Bunsen Burner and Bernoulli

It occurred to me yesterday that the Bunsen Burner works by way of the Bernoulli effect. That says that when a gas flows, it has lower pressure. Thus when the hole at the bottom is opened, the air outside will have higher pressure than the flowing gas inside. The pressure difference pushes the air into the flowing gas so you get a mixture of air and gas at flowing up the middle of the Bunsen Burner. A proper gas/air mixture burns at a higher temperature hence the blue flame. I suspect that with the air hole closed the gas reacts with the oxygen around the outside so combustion is not as complete.