I have repaired a few bows, and used several types of glue including Loctite 420 for wood to glass and wood to wood. It is very thin. A slow curing epoxy can work, if it is thin enough. Titebond II is very strong, but thick.
If you are making the repair, yourself do a dry run without the adhesive. Use painters masking tape which can be removed several days later without curing. Cut two strips of rubber from an old bike inner tube, cutting through the sides will give you a top and bottom, cut so they lay flat without twisting. You might use two wood strips for compression on both sides. The strips should be narrower than the limb. About 1" x 1/4" and long enough to span the separation, but not overlap the edge of the bow. Ideally, the strips should be inside the width of the limb. The purpose of the wood strips is to equalize pressure when using the rubber to wrap the bow. Try wrapping from the center out to see if it compresses the separations before using the glue. The glue line should look like the others.
If the gap does not close, you could try to clear it out with a thin metal tool, paint scrapper, metal putty knife, or a dentist tool (pick).
When your ready to use the glue, make sure it is taped off to keep the glue off as much as possible. Work the glue into the end areas where there is some adhesion, and then towards the middle. You can keep it open with the scrapper while adding glue. Wrap with clear wrap, or wax paper, before laying on the wood strips and wrapping with the rubber. Wrap tightly from the center out towards the ends. Use tape to hold the ends of the rubber wrap. Use the second strip to wrap the other end from the center out. This will pushing glue into the end areas (important). Wrap it tight. You will get glue running out, so be prepared, cover the work area, and wear rubber gloves, face mask and work in a ventilated area.
Avoid contact with the Loctite 420 or epoxy.
Let it dry several days, in the sun, or where it is warm. If it is cold, let it dry much longer.
You can refer to the manufacturer's website for specifications on drying time at ideal temperature.
After it is dry, remove the rubber, wood compression strips, clear wrap/wax paper, masking tape and to see what work is need to refinish, or touch up the bow.
If this sounds like more than you want to do, then it would be better to contact a shop that repairs bows. Good luck.