PS5 - way from u x + v y = 0. Consider now a more general...

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MATH 505: Mathematical Fluid Dynamics Problem Set 5 due Wednesday, December 12, 2007 1. In class we showed that the velocity potential φ for a slightly compressible fluid satisfied the linear wave equation to leading order, φ tt = c 2 2 φ . Derive the nonlinear terms which we neglected. 2. For an oscillating spherical air bubble, we used the normal stress boundary condition in class to connect the bubble pressure p b to the fluid pressure p ( R, t ), - p ( R, t ) + 2 μ ∂u ∂r ± ± ± ± R + p b ( t ) = γκ this led to the Rayleigh-Plesset equation for R . Here set the surface tension γ = 0 and the viscosity μ = 0, but replace the viscous stress with an elastic stress which depends on the displacement (the integral of the velocity): G 0 Z t 0 u ( R, t 0 ) dt 0 Using the known form for u write this in terms of R . Then use the normal stress boundary condition to rederive a Rayleigh-Plesset equation including this elastic effect. 3. Consider the steady boundary layer equation of Prandtl for ~u = ( u, v ) uu x + vu y = νu yy where ν is the kinematic viscosity. A streamfunction ψ can be defined in the usual
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Unformatted text preview: way from u x + v y = 0. Consider now a more general approach to solving this equation in terms of the similarity solution ( x, y ) = F ( x ) f ( ) , where = y/g ( x ) The more general far eld boundary condition ~u ( U ( x ) , 0) as y/ leads to the condition F ( x ) = U ( x ) g ( x ) (in class we had U ( x ) = U = const). Show that this leads to the equation f 2- 1 + U U g g ff 00 = 1 + f 000 g 2 U , where the prime is the derivative with respect to the appropriate single variable ( x or ). Show that a similarity solution is only possible if either U ( x ) ( x-x ) n or U ( x ) e x , where x , n, and are constants. 4. For the case U ( x ) = Ax n ( A > 0) in the previous problem, show that g ( x ) x (1-n ) / 2 . By choosing g ( x ) = s 2 ( n + 1) Ax n-1 derive an ordinary dierential equation for f ( ) (the Falkner-Skan equation)....
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This note was uploaded on 07/23/2008 for the course MATH 505 taught by Professor Belmonte,andrewl during the Fall '07 term at Pennsylvania State University, University Park.

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