to the 2/15 power.

Effects of shape and stroke parameters on the propulsion performance of an axisymmetric swimmer
J. Peng and S. Alben,
Bioinspir. Biomim. 2012
We consider a model jellyfish which swims by periodic body contractions, and sheds vorticity into the fluid at its outer edge according to the unsteady Kutta condition.  We identify kinematic parameters which lead to swimming with high speed and small "cost of locomotion."
The attraction between a flexible filament and a point vortex
S. Alben, J. Fluid Mech., 
A flexible filament is attracted to a point vortex when they move together as a coupled system. The point vortex collides with the filament at a finite time, with the separation distance scaling as the square root of time. We derive the power laws describing the collision.

Flapping propulsion using a fin ray
S. Alben, J. Fluid Mech., 2012
We calculate optimal driving motions for a fin ray, the skeletal structure of a fin. The fin ray is driven by heaving, pitching, and a less-studied motion called "shifting." We calculate the phases of shifting relative to heaving and pitching that maximize thrust power and efficiency.

Model problems for fish schooling
S. Alben, IMA Volume on Natural Locomotion, 
We review recent work on model systems for body-vortex and body-body interactions in a fluid, related to fish schooling.

Edge Effects Determine the Direction of Bilayer Bending
S. Alben, B. Balakrisnan, E. Smela, Nano Letters, April 29, 2011

We explain the reason for preferential bending along the long edge in thin rectangular bilayers. The bending direction is determined by the existence of doubly curved regions at the curled edges, which lower the elastic energy. Thus "spirals" are favored over "cigars."

44 Interactions between vortices and flexible walls
S. Alben, Int. J. Nonlinear Mech.
, 46, 586-591 (2011), Special issue 
We calculate the motion of a point vortex near a flexible wall. The vortex moves tangentially to the wall, and induces an outward "bump" on it. The force on the wall scales as the inverse cube of the distance to the point vortex, and the speed of the point vortex scales as the inverse fourth power of the same distance.

Self-similar bending in a flow: The axisymmetric case
S. Alben, Physics of Fluids
, 22, 081901 (2010).

We study how sheets roll up into conical configurations when exposed to fluid flows. We find power-law scalings for the cone angles and drag coefficients which result from a self-similar behavior of the flows at the outer edges of the cones. The similarity length scale is the radius of the vortex ring in the wake.

Flexible sheets falling in an inviscid fluid S. Alben, Physics of Fluids, 22, 061901 (2010). Click for movie (14 Mb).
We use inviscid simulations to study falling flexible sheets in the two-parameter space of sheet density and bending rigidity. The basic behavior is a repeated series of accelerations to a critical speed at which the sheet flexes, and rapidly decelerates, shedding large vortices. The maximum and average speeds of the sheet are closely related to the critical flutter speed. The sheet trajectories also show persistent circling, quasi-periodic flapping, and more complex repeated patterns.  aaa45aaa3
Coordination of multiple appendages in drag-based swimming S. Alben, K. Spears, S. Garth, D. Murphy, and J. Yen, J. R. Soc. Interface (2010).
Krill are aquatic crustaceans that engage in long distance migrations. Hence efficient locomotory performance is important for their survival. Krill locomote using kinematics that are very nearly metachronal. We study a drag coefficient model which compares metachronal,synchronous and intermediate motions for a model krill. With fixed leg velocity amplitude, and with fixed output power, metachronal kinematics give the highest average body speed for both linear and quadratic drag laws.
pollen pollen
Foldable structures and the natural design of pollen grains E. Katifori, S. Alben, E. Cerda, D.R. Nelson, and J. Dumais, PNAS, 107, 7635-7639 (2010).
pollen Upon release from the anther, pollen grains of angiosperm flowers are exposed to a dry environment and dehydrate. To survive this process, the pollen wall can fold inward to prevent further desiccation. Here we demonstrate that simple geometrical and mechanical principles explain how wall structure guides pollen grains toward distinct folding pathways.  pollen cover
Regularizing a vortex sheet near a separation point
S. Alben, Journal of Computational Physics (2010).

