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EML 5709 Homework 2 Spring 1995

1. Find the streamlines, pathlines, and streaklines for the flow with velocity field

   

   Sketch a bundle of streamlines at a typical time t=0.5 both in 3D and their projections onto the x,y- and z,x-planes. Sketch the projections of a bunch of path lines onto the x,y-plane. Sketch the projection of a typical pathline on the z,x-plane and in 3D. Draw the streakline generated by a smoke generator at the point (1,0,0) both at time t=0 and at .

2. Find the streamlines, pathlines, and streaklines for the flow with velocity field

   

   Describe and draw the three families of lines.

3. Find the streamlines and pathlines for the flow with velocity field

   

   Describe and draw both families of lines. Also find and draw the streakline generated by a smoke generator positioned at x=1, y=2, for time t=0.

4. Find the streamlines, pathlines, and streaklines for the flow with velocity field

   

   where Q>0 and are constants. Express in the form of two equations for the three variables .

5. Make question 2.3 in the book.
6. Make question 2.6 in the book. Compute the vorticity for both flows. Integrate the velocity around a circle of radius r around the origin and check Stokes theorem. Comment.
7. Compute the divergence of the velocity for both flows listed in question 2.6 in the book and for

   

   Integrate the volume of fluid flowing out of a cylinder around the z-axis normal to the -plane, taking the radius of the cylinder r and the height 1, and check the divergence theorem. Comment.

8. Consider flow through a pipe of arbitrary cross-section and length. Show that the product of cross sectional area times average axial component of vorticity, , is constant along the pipe, the same way that the volume flow u A is constant for an incompressible flow. What is the value of the constant?

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