Extension and Complex Flows

Suppose we have designed our geometry such that the radius r is proportional to the length / via a constant k  

In our thought experiment we can force fluid to flow through this die at a certain volume per unit time or volumetric flow rate Q and allow no drag with the walls. This is a total slip condition so no shear will be applied to the fluid. The fluid will travel faster as the geometry contracts so the velocity V(l) in a plane across the geometry simply depends upon the area at any length /  

Now the extension rate e in the geometry is given by the rate of change of velocity with distance  

Evaluating the integral and using the extension rate we obtain the following expression  

In the opposed jets design fluid is sucked or pumped into a beaker. The profile which develops is dominantly extensional. In the profiled slot design a rectangular channel is designed such that in the total slip condition an extensional flow develops with a constant rate. The pressure is measured at the stagnation point. Other designs include the open syphon, where fluid is sucked from a beaker through a nozzle which is 

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