When double emulsion droplets flow through a tapered nozzle, the droplets may break up and cause the core to be released. We model the system on the basis of the capillary instability and Show that a droplet will not break up when tilt angle of the nozzle is larger than 9 degrees. For smaller tilt angles, whether the droplet breaks up also depends an the diameter ratio of the core of the droplet to the orifice of the nozzle We verified this mechanism by experiments. The ideas are useful for the design of nozzles not only to break droplets for controlled release but also to prevent the droplet from rupturing in applications requiring the reinjection of an emulsion.