3.1 Complex Representation of Fluid Motion

Fluid motion that is the same in all planes parallel to the xy plane, say, with no velocity in the z direction, is said to be two dimensional, with all flow quantities dependent on only x, y, and possibly the time t. Fully three-dimensional flow problems can often be regarded as locally two dimensional in small, confined neighbourhoods, and the two-dimensional solutions can sometimes be pieced together to provide a useful representation of the three-dimensional flow. Furthermore, considerable insight into properties of more general flows can frequently be obtained by a study of the solutions of analogous two-dimensional flows. Many of these solutions can be obtained in analytic closed form by methods based on the theory of complex variables; these methods are studied in this chapter for an ideal, incompressible fluid.

3.1.1 The stream function

Let the velocity v have components u, ? in the x and y directions, respectively. For incompressible motion, the equation of continuity

can be satisfied by introduction of the stream function ?( x, y), where

Consider two points, 1 at ( x ₁ , y ₁) and 2 at ( x ₂ , y ₂), lying on curve C of Figure 3.1.1. The volume flux from left to right through C (per unit span , i.e. per unit length normal to the plane of the paper) is ? _C v n ds, where s is arc...