Midterm Exam 2004 solutions.doc

Again with no changes in the expression for r

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Again, with no changes in , the expression for r simplifies to r v r r r , and with no z component of velocity, the expression for z simplifies to z v z . When these two shear stresses are introduced into Eq. 13-1, the result is: z v z r v r r r r r 2 2 1 0 On dividing through by (allowable because the fluid is Newtonian), this becomes: z v z r v r r r r 3 2 1 0 . The second term matches answer c, but the first term does not. 7 Eq. 13-1
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Steven A. Jones BIEN 501 Physiological Modeling Midterm Exam April 7, 2004 The correct answer is e. none of the above. 14. Assuming that the momentum balance in the direction for a rotating flow is: 0 1 2 z v z r v r r r r , let z r R v . Then the equation for the z dependence of v is: a. non-linear b. 0 2 2 kZ dz Z d c. k dz Z d 2 2 d. r G dz Z d 2 2 , where r G is an arbitrary function of r . e. None of the above. This question illustrates the technique of “separation of variables” for solving partial differential equations. Separation of variables can be used when the boundary conditions are separable, meaning that they have the form: z Z r R z r v r , , on the surface 0 , , z r f . For example, consider the Stokes-sphere problem in which the boundary conditions are: 0 r v at R r 0 v at R r 2 2 2 1 sin r v The first two of these boundary conditions are separable because 0 can be expressed as r R , where one or the other of r R or is zero. The third can be expressed as r R , where 2 2 1 r v r R and 2 sin . Problem 14 therefore assumes that we have examined the boundary conditions and found them to be separable. The next step is to substitute the separated version of v into the differential equation to obtain: 0 1 2 z z Z r R z r z r R r r r r Since r R depends on r only, it can be treated as constant with respect to the derivative in z . Similarly, z Z is constant with respect to derivatives in r . These functions can be taken out of the derivatives with respect to the variables they do not depend on to yield: 8
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Steven A. Jones BIEN 501 Physiological Modeling Midterm Exam April 7, 2004
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