Current methods for computing vortex sheet separation use a regularization parameter which is discontinuous from the body to the vortex sheet. We propose two methods for reducing the errors associated with the regularization parameter: the ``velocity smoothing" method and the ``tapered smoothing" method. In a model problem and a benchmark problem, both methods are found to converge much more rapidly (with exponents 3/2 and 2 versus 1/2 for the standard method) for similar computational expense.
sheet Optimizing a fin ray for stiffness
S. Alben and R.L. McGee, Journal of the Mechanics and Physics of Solids, 58, 656-664 (2010).

We optimize the elastic constants of the supporting structure in flexible fish fins--the fin ray--to obtain the least deflection under loading. We first solve the problem numerically for rays with shear layers of uniform thickness. The optimal distributions of bending and shear moduli are nonzero on disjoint regions. The numerical solution suggests the form of the analytical solution, which we obtain using calculus of variations on two intervals with continuity conditions at the interface. The deflection of the optimal ray is less than half that of the uniform ray.
Passive and active bodies in vortex-street wakes
S. Alben,
Journal of Fluid Mechanics, 642, 95-125 (2010).

We model the swimming of a finite body in a vortex street using vortex sheets distributed along the body and in a wake emanating from its trailing edge. We consider the motion of a flexible body clamped at its leading edge in the vortex street as a model for a flag in a vortex street, and find alternating bands of thrust and drag for varying wave number. We consider a flexible body driven at its leading edge as a model for tail-fin swimming, and determine optimal motions with respect to the phase between the body's trailing edge and the vortex street.
Wake-mediated synchronization and drafting in coupled flags
S. Alben, J. Fluid Mech., 641, 489-496 (2009).

Recent experiments have shown "inverted drafting" in flags: the drag force on one flag is increased by excitation from the wake of another. Here we use vortex sheet simulations to show that inverted drafting occurs when the flag wakes add coherently to form strong vortices. By contrast, normal drafting occurs for higher-frequency oscillations, when the vortex wake becomes more complex and mixed on the scale of the flag. The types of drafting and dynamics (synchronization and erratic flapping) depend on the separation distance between the flags.

Movie of tandem flags, synchronized 
Movie of tandem flags, unsynchronized
Movie of side-by-side flags
On the swimming of a flexible body in a vortex street S. Alben, Journal of Fluid Mechanics, 635, 27-45 (2009).
We consider periodic travelling-wave swimming motions of a flexible body in a vortex street, in the limit of small amplitude. We determine the body wave which provides maximum output power for fixed amplitude and the body wave which maximizes efficiency for a given output power. We compare our results with previous experiments and simulations and give physical interpretations for agreements and disagreements in terms of the phase between the body wave and vortex street. sheet
Collapse and folding of pressurized rings in two dimensions
 E. Katifori, S. Alben, D.R. Nelson, Physical Review E, 79, 056604 (2009).
buckled Hydrostatically pressurized circular rings confined to two dimensions (or cylinders constrained to have only z-independent deformations) undergo Euler-type buckling when the outside pressure exceeds a critical value. We perform a stability analysis of rings with arclength-dependent bending moduli and determine how weakened bending modulus segments affect the buckling critical pressure. Rings with a fourfold symmetric modulation are particularly susceptible to collapse. In addition we study the initial postbuckling stages of the pressurized rings to determine possible ring folding patterns.
Simulating the dynamics of flexible bodies and vortex sheets
 S. Alben, Journal of Computational Physics, 228, 2587-2603 (2009) .
Resonances We present a numerical method for the dynamics of a flexible body in an inviscid flow with a free vortex sheet. The formulation is implicit with respect to body variables and explicit with respect to the free vortex sheet. We apply the method to a flexible foil driven periodically in a steady stream. We give numerical evidence that the method is stable and accurate for a relatively small computational cost. A continuous form of the vortex sheet regularization permits continuity of the flow across the body's trailing edge. Nonlinear behavior arises gradually with respect to driving amplitude, and is attributed to the rolling-up of the vortex sheet. Flow quantities move across the body in traveling waves, and show large gradients at the body edges. We find that in the small-amplitude regime, the phase difference between heaving and pitching which maximizes trailing edge deflection also maximizes power output; the phase difference which minimizes trailing edge deflection maximizes efficiency. Streamlines
A cascade of length scales in elastic rings under confinement
 K. Spears and S. Alben, Chaos, 18, 041109 (2008).
MylarRing Elastic objects under confinement are common in mechanics and biology. Examples include mitochondria and chromosomes, for which conformation and function are strongly determined by confining forces. When elastic objects grow in a confined space, minimization of elastic energy creates a complex spatial configuration and force network. To simulate a two-dimensional ring growing within a rigid circular boundary with a fixed radius, we take a long strip of elastic material (mylar) of fixed length, join the ends to form a closed loop, and then shrink the confining ring boundary. We find a distribution of curvatures which is inversely proportional to the local number of overlapping layers. We give a simple quantitative argument for this relationship. PaperRing

