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Stability lecture 5

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

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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

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The Kinematic B.C. gives 12 1,2 c o sc o s c o o s c o o s
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Stability lecture 5 - Lecture 5 Solve Laplace's equation...

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