hw6 - h t uh x = 0 u t uu x = g sin θ-fu 2/h where f is an...

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Department of Chemical Engineering University of California, Santa Barbara ChE 230A Fall 2006 Instructor: Glenn Fredrickson Homework #6 Homework Due: Wednesday, November 22, 2006 1. In high-speed aerodynamics, the variation of air density due to motion is impor- tant. For one-dimensional isentropic flow of an ideal gas the equations governing the density, ρ , pressure p , and velocity u are: ρ t + ( ρu ) x = 0 ρ ( u t + uu x ) = - p x p/p 0 = ( ρ/ρ 0 ) γ where p 0 and ρ 0 are constant reference values of p and ρ , and γ (constant) is the ratio of specific heats. Determine the differential equations for the characteris- tics appropriate to these equations and the so-called Riemann invariants. The latter are combinations of the dependent variables that remain constant along the characteristic directions. 2. Consider a one-dimensional flood wave in a wide river of bed inclination θ . The depth h and velocity u are governed by the following equations:
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Unformatted text preview: h t + ( uh ) x = 0 u t + uu x = g sin θ-fu 2 /h where f is an empirical constant of bed friction and g is the gravitational con-stant. Determine the characteristics. 3. The following equation describes the time evolution of the momentum distribu-tion function, W ( p,t ), in the theory of Brownian motion: W t = ( pW ) p + W pp Consider an initial value problem, where W ( p, 0) = f ( p ) is a prescribed initial velocity distribution function for-∞ < p < ∞ . Adopting the boundary con-ditions at infinity W, W p → 0 for p → ±∞ , solve this equation by first Fourier transforming over p to reduce the order of the PDE, then applying the method of characteristics, and finally back-transforming. 4. Prepare solutions to the following problems in your RHB text: 18.8, 18.9, 18.10....
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