me Silas Alben

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

Email: last name at

phone: 734-647-5518
Math 654: Introduction to Fluid Dynamics
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.

Optimal wall shapes and flows for steady planar convection S. Alben, J. Fluid Mech. (984) A43, 2024
We compute steady planar incompressible flows and wall shapes that maximize the rate of heat transfer (Nu) between hot and cold walls, for a given rate of viscous dissipation by the flow. We show theoretically that flat walls are optimal in the case of no flow, and up to a critical nonzero flow magnitude. Beyond this value, our computed optimal flows and wall shapes converge to a set of forms that are invariant except for a scaling of horizontal lengths as Pe-1/3, where Pe is the Peclet number. The corresponding rate of heat transfer Nu scales as Pe2/3. We show that these scalings result from flows at the interface between the diffusion-dominated and convection-dominated regimes. We also show that the separation distance of the walls remains at its minimum value at large Pe.

Transition to branching flows in optimal planar convection S. Alben, Phys. Rev. Fluids (8) 074502, 2023
Fluid flow structures that enhance heat transfer are important in many natural and technological settings. Here we compute the steady planar incompressible fluid flows that maximize the rate of heat transfer from a solid surface, for various rates of viscous energy dissipation. We find a transition from rectangular convection rolls to flows that branch near the boundary. These flows may be related to branching flows that have been predicted theoretically and computed in three dimensions.

Membrane flutter in three-dimensional inviscid flow C. Mavroyiakoumou and S. Alben, J. Fluid Mech. (953) A32--1-38, 2022
We study the large-amplitude flutter of rectangular membranes (of zero bending rigidity) that shed a trailing vortex-sheet wake in a three-dimensional (3-D) inviscid fluid flow. For 12 combinations of boundary conditions at the membrane edges we compute the stability thresholds and the subsequent large-amplitude dynamics across the three-parameter space of membrane mass density, pretension and stretching rigidity. We find that the 3-D dynamics in the 12 cases naturally forms four groups based on the conditions at the leading and trailing edges. The conditions at the side edges, although generally less important, may have small or large qualitative effects on the membrane dynamics – e.g. steady vs unsteady, periodic vs chaotic or the variety of spanwise curvature distributions – depending on the group and the physical parameter values. (ArXiv)

Efficient bending and lifting patterns in snake locomotion S. Alben, Proc. Roy. Soc. A (478) 20220312, 2022
We optimize three-dimensional snake kinematics for locomotor efficiency. We assume a general space-curve representation of the snake backbone with small-to-moderate lifting off the ground and negligible body inertia. The cost of locomotion includes work against friction and internal viscous dissipation. When restricted to planar kinematics, our population-based optimization method finds the same types of optima as a previous Newton-based method. With lifting, a few types of optimal motions prevail. We have an s-shaped body with alternating lifting of the middle and ends at small-to-moderate transverse friction. With large transverse friction, curling and sliding motions are typical at small viscous dissipation, replaced by large-amplitude bending at large viscous dissipation. With small viscous dissipation we find local optima that resemble sidewinding motions across friction coefficient space. They are always suboptimal to alternating lifting motions, with average input power 10-100% higher. (ArXiv)

Efficient sliding locomotion of three-link bodies with inertia A. Earnst and S. Alben, Phys. Rev. E (106) 044404, 2022
Many previous studies of sliding locomotion have assumed that body inertia is negligible. Here we optimize the kinematics of a three-link body for efficient locomotion and include among the kinematic parameters the body inertia. The optimal inertia is nonnegligible when the coefficient of friction for sliding transverse to the body axis is small. Inertia is also significant in a few cases with relatively large coefficients of friction for transverse and backward sliding, and here the optimal motions are less sensitive to the inertia parameter. For some of the optimal motions with significant inertia we find dramatic reductions in efficiency when the inertia parameter is decreased to zero. For the motions that are optimal with zero inertia, the efficiency decreases more gradually when we raise the inertia to moderate and large values.

