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To prevent the swirhng motion, devices consisting of specially shaped blades may be placed at the outlet or in the approach to it. We assume t h a t in the stream there is an isolated vortex tube of finite dimensions such t h a t outside it the angular velocity of the fluid particles is zero. Clearly, in this case Stokes's theorem holds not only for a contour lying on the surface of the tube but for any other contour which encircles the tube once. If (Fig. 14) several isolated vortex tubes are given in space with strengths iv i2i.

Ir, c c (51) which expresses Kelvin's theorem stated above. From Kelvin's theorem it follows that the circulation along a closed contour, or according to Stokes's theorem the strength of the vortex tube, changes, in the general case, with time and depends on the distribution of acceleration in the fluid. Recalling Newton's second law which relates acceleration to the applied forces, we conclude that the change of the strength of the vortex tube with time depends on the distribution of the forces applied to the fluid.

Charnyi, S. N. Numerov, G. I. Barenblatt and others. L. S. Leibenzon originated a theory of the flow of gas through a porous medium and solved a number of problems in the theory and practice of oil extraction. Steady flow of aerated liquids was studied by Kristianovich. During the last fifty years, developments in the theory of viscous motion were concerned mainly with the study of the flow in the boundary layer which is formed near the surface of a body under most circumstances of practical interest.

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