Consider the following steady, three-dimensional velocity field:                        V-›= (u, ? , w)                                                                                                              = (2.5 + 2.0x − y) i \-›+ (2.0x − 2.0y) j-›+ (0.8xy)k-›Calculate the vorticity vector as a function of space (x, y, z).

A

a. Given the velocity field u=(u,v,w) in Cartesian coordinates with u=2x+y, v=2zt, w=0. i. Find the equations of the corresponding streamlines (Eulerian concept) ii. Find the equations of the corresponding particle paths, i.e., the pathlines (Lagrangian concept). b. Show that the Vu=0 everywhere implies that volumes are conserved, i.e., the volume of red particles at t-0 is the same as at t=t. Hint: Write out what you must prove and use the theorems to get there.

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