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Unformatted text preview: MAE 150A, Winter 2009 J. D. Eldredge Homework 1, Due Friday, January 16
0. Read chapters 4 and 5 of Wilcox. 1. (Problem 4.1, Wilcox) Compute the acceleration vector for the following velocity vectors, where U , and a are constants. (a) u = U cos(x + at)i (b) u = U 1 - e-y/ i (c) u = f (y)i + g(x)j 2. (Problem 4.12/13, Wilcox modified) In cylindrical coordinates, the components of the acceleration vector in two-dimensional flow are u2 ur ur u ur + ur + - t r r r u u u u u ur a = + ur + + t r r r ar = Find the acceleration vector for the following flows. (a) Potential vortex, u = 2r e , where is constant. (b) Rigid-body rotation, u = re , where is constant. (c) Rigid-body rotation plus a mass sink, u = re -
Q 2r er , where m is constant. 3. (Problem 4.10, Wilcox) For flow near the stagnation point of an accelerating cylinder, the velocity is u= 2U (t) (xi - yj) R where R is the cylinder's radius and U (t) is the cylinder's speed. (a) Compute the unsteady, convective and total accelerations. (b) Determine U (t) and ax if ay = 0. Assume U (0) = U0 .
y U x 4. (Problem 5.12, Wilcox) The y component of the velocity vector for a two-dimensional, incompressible, irrotational flow is v= x2 Bx , + y2 B = constant What must the x component, u(x, y), be? 5. A type of braking system consists of a cylindrical ram of cross-sectional area Ar that is pushed into a slightly larger cylindrical cavity of area Ac (see figure). Suppose that the ram velocity is constant, V , and the fluid in the cavity is incompressible. Find an expression for the velocity Vj , exiting the gap between ram and cavity wall.
Vj Ac V Ar 6. Consider again a stagnation flow, similar to problem 4.10, but in this case the flow is radially symmetric (see below). Suppose that the velocity field is given by u(x, y, z) = 1 Ax i + 1 Ay j + 2 2 Az z k, where A and Az are constants.
y z n a a x Side view Top view (a) If the flow is incompressible, then what must Az be in terms of A? (b) Find an expression for the local mass flux per unit area, u n, out of the hemispherical surface of radius a, shown below. (Note: Appendix E of Wilcox has some potentially useful formulas). (c) What is the angle at which the mass flux switches from being locally inward to locally outward. ...
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