Stability lecture 5

Stability lecture 5 - Lecture 5 Solve Laplace's equation...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 5 Solve Laplace’s equation for velocity above and below the boundary condition. Assume Fourier series expansion () ( ) ) ,0 ,, c o s c o s ,,, , c o o s ii m m im m xyt A t x m y xyzt F zt x my η φ = = = = A A A A A A Conservation of mass is guaranteed by LaPlace’s equation. Plugging into LaPlace’s equation on each side of interface: () 2 22 2 0 mm Fm F z −+ = AA A has solution ( upper layer) 1, i = ) 11 , m kz m Fz tCt e = A A in the upper fluid layer, , the positive root. 22 2 0, with , using 0 zk m k >= + A > Which satisfies the condition at . z →+∞ In the lower fluid layer, ( ) 2, , kz iF z t C t e + == since 0 z < and this satisfies the condition as . z →−∞ Now drop the summation for all components of the Fourier series, and work with a single particular mode. Whatever is found for one mode is valid for all modes. m A
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The Kinematic B.C. gives 12 1,2 c o sc o s c o o s c o o s
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/07/2011 for the course EOC 6934 taught by Professor Staff during the Fall '08 term at University of Florida.

Page1 / 2

Stability lecture 5 - Lecture 5 Solve Laplace's equation...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online