problem 1.1

Pattern of ink-drift

Suppose that some amount of water is contained in a vessel, and the water is set in motion and its horizontal surface is in smooth motion. Let a liquid-drop of Chinese ink be placed quietly on the flat horizontal surface maintaining a flow with some eddies. The ink covers a certain compact area of the surface.

After a while, some ink pattern will be observed. If a sheet of plain paper (for calligraphy) is placed quietly on the free surface of the water, a pattern will be printed on the paper, which is called the ink-drift printing (Fig. 1.5). This pattern is a snap-shot at an instant and consists of a number of curves. What sort of lines are the curves printed on the paper? Are they stream-lines, particle-paths or streak-lines, or other kind of lines?

Figure 1.5: Ink-drift printing.

problem 1.2

Divergence operator div

Consider a small volume of fluid of a rectangular parallelepiped in a flow field of fluid velocity v = ( v _x , v _y , v _z). The fluid volume V changes under the straining motion. Show that the time-rate of change of volume V per unit volume is given by the following,

problem 1.3

Acceleration of a fluid particle

Given the velocity field v( x , t) with x = ( x, y, z) and v = ( u, v, w