Packings of a charged line on a sphere
S. Alben, Physical Review E, 78, 066603 (2008).

We find equilibrium configurations of open and closed lines of charge on a sphere, and track them with respect to varying sphere radius. Closed lines transition from a circle to a spiral-like shape through two low-wave-number bifurcations ("baseball seam" and "twist") which minimize Coulomb energy. The spiral shape is the unique stable equilibrium of the closed line. Other unstable equilibria arise through tip-splitting events. An open line transitions smoothly from an arc of a great circle to a spiral as the sphere radius decreases. Under repulsive potentials with faster-than-Coulomb power-law decay, the spiral is tighter in initial stages of sphere shrinkage, but at later stages of shrinkage the equilibria for all repulsive potentials converge on a spiral with uniform spacing between turns. Multiple stable equilibria of the open line are observed.
Optimal flexibility of a flapping appendage in an inviscid fluid
S. Alben, Journal of Fluid Mechanics, 614, 355 - 380 (2008).aa
We study propulsive forces generated by a flexible body with a vortex-sheet wake pitched periodically at the leading edge. We find that the thrust power generated by the body has a series of damped resonant peaks with respect to rigidity, the highest of which corresponds to a body flexed upwards at the trailing edge in an approximately one-quarter-wavelength mode of deflection.  Subsequent peaks in response have power-law scalings with respect to rigidity and correspond to higher-wavenumber modes of the body. We derive the power-law scalings by analysing the fin as a damped resonant system. In the limit of small driving frequency, solutions are self-similar at the leading edge. In the limit of large driving frequency, we find a power-law distribution of resonant rigidities. We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.
The flapping-flag instability as a nonlinear eigenvalue problem
S. Alben, Physics of Fluids, 20, 104106 (2008).
We reconsider the classical problem of the instability of a flapping flag in an inviscid background flow with a vortex sheet wake, and reformulate it as a nonlinear eigenvalue problem. We solve the problem numerically for the 20 lowest wave number modes in the parameter space of flag mass and flag rigidity. We study the connection between the modes and the growth rates in the eigenvalue and initial value problems. Using an infinite flag model we compute the parameters of the most unstable flag and show that a classical mechanism for the instability correlating pressure lows to flag amplitude peaks does not hold.
aa Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos
S. Alben and M.J. Shelley, Physical Review Letters, 100, 074301 (2008).

We investigate the "flapping flag" instability through a model for an inextensible flexible sheet in an inviscid 2D flow with a free vortex sheet. We solve the fully-nonlinear dynamics numerically and find a transition from a power spectrum dominated by discrete frequencies to an apparently continuous spectrum of frequencies. We compute the linear stability domain which agrees with previous approximate models in scaling but differs by large multiplicative factors. We also find hysteresis, in agreement with previous experiments.

Erratum: correction of parameters and Fig. 2

Movies of Flapping Flags (.avi files):
First Periodic State                  Second Periodic State         
Third Periodic State                Chaotic State
An implicit method for coupled flow-body dynamics
S. Alben, Journal of Computational Physics, 227, 4912--4933 (2008).

