Two-dimensional flow

Consider flow in two dimensions only. The flow is the same as that between two planes set parallel and a little distance apart. The fluid can then flow in any direction between and parallel to the planes but not at right angles to them. This means that in the subsequent mathematics there are only two space variables, v and у in Cartesian (or rectangular) coordinates or r and в in polar coordinates. For convenience, a unit length of the flow field is assumed in the z direction perpendicular to x and y. This simplifies the treatment of two-dimensional flow problems, but care must be taken in the matter of units.

In practice if two-dimensional flow is to be simulated experimentally, the method of constraining the flow between two close parallel plates is often used, e. g. small smoke tunnels and some high-speed tunnels.

To summarize, two-dimensional flow is fluid motion where the velocity at all points is parallel to a given plane.

We have already seen how the principles of conservation of mass and momentum can be applied to one-dimensional flows to give the continuity and momentum equations (see Section 2.2). We will now derive the governing equations for two-dimensional flow. These are obtained by applying conservation of mass and momentum to an infinitesimal rectangular control volume – see Fig. 2.8.