Consider the fluid region inside a spherical volume

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Unformatted text preview: orem f V .UdV=:I u.dA, by integrating over the sphere. 21 Problem 3.9 22 E.i'erci.~e Problem 3.10 10. Show that the vorticity field for any flow satisfies V..w=o. i 1. A flow field on the xy-plane has the velocity componen U = 3x + y v = 2x - 3y. Show that the circulation around the circle (x - 1)2 + (y - 6)2 = 12. Consider the solid-body rotation Ue = wor Ur = O. 23 Problem 3.10 24 10. Show that the vorticity field for any flow satisfies Problem 3.11 V..w=o. i 1. A flow field on the xy-plane has the velocity components U = 3x + y v = 2x - 3y. Show that the circulation around the circle (x - 1)2 + (y - 6)2 = 4 is 4rr. 12. Consider the solid-body rotation Ue = wor Ur = O. Take a polar element of dimension r de and dr, and verify that the circulat vorticity times area. (In Section i I we performed such a verification for a cir element surounding the origin.) 13. Using the indicial notation (and without using any vector identity) show the acceleration of a fluid particle is given by au ( I ) 25 Problem 3.11 26 i 1. A flow field on the xy-plane has the velocity components Problem 3.12 U = 3x + y v = 2x - 3y. Show that the circulation around the circle (x - 1)2 + (y - 6)2 = 4 is 4rr. 12. Consider the solid-body rotation Ue = wor Ur = O. Take a polar element of dimension r de and dr, and verify that the circulation is vorticity times area. (In Section i I we performed such a verification for a circular element surounding the origin.) 13. Using the indicial notation (and without using any vector identity) show that the acceleration of a fluid particle is given by au ( I ) at 2 a = - + V - q2 + w x u, where q is the magnitude of velocity u and w is the vorticity. 14. The definition of the streamfunction in vector notation is u = -k x V1/, 27 Problem 3.12 28 Ue = wor Ur = O. Take a polar element of dimension r de and dr, and verify that the circulation is vorticity times area. (In Section i I we performed such a verification for a circular element surounding the origin.) Problem 3.13 13. Using the indicial notation (and without...
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This document was uploaded on 02/28/2014 for the course PHYS 4200 at Columbia.

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