We propose an efficient method for computing coupled flow-body dynamics. The time-stepping is implicit, and uses an iterative method (preconditioned GMRES) to solve the flow-body equations. The preconditioner solves a decoupled version of the equations which involves only the inversion of banded matrices, and requires a small number of iterations per time step. We use the method to probe the instability to horizontal motions of an elliptical body with simple vertical motions: flapping and rising. 
How bumps on whale flippers delay stall: an aerodynamic model
E.A. van Nierop, S. Alben, and M.P. Brenner, Physical Review Letters, 100, 054502 (2008).
Wind tunnel experiments have shown that bumps on the leading edge of model humpback whale flippers cause them to "stall" (i.e., lose lift dramatically) more gradually and at a higher angle of attack. Here we develop an aerodynamic model which explains the observed increase in stall angle. The model predicts that as the amplitude of the bumps is increased, the lift curve flattens out, leading to potentially desirable control properties.

See also: "Whale-Inspired Windmills," MIT Technology Review Mar. 6, 2008
"Fluid dynamics: Lifting a whale," Nature, Research Highlights Feb. 21, 2008
The mechanics of active fin-shape control in ray-finned fishes
S. Alben, P.G. Madden and G.V. Lauder, Journal of the Royal Society Interface, 4, 243--256 (2007).
We have studied the mechanical properties of fin rays, which are a fundamental component of fish fin structure. We have derived a linear elasticity model which predicts the shape of fin rays given the input muscle actuation and external loading. The model agrees well with experiments: both show a concentration of curvature at the ray base or at the point of an externally-applied force, and a variation in ray stiffness over more than an order of magnitude depending on actuation at the bases of the fin rays.

aa The self-assembly of flat sheets into closed surfaces
S. Alben and M.P. Brenner, Physical Review E, 75, 056113 (2007).
A recent experiment (Boncheva et al. PNAS 102, 3924-3929 (2005)) introduced the possibility of initiating the self-assembly of a 3D structure from a flat elastic sheet. The ultimate utility of this method for assembly depends on whether it leads to incorrect, metastable structures. Here we examine how the number of metastable states depends on the sheet shape and thickness. Using simulations and theory we have identified out-of-plane buckling as the key event leading to metastability. The buckling strain that arises from joining edges of a planar sheet can be estimated using the theory of dislocations in elastic media. The number of metastable states increases rapidly with increasing variability in the boundary curvature and decreasing sheet thickness.

See also: Self-assembly could simplify nanotech construction, New Scientist, June 7, 2007
Coherent locomotion as an attracting state for a free flapping body
PNAS, 2005, 102 (32), 11163-11166;  S. Alben and M.J. Shelley.
We study numerically a fluid flow problem at the transition between low and high Reynolds number locomotion, motivated by an experiment at the Courant Institute Applied Math Lab. In our study, a 2-D rigid body is flapped in the vertical direction and is free to move horizontally. Above a critical flapping frequency, the wing becomes unstable to horizontal motion. For certain ranges of wing shape and mass, this instability saturates to unidirectional flapping flight. We have found that the typical event which triggers "take-off" is a fortuitous collision of the body with vortices shed on previous flapping strokes.

Dynamics of a free flapping body

Take-off (.mov)   Take-off (.avi)  
Back-and-forth (.mov)     Back-and-forth (.avi)
Chaotic (.mov)    Chaotic (.avi)        Thin body (.mov)     Thin body (.avi)
How flexibility induces streamlining in a two-dimensional flow
Physics of Fluids 16 (5): 1694-1713 (2004); S. Alben, M. Shelley, and J. Zhang

Drag reduction through self-similar bending of a flexible body
Nature 420, 479-481 (2002); S. Alben, M. Shelley, and J. Zhang
Nature abounds with organisms utilizing body flexibility in order to survive in flowing fluids.  An experiment in the Applied Mathematics Lab at Courant studied aspects of this using a length of fiber optic glass -- a flexible body -- immersed in the the quasi- two-dimensional flow of a running soap film.  As the flow speed increases the shape of the flexible body bends and becomes more and more streamlined -- the two left panels -- and consequently the fluid drag on the body grows much more slowly than if it were rigid.  The rightmost figure shows the numerical solution of our model of a flexible body deformed by an surrounding  flow and wake.  This theory shows an emerging self-similarity in shape arising from a balance of fluid and elastic forces at the tip.  This self-similarity yields a new, reduced drag law where drags grows as the 4/3 power, rather than the square, of the flow velocity.

See also: Nature's Secret to Building for Strength: Flexibility, New York Times, Dec. 17, 2002