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For this you will need anything that could plausibly pass as an artery. I used rubber tubing, but at a stretch you could use long macaroni, or even a drinking straw.
In order to make this look most convincing, you want to apply the 'skin' first, let it dry, then tear it and stick the artery under later. This will prevent you looking like you have a massive excess of neck skin.
Paint on 2 to 3 layers of latex covering a fairly large (Shaved!) area of the neck, with a single ply of thin tissue paper in between to add some strength and manageability to the latex. Take care to tease the fibres apart at the edge of the tissue to avoid an obvious join. Allow this to dry until the latex is not sticky to the touch.
While the latex is drying, prepare the 'jugular'. If using plastic tubing, you may need to apply something to help the blood stick, acrylic paint, or even greasepaint can work for this. Make two equal lengths, leaving about an inch to go under the 'skin', and an inch protruding.
If possible with what you are using, bend it into an 'r' shape, so that when lying against the skin, it will stick out. If you are using plastic, a candle may help make it more malleable, but watch out for fumes, melting the plastic or burning your fingers. When this is done, cut a cross section across the inch that will be under the skin, so the pipe will gradually emerge from your neck.
When the latex is dry to the touch, pick a small hole in the centre, where you want the artery protruding, then carefully cut a tear in it, running across the neck. Then pull back the 'skin' above and below using your finger to get underneath as far as the tubing needs to go in. Using either spirit gum or latex, paint the hidden part of the artery and the neck under the latex 'skin', and glue the artery in place under the skin. Push the rest of the 'skin' back down around the artery, so there is no loose skin left.
Rub some black and red greaspaint into the wound, and conceal any edges of latex with bruising.
Carefully put blood into the tubing, into the cut, and let a fair bit dribble down the neck.
For a more advanced effect, make the bottom tube extend down to a hidden blood-squirting device concealed under the jacket, and pump it out at select moments.
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