Looking for an equation to find resulting pressure

[TSN79]

Picture a pipe reduction from a diameter A down to B. I'm looking for an equation that can tell me the resulting pressure and velocity out of B from input data at point A such as flow and pressure. I've would really appreciate it :)

 

[FredGarvin]

Bernoulli equation.

If you want to get more detailed with the pressure losses, you'll need to have a source for resistance coefficients of reducers, etc...