A long thin no stretch no stiffness weightless film is supported by horizontal parallel beams and filled with a liquid. What is the shape of the cross section of the trough?
Define:
D: depth of liquid in the center
T: tension or force in film at any point x
Tx: horizontal component of T
Ty: vertical comp. of T
Doing just the right half of the curve:
Weight of liquid from 0 to x = Ty = area of liquid from surface to curve:
Ty = integral of [D - y(x)] dx
but since
y' = Ty/Tx
then
y' = [int (D-y) dx]/Tx
Tx doesn't change with x so taking the derivative of both sides:
y" + Ay + B = 0
Where D/Tx = -B and 1/Tx = A
Is there an analytical solution?
Bret Cahill