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The situation when the mass is created in the whole bulk of the fluid is rather unusual. Usually the density A of a distribution of sources is non-zero only in some localized volume L\ V, while it vanishes in the rest of the fluid. If one has no interest in the details of motion inside the volume L\ V and studies the motion only outside it then the useful mathematical idealization of a point source of mass can be introduced. This can be done by means of a limiting procedure. Consider the simple case when in the fluid of homogeneous density A = const =F 0 only in the small volume L\ V, while A = 0 outside this 48 INTRODUCTION TO VORTEX DYNAMICS region.

Instead of salt and fresh water, one can use immiscible liquids to create a two-layer stratification. 5 g cm - 3) carbon tetrachloride (CCI 4 ) with low viscosity (v*=5 x 1O- 3 cm 2 s- 1) and lighter but more viscous aqueous glycerol. The viscosity of an aqueous solution of glycerol depends strongly on the concentration of glycerol and can easily be varied over a wide range. A suitable value for our experiments is v = (3-5) X 10- 2 cm 2 S-l, corresponding to a dilute solution. 5 cm) layer of the prepared solution lying on a thick (2-3 cm) layer of carbon tetrachloride allows one to generate a nearly two-dimensional flow in the upper layer.

There are many reasons why it is convenient to describe rotational flows in terms of the vorticity. 14): oco -+ ot Vx(u·V)u=v~co. INTRODUCTION TO VORTEX DYNAMICS 54 Note that the pressure is eliminated in this operation because the curl of a gradient vanishes. Since the divergence of a curl also vanishes, the vorticity of any flow must satisfy V· w= 0. 21) which is called the Helmholtz equation for the vorticity. In planar flows when u = (u 1, U2' 0) and the motion does not depend on the third coordinate X3 the vorticity vector has only one component w = (0,0, w), normal to the plane of motion.