Fluid dynamics of animal locomotion
Topics in Fluid Dynamics G63.2861)
Professor Stephen Childress.
Wednesday 9:30-11:20am, Room 613 WWH.
Office Hours: By appointment (83135 or childres@cims).
This course will study the dynamics of animal locomotion
in a dissipative fluid environment.
Some familiarity with elementary fluid dynamics is desired. However
the basic material will be
reviewed, see also the suggested preparatory reading below. Most
of the important concepts
in the unsteady dynamics of incompressible fluids will come into
play. Emphasis will be on the
role of fluid dynamics in determining modes of locomotion in the
natural world, including
microorganisms, insects, birds, and fish.
Materials and content
- Lecture 1, September 8: Background
and overview Anyone who missed the
first class may download a pdf file with a
summary of this lecture.
Lecture Notes in pdf format. The
individual chapter files will lack proper cross-referencing. A final compilation
with bibliography and index will be made available at the end of the course.
- Lecture 2, September 15: Formulation and derivation of the Navier-Stokes equation
The continuum description using Lagrangian and Eulerian coordinates.
Conservation laws for mass and momentum. Vorticity, and Kelvin's theorem.
Potential flows. The stress tensor and the Navier-Stokes equations. Incompressible
flows. The dimensionless form and the Reynolds numbers.
- Lecture 3, September 22: Formulation and derivation of the Navier-Stokes equation continued.
Vorticity equation. Amplification of vorticity by stretching. Vortex
tubes. Velocity from vorticity and the Biot-Savart law. Hodge decomposition.
Some simple solutions of the Navier-Stokes equations: Poiseuille flow
and the Rayleigh problem. The Stokes equations.
- Lecture 4, September 29: Stokes flows and the swimming of a stretching sheet. Calculation
of the swimming velocity in Stokes flow. Extension to finite Reynolds
number and the paradox of the inviscid limit as a boundary-layer
phenomenon. Envelope models of ciliates. Stokes as instantaneous conformatioin
of the fluid to boundary conditions, and the uniqueness of the flow. The
Stokeslet solution as the fundamental solution of the Stokes equations
for exterior problems.
- Lecture 5, October 6: Stokesian locomotion. The general problem of locomotion
in fluids. Standard versus current shape. The complete family of solutions
of Stokes equations for a 3D body. The poloidal-toroidal decomposition of
a divergence-free field. Solutions with symmetry. Force and moment. Time-reversal
symmetry and the scallop theorem. The swimming sheet reconsidered. Efficiency
in Stokesian locomotion.
- Lecture 6, October 13: Flagellar locomotion I. The local theory of a thin
straight stalk. Local resistive force approximation. Gray-Hancock theory.
Zero thust swimming. Helical waves. The problem of calibrating resistive
- Lecture 7, October 20: Flagellar locomotion II. Ciliary locomotion I. The formal theory and the main localization theorem. Lighthill'
application to the helical flagellum. Results for the calibration problem.
Motioin of an individual cilium. Metachronal waves. The envelope model.
- Lecture 8, October 27: Ciliary locomotion II. The Eulerian realm. The macrostructure
of ciliary propulsiion. Envelope stresses. The locomotion of a ciliated sphere.
Euler's equation of an inviscid fluid. Inertial versus vortex
forces. The unsteady Bernoulli theorem. Recoil swimming and squirming in an
inviscid fluid.motion of anlocal theory of a thin straight stalk.
- Lecture 9, November 3: The swimming of slender fish.
Lighthill's theory for slender fish. The energy
method for small-amplitude swimming. Large amplitude theory of slender fish
by momentum balance. The wake structure and vortex shedding. Effect of mid-body
- Lecture 10, November 10: The vortex force. Bound versus free vorticity. The non-uniqueness of ideal flow past
a circular cylinder and the arbitrary vortex force which results. The Kutta-Joukowsky
condition and the calculation of lift for a 2D airfoil. Quasi steady aerodynamics.
The non-steady theory of a flapping flat plate: small amplitude theory od
thrust generation. Interpretation of the small-amplitude thrust generated
by a slwnder fish as a result of vortex forces.
- Lecture 11, November 17: Hovering flight. Normal hovering. The hovering wake: actuator disc theory. The clap
and fling mechanism. Potential theory of the generation of lift. Viscous effects
and the role of shed vorticity. Impulse and the calculation of lift from
shed vorticity. The motion of a flexible body in an oscillatory flow.
Role of vortex versus inertial forces.
- Lecture 12, November 24: Intermediate Reynolds numbers.
Flapping flight as a bifurcation in frequency Reynolds
number. Simple models of thrust generatioin: Oseen model, the flapping venetian
blind. Potential theory of the venetian blind. The role of mass. Models
of the bifurcation.
- Lecture 13, Dec. 1: Some miscellaneous topics
Chapter 3. (Through lecture 9)