I sometimes fly to Europe, arriving late morning, and go direct to a meeting. Even though I've slept on the plane, the eyes may start to drop at about 3pm. At that stage, I excuse myself, find a quiet office, and put my head down for about half to one hour (don't get too comfortable). After that I'm right to go for another few hours. It's better than falling asleep and being totally non-productive in the meeting, and when explained, people understand. Note, I would never attempt this after flying economy.
My other tip is, if you find yourself in bed still awake at midnight after a long flight, take a sleeping pill.