curiosity about pressure difference

 lets say if i have a test tube (diameter 3cm and height 30cm)close with a rubber. in the test tube there 100ml of water= around 10 cm height. if air was put into the test tube with flow rate around 20mL/min, and after a few sec, the rubber was push out. in general i understand that the pressure inside the test tube is higher than atmospheric pressure. but i cant find any mathematical solution that relate flow rate and pressure to proof that the pressure inside the test tube is higher than outside. can anyone explain it to me