Packing of elastic rings with friction S. Alben, Proc. Roy. Soc. A (478) 20210742, 2022
We study the deformations of elastic filaments confined within slowly shrinking circular boundaries, under contact forces with friction. We perform computations with a spring-lattice model that deforms like a thin inextensible filament of uniform bending stiffness. Early in the deformation, two lobes of the filament make contact. If the friction coefficient is small enough, one lobe slides inside the other; otherwise, the lobes move together or one lobe bifurcates the other. There follows a sequence of deformations that is a mixture of spiralling and bifurcations, primarily the former with small friction and the latter with large friction. With zero friction, a simple model predicts that the maximum curvature and the total elastic energy scale as the wall radius to the −3/2 and −2 powers, respectively. With non-zero friction, the elastic energy follows a similar scaling but with a prefactor up to eight times larger, due to delayering and bending with a range of small curvatures. For friction coefficients as large as 1, the deformations are qualitatively similar with and without friction at the outer wall. Above 1, the wall friction case becomes dominated by buckling near the wall. (ArXiv)

Dynamics of flags over wide ranges of mass and bending stiffness S. Alben, Phys. Rev. Fluids (7) 013903, 2022
There have been many studies of the instability of a flexible plate or flag to flapping motions, and of large-amplitude flapping. Here we use inviscid simulations and a linearized model to more generally study how key quantities—mode number (or wave number), frequency, and amplitude—depend on the two dimensionless parameters: flag mass and bending stiffness. In the limit of small flag mass, flags perform traveling wave motions that move at nearly the speed of the oncoming flow. The flag mode number scales as the −1/4 power of bending stiffness. The flapping frequency has the same scaling, with an additional slight increase with flag mass in the small-mass regime. The flapping amplitude scales approximately as flag mass to the 1/2 power. In a linearized model, the fastest growing modes have somewhat different power law scalings for wave number and frequency. We discuss how the numerical scalings are consistent with a weakly nonlinear model. (ArXiv)

Inverse design of self-oscillatory gels through deep learning D. Aksoy, S. Alben, R.D. Deegan, and A.A. Gorodetsky, Neural Computing and Applications, 2022
We develop a deep learning architecture for inverse design of a self-oscillating sheet propelled by an embedded chemical reaction. The dynamics of our problems are nonlinear and exhibit chaotic behavior, a challenging setting for existing deep-learning-based inverse design approaches. The aim is to explore data-driven design of soft robots using a novel locomotion mechanism. We train the architecture using a forward model of the locomotion mechanism. The architecture is shown to successfully map a snapshot of target motions of the gel into geometric and reaction parameters. Our inverse design setting is unique in that it considers both discrete and continuous outputs, requiring an architecture capable of classification and regression. Because the motion has a chaotic quality, our demonstration is able to show quantitative agreement for a small time horizon and qualitative agreement over longer time horizons.

Dynamics of tethered membranes in inviscid flow C. Mavroyiakoumou and S.Alben, J. Fluids Struct. (107) 103384-1--30, 2021
sheet3We investigate the dynamics of membranes that are held by freely-rotating rigid tethers in fluid flows. The tethered boundary condition allows periodic and chaotic oscillatory motions for certain parameter values. We characterize the oscillations in terms of deflection amplitudes, dominant periods, and numbers of local extrema of deflection along the membranes across the parameter space of membrane mass density, stretching modulus, pretension, and tether length. We determine the region of instability and the small-amplitude behavior by solving a nonlinear eigenvalue problem. We also consider an infinite periodic membrane model, which yields a regular eigenvalue problem, analytical results, and asymptotic scaling laws. We find qualitative similarities among all three models in terms of the oscillation frequencies and membrane shapes at small and large values of membrane mass, pretension, and tether length/stiffness.(ArXiv)

Efficient sliding locomotion of three-link bodies S. Alben, Phys. Rev. E (103) 042414-1--18, 2021
sheet3We study the efficiency of sliding locomotion for three-link bodies with prescribed joint angle motions. The bodies move with no inertia, under dry (Coulomb) friction that is anisotropic (different in the directions normal and tangent to the links) and directional (different in the forward and backward tangent directions). Friction coefficient space can be partitioned into regions with distinct types of efficient kinematics: lateral undulation with very anisotropic friction, small-amplitude reciprocal kinematics, very large amplitude kinematics near isotropic friction, and kinematics that are very asymmetric about the flat state. A stochastic optimization algorithm finds that optimal kinematics are close to elliptical trajectories in most cases, except for small normal friction. With a linear (viscous) resistance law, the optimal trajectories are similar, and relative efficiencies are much lower except with very large normal friction. (ArXiv)

sheet3Eigenmode analysis of membrane stability in inviscid flow
C. Mavroyiakoumou and S.Alben, Phys. Rev. Fluids (6) 043901-1--32, 2021
We study the instability of a thin membrane (of zero bending rigidity) to out-of-plane deflections, when the membrane is immersed in an inviscid fluid flow and sheds a trailing vortex-sheet wake. We find instability by divergence or flutter (particularly at large mass density, or with one or both ends free). The most unstable eigenmodes generally become ``wavier" at smaller mass density and smaller tension, but with regions of nonmonotonic behavior. We find good quantitative agreement with unsteady time-stepping simulations at small amplitude, but only qualitative similarities with the eventual steady-state large-amplitude motions.(ArXiv)

