This preview shows page 1. Sign up to view the full content.
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 xyplane 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 solidbody 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 xyplane 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 solidbody 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 xyplane 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 solidbody 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...
View
Full
Document
This document was uploaded on 02/28/2014 for the course PHYS 4200 at Columbia.
 Spring '14
 MichaelMauel
 Physics

Click to edit the document details