Transport Phenomena - Semester Project

Advanced Variational Methods

Chaplygin's Problem for a Swimmer in a Constant Flow

A swimmer traverses a closed loop in a river by starting and returning to the same point, swimming at a constant velocity v and the swimmer's position can be given in 2-dimensions as time-parametrized Cartesian coordinates (Y(t) = (x(t), y(t))). The river is unidirectional and of constant velocity, w, and all energy transfer between the swimmer and river is inertial (i.e. no viscous dissipation of energy). Furthermore, the swimmer is given a finite-time, T , to enclose an area, A, which should be maximized.

a) Find the shape of the swimmers path if w = 0. That is, find the shape of a closed loop that maximizes area if traversed in a finite-time.

b) Do some leg-work and research Chaplygin's problem to find out what the con- strained functional is for this problem when w ≠ 0.

c) What is the Euler-Lagrange eqn(s) for this system? (You'll need to set the Gateaux derivative equal to zero in the limit of the perturbation also going to zero (i.e. as we did in class If you simply copy/transfer your notes over here, you will be docked this part and the rest of the problem as it will be impossible to come up with the right answer!!!!))

d) Using your E-L eqns, come up with the system of ODEs/eqns that predicts the shape of the closed-curve for a swimmer moving at velocity v in a river moving at velocity w such that v > w.

e) Using the additional constraint (x'(t) - w)² + (y'(t))² ≡ 1, prove that the minimizing functions for the functional satisfy:

2λ(x'(t) - w) = -y + c₂ (1)

and

2λy(t)' = x - c₁ (2)

for some constants c₁ and c₂.

f) Show that (2λ(t, Y ))² = (x - c₁)² + (y - c₂)² and differentiate along the trajectory to get:

(2λ(t))t' = wy'(t) (3)

and

2λ = wy(t) + r(4)

g) Conclude that the path of the swimmer is an ellipse with eccentricity.

BONUS to be added to your first mid-term grade

1) Give me a concise mathematical description of Chaplygin's variational application to airfoils that revolutionized airplane flight. (i.e. what were the equations that governed the physics, the resulting functional, constraints and interpretation of the equation terms. Also, how did Chaplygin go about his solution?)

2) Derive the Euler-Lagrange equation for a functional that depends on (x, y(x), y'(x) and y"(x)). Generalize this result for an n-th order derivative equation.