(5.1) |
Let be a fixed point in the - plane, and let and be two curves, also in the - plane, that join to an arbitrary point . (See Figure 5.1.) Suppose that fluid is neither created nor destroyed in the region, (say), bounded by these curves. Because the fluid is incompressible, which essentially means that its density is uniform and constant, fluid continuity requires that the rate at which the fluid flows into the region , from right to left (in Figure 5.1) across the curve , is equal to the rate at which it flows out the of the region, from right to left across the curve . The rate of fluid flow across a surface is generally termed the flux. Thus, the flux (per unit length parallel to the -axis) from right to left across is equal to the flux from right to left across . Because is arbitrary, it follows that the flux from right to left across any curve joining points and is equal to the flux from right to left across . In fact, once the base point has been chosen, this flux only depends on the position of point , and the time . In other words, if we denote the flux by then it is solely a function of the location of and the time. Thus, if point lies at the origin, and point has Cartesian coordinates ( , ), then we can write
(5.2) |
Consider two points, and , in addition to the fixed point . (See Figure 5.2.) Let and be the fluxes from right to left across curves and . Using similar arguments to those employed previously, the flux across is equal to the flux across plus the flux across . Thus, the flux across , from right to left, is . If and both lie on the same streamline then the flux across is zero, because the local fluid velocity is directed everywhere parallel to . It follows that . Hence, we conclude that the stream function is constant along a streamline. The equation of a streamline is thus , where is an arbitrary constant.
Let be an infinitesimal arc of a curve that is sufficiently short that it can be regarded as a straight-line. The fluid velocity in the vicinity of this arc can be resolved into components parallel and perpendicular to the arc. The component parallel to contributes nothing to the flux across the arc from right to left. The component perpendicular to contributes to the flux. However, the flux is equal to . Hence,
(5.3) |
(5.4) |
The vorticity in two-dimensional flow takes the form
(5.8) |
(5.9) |
When expressed in terms of cylindrical coordinates (see Section C.3), Equation (5.7) yields
(5.12) |