Lets say if i have a test tube (diameter 3cm and height 30cm) seal with a rubber. in the test tube there 100ml of water = around 10 cm height. if air was flow into the test tube from bottom of the tube with flow rate around 20L/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.

Bernoulli's equation, perhaps?

http://hyperphysics.phy-astr.gsu.edu/hbase/pber.html

 

Because the fluid's speed decreases when it goes from the small tube into the test tube, pressure must increase in order to conserve energy.

Lets say if i have a test tube (diameter 3cm and height 30cm) seal with a rubber. in the test tube there 100ml of water = around 10 cm height. if air was flow into the test tube from bottom of the tube with flow rate around 20L/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.

 

The pressure inside the tube must be higher than the outside pressure in order for flow from inside to outside to exist.

 

To a first order, the flow rate will be proportional to the pressure difference. See Poiseuille's Law. http://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation