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### hw8_08

Course: ENGN 2210, Fall 2009
School: Sanford-Brown Institute
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Word Count: 303

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EN221: HW #8, Due Wednesday, 11/19. 1. (a) Consider a static situation in which a body occupies the region B0 for all time. Assume that the body force b = 0 and that the body is bounded by surfaces S0 and S1 as shown in Fig. (a) below. Assume further that S0 and S1 are acted on by uniform pressures 0 and 1. Show that the average stress in B0 is a pressure of amount 1 v1 - 0 v0 , (1) v1 - v0 where v0 and v1 are,...

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EN221: HW #8, Due Wednesday, 11/19. 1. (a) Consider a static situation in which a body occupies the region B0 for all time. Assume that the body force b = 0 and that the body is bounded by surfaces S0 and S1 as shown in Fig. (a) below. Assume further that S0 and S1 are acted on by uniform pressures 0 and 1. Show that the average stress in B0 is a pressure of amount 1 v1 - 0 v0 , (1) v1 - v0 where v0 and v1 are, respectively, the volumes enclosed by S0 and S1 . (b) Consider a steady, irrotational flow of an ideal fluid of density 0 over an obstacle R, where R is a bounded regular region whose interior lies outside the flow region B0 (see Fig. (b)). Assume that the body force is zero. Show that the total force exerted on R by the fluid is equal to 0 2 where R is the boundary of R. R v2ndA, (2) 2. Derive the boundary conditions (26.2c) for the flow problem sketched in Fig. 2.5 of the notes from the OCAIM institute distributed in class. 3. Using the traveling wave solutions (2.63 of the OCAIM notes) derive the dispersion relation (relation between and k, 2.64 in the notes) and verify the conditions (2.67) and (2.68) for the Rayleigh-Taylor and Kelvin-Hemholtz instabilities, respectively. 4. In class we derived the Rayleigh-Plesset equation for bubble dynamics in an <a href="/keyword/incompressible-fluid/" >incompressible fluid</a> using the cauchy equation of motion (you can find a similar analysis in Chadwick, Problem 7, page 101). Derive this equation using the Bernoulli equation for <a href="/keyword/incompressible-fluid/" >incom...

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