Let's say water is flowing from an arbitrary Source to some arbitrary Sink via a small pipe:
I'm curious as to what actually happens when we change the diameter of the pipe:
If all other things are held constant, does the speed of the water increase through the pipe increase, or does only the volume increase?
BTW, this is not homework I'm a 30-yr old software developer/homeowner trying to figure out a problem with my plumbing.
Consider the fluid in the source tank to be of density $\phi$. Let the height of the source tank be $h$ and the length be $l$. Pressure i.e force per unit area exerted by the fluid molecule is directly proportional to the volume of the fluid. Let the opening of the source tank be of diameter $x$ (assume it to be at the bottom most part of the tank) such that $a+x=h$, here $a=h-x$ i.e height of the fluid portion above the opening.
Thus, $$\text{P } \alpha hl \implies F/A=k(hl)\implies F/A=k(a+x)l=kal+klx$$
Here constant $k$ depends on the density $\phi$ of the fluid.
From the above expression it follows that the force per unit area is directly proportional to the diameter of the opening. If force per unit area increases, speed at which fluid flows also increases.