Collective locomotion of two-dimensional lattices of flapping plates. Part 2. Lattice flows and propulsive efficiency S. Alben, J. Fluid Mech. (915) A21-1--25, 2021
sheet3We study propulsion of rectangular and rhombic lattices of flapping plates at O(10–100) Reynolds numbers in incompressible flow. We classify the propulsive performances of the lattices and the periodicities of the flows with respect to flapping amplitude and frequency, horizontal and vertical spacings between plates, and oncoming flow velocity. Non-periodic states are most common at small streamwise spacing, large lateral spacing and large Reynolds number. Lattices that are closely spaced streamwise shed intense vortex dipoles. The flows transition sharply from drag- to thrust-producing at critical flow speeds, and pass through a variety of periodic and non-periodic states. Multiple stable self-propelled speeds can occur. As the streamwise spacing increases (and with large lateral spacing), the plates may shed vortex streets that impinge on downstream neighbors. With small lateral spacing, the rectangular lattices yield net drag, while the rhombic lattices may generate net thrust efficiently. As lateral spacing increases, the two types of flows converge. The lattices’ maximum Froude efficiencies exceed those of an isolated plate by a factor that grows with decreasing Re, exceeding two at Re = 70. (ArXiv)

Collective locomotion of two-dimensional lattices of flapping plates. Part 1. Numerical method, single-plate case and lattice input power S. Alben, J. Fluid Mech. (915) A20-1--28, 2021
We propose a model and numerical method for the propulsion of rectangular and rhombic lattices of flapping plates at O(10–100) Reynolds numbers in incompressible flow. We use an adaptive mesh to mitigate singularities at the plates’ edges, and find convergence rates in a test problem. We then establish benchmark results for a single flapping plate, including vortex wake characteristics and Froude efficiency over ranges of flapping amplitude, frequency and Strouhal number. As a prelude to a study of propulsive efficiency in Part 2, we study a key ingredient: the time-averaged input power in lattices of plates. Scaling laws for the mean input power are estimated in the limits of small and large streamwise spacings, using steady flow models with small-gap and Poiseuille-like flows between the plates respectively in the two limits. (ArXiv)

Intermittent sliding locomotion of a two-link body sheet3S. Alben and C. Puritz, Phys. Rev. E (101) 052613-1--20, 2020
We study the possibility of efficient intermittent locomotion for two-link bodies that slide by changing their interlink angle periodically in time. The anisotropy ratio of the sliding friction coefficients is a key parameter. With very anisotropic friction, efficient motions involve coasting in low-drag states, with rapid and asymmetric power and recovery strokes. As the anisotropy decreases, burst-and-coast motions change to motions with long power strokes and short recovery strokes. We find these motions in the spaces of sinusoidal and power-law motions described by two and five parameters, respectively, and with an optimization search in the space of more general periodic functions (truncated Fourier series). When we increase the resistive force's power-law dependence on velocity, the optimal motions become smoother, slower, and less efficient, particularly near isotropic friction. (ArXiv)

Large-amplitude membrane flutter in inviscid flow C. Mavroyiakoumou and S. Alben, J. Fluid Mech. (891) A23--1-24, 2020
sheet3We study the large-amplitude flutter of membranes (of zero bending rigidity) with vortex sheet wakes in two-dimensional inviscid fluid flows. The dynamics depend on three dimensionless parameters: membrane pretension, mass density, and stretching modulus. With both ends fixed, all the membranes converge to steady deflected shapes with single humps that are nearly fore-aft symmetric. With leading edges fixed and trailing edges free to move in the transverse direction, the membranes flutter periodically or aperiodically (at larger mass density). With both edges free to move in the transverse direction, the membranes flutter similarly to the fixed–free case, but also translate vertically with steady, periodic or aperiodic trajectories, and with non-zero slopes that lead to small angles of attack with respect to the oncoming flow. (ArXiv) Movie

