Let t be time. Confirm that o(r,t) = F(t)r-1 satisfies Laplace's equation in spherical polar coordinates...

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Let t be time. Confirm that o(r,t) = F(t)r-1 satisfies Laplace's equation in spherical polar coordinates (r, 0, a). A spherical bubble, centered at r = 0, has radius R(t). It is surrounded by water of constant density p moving according to the above velocity potential. Derive the following boundary condition on r = R(t): = Ř(t), ar where the dot denotes a derivative with respect to time. Hence show that F(t) = -R R. The pressure inside the bubble p1(t) everywhere equals the fluid pressure p on r = R, and as r → 0, p→ Poo, a constant. Derive and ordinary differential equation for R(t) by using the additional assumption that the bubble obeys an isothermal gas law R°p1 = a p 3. where a is the static radius of the bubble. Assume R(t) smaller, show that w? = 3po0/(pa?). a(1 + esin wt) where e « 1 and you may neglect terms of size e? and %3D Let t be time. Confirm that o(r,t) = F(t)r-1 satisfies Laplace's equation in spherical polar coordinates (r, 0, a). A spherical bubble, centered at r = 0, has radius R(t). It is surrounded by water of constant density p moving according to the above velocity potential. Derive the following boundary condition on r = R(t): = Ř(t), ar where the dot denotes a derivative with respect to time. Hence show that F(t) = -R R. The pressure inside the bubble p1(t) everywhere equals the fluid pressure p on r = R, and as r → 0, p→ Poo, a constant. Derive and ordinary differential equation for R(t) by using the additional assumption that the bubble obeys an isothermal gas law R°p1 = a p 3. where a is the static radius of the bubble. Assume R(t) smaller, show that w? = 3po0/(pa?). a(1 + esin wt) where e « 1 and you may neglect terms of size e? and %3D

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It is important to know how to solve Laplaces equation in various coordinate systems The coordinate ... View the full answer