Midterm Exam 2004 solutions.doc

# Answer b is also incorrect refer to table 17 1 which

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Answer b is also incorrect. Refer to Table 1.7-1 which demonstrates that the direction of the momentum flux is the same as the direction of the velocity vector. Since the surface itself can be arbitrarily selected, the velocity vector, and hence the momentum flux may not be perpendicular to the surface. Answer c is correct since it is the definition of momentum flux. Answer d is also correct. The nonlinear terms (on the left hand side of the momentum equations) have the form u u , and this is a nonlinear term. Therefore, both c and d are true, so the correct answer is e. 18. The momentum equation used in the book and typically used in fluid mechanical applications: a. is written in terms of a Lagrangian frame of reference. b. is written in terms of an Eulerian frame of reference. c. is linear in all practical applications. d. contains 3 unknown variables. 11

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Steven A. Jones BIEN 501 Physiological Modeling Midterm Exam April 7, 2004 e. none of the above. In solid mechanics, the equations are typically written in a Lagrangian frame of reference, where the coordinate system follows the motion of materials. A classic example is the equation for a falling body 2 2 1 0 at vt x x . Note that the coordinate x is chaging in time. In fluid mechanics, however, the equations are written with respect to a stationary coordinate system and measurements are made at a point in space. This frame of references is the reason for the use of the substantial derivative rather than the time derivative in the momentum equations. The momentum equation is certainly not linear in all practical applications. It is inherently non- linear, but sometimes reduces to a linear equation under special circumstances. The momentum equation contains 4 unknown variables. In Cartesian coordinates these are x v , y v , z v , and p . Therefore the correct answer is b. 19. The expression u a. is the continuity equation for incompressible flow. b. is a single equation. c. involves 3 unknown variables. d. contains y v y as one term. e. all of the above. I confess, this is another mistake on my part. The statement of the equation should have been, “The expression 0 u ….” In this case, it is the continuity equation for incompressible flow, and it is a single equation since the dot product of the vector and the vector u yields a scalar. (The expression contrasts with that in question 20 below, where the vector formulation of the momentum equation is three separate equations, one for x v momentum, one for y v momentum, and one for momentum). It also involves 3 unknown variables, which are x v , y v , and z v . It also includes the term y v y . Therefore, the correct answer is e. However, since some of you may have been confused by my suggesting that u is a complete equation, I accepted also either answer c. or answer d.
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