Semi-implicit methods for the dynamics of elastic sheets S. Alben, A. A. Gorodetsky, D. Kim, and R.D. Deegan, J. Comp. Phys. (399) 108952--1-17, 2019
Recent applications (e.g. active gels and the self-assembly of elastic sheets) motivate the sheet3need to efficiently simulate the dynamics of thin elastic sheets. We present semi-implicit time stepping algorithms to improve the time step constraints of explicit methods. For a triangular lattice discretization, our semi-implicit approach is stable for all time steps in the case of overdamped dynamics. For a more general finite-difference formulation that can allow for general elastic constants, we use the analogous approach on a square grid, and find that the largest stable time step is two to three orders of magnitude greater than for an explicit scheme. For a model problem with a radial traveling wave form of the reference metric, we find transitions from quasi-periodic to chaotic dynamics as the sheet thickness is reduced, wave amplitude is increased, and damping constant is reduced. (ArXiv)

Efficient sliding locomotion with isotropic friction S. Alben, Phys. Rev. E (99) 062402--1-17, 2019
Snakes' bodies are covered in scales that make it easier to slide in some directions than in others. This frictional anisotropy allows for sliding locomotion with an undulatory gait, one of the most common for snakes. Isotropic friction is a simpler situation (that arises with snake robots, for example) but is less understood. In this work we regularize a model for sliding locomotion to allow for static friction. We then propose a robust iterative numerical method to study the efficiency of a wide range of motions under isotropic Coulomb friction.We find that simple undulatory motions give little net locomotion in the isotropic regime. We compute general time-harmonic motions of three-link bodies and find three local optima for efficiency. The top two involve static friction to some extent. We then propose a class of smooth body motions that have similarities to concertina locomotion (including the involvement of static friction) and can achieve optimal efficiency for both isotropic and anisotropic friction. (ArXiv)

Dynamics and locomotion of flexible foils in a frictional environment X. Wang and S. Alben, Proc. Roy. Soc. A (474) 2209:20170503--1-20, 2018
We study the dynamics of snake-like bodies (flexible foils) under frictional forces to understand the physics of locomotion by passive appendages in terrestrial environments. When a flexible foil is oscillated by heaving at one end but is not free to locomote, the dynamics change from periodic to non-periodic and chaotic as the heaving amplitude increases or the bending rigidity decreases. Resonant peaks and bistable states are observed. When the foil is free to locomote, the horizontal motion smoothes the resonant peaks.
Locomotion is steady but slow at moderate frictional coefficients, and faster at larger transverse friction and small tangential friction corresponding to wheeled snake robots. Here travelling wave motions arise spontaneously, and move with horizontal speeds that scale as transverse friction coefficient to the power 1/4 and input power that scales as the transverse friction coefficient to the power 5/12. These scalings correspond to boundary layers near the foil’s leading edge. (ArXiv)

Intracellular localization of nanoparticle dimers by chirality reversal
M. Sun, L. Xu, J.H. Bahng, H. Kuang, S. Alben, N.A. Kotov, and C. Xu. Nature Communications (8), 1847--1-10, 2017

The intra- and extracellular positioning of plasmonic nanoparticles (NPs) can dramatically alter their curative/diagnostic abilities and medical outcomes. Here we show that the chiroptical activity of DNA-bridged NP dimers allows one to follow the process of internalization of the particles by mammalian cells and to distinguish their extra- vs intra-cellular localizations by real-time spectroscopy in ensemble. This finding opens the door for spectroscopic targeting of plasmonic nanodrugs and quantitative assessment of nanoscale interactions.

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.

Enhanced convection heat transfer using small-scale vorticity concentrations effected by flow-driven, aeroelastically vibrating reeds
A. Glezer, R. Mittal, and S. Alben, Air Force Research Laboratory Technical Report, 2016
We use experimental/modeling/numerical approaches to study the formation, shedding, and advection of small-scale vortical motions induced by autonomous, aeroelastic fluttering of a cantilevered thin-film reed at the centerplane of a rectangular air channel. The flow mechanisms and scaling of the interactions between the reeds and the channel flow were explored to develop the fundamental knowledge needed to overcome the limits of forced convection heat transport from air-side heat exchangers at low (laminar or transitional) Reynolds numbers.

Fluid–structure interactions with applications to biology W.-X. Huang and S. Alben, Acta Mechanica Sinica 32 (6): 977-979, 2016
Introduction to a thematic issue.

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