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A local 2-D model of a singular strain field for a finite time singular vortex was employed to investigate the effect of an external source of matter. This line source is assumed to be nested at the core of the singularity which can handle a forced 2-D singularity with infinite energy. Hence, there is no necessity to speak about any inner and outer region relative to a singular bihavior. Two different geometries of flows is considered with some solutions for each of them. Some of these solutions is shown to prevent the singularity to collapse on the source which is a new effect only due ti the source. A finite area of singularity around the source can be created. The effect of a line sink is also found on the singularity. Some other solutions turned out to possess a structure of maximums and minimums for the vorticity. These are the generalization of so called Burgers vortices.
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