Continuity Equation - For a vertical pipe

[hermish]

I don't understand the ideas behind the continuity equation when applied to a vertical pipe. In all the questions I see regarding a vertical pipe of constant diameter, I see that the fluid's velocity will remain constant while traveling through the pipe. Common sense will tell you this isn't true despite what the formula tells you. The fluid will accelerate because of gravity, giving it a different velocity the further you go down the pipe.

I'm well aware of the two assumptions required for the continuity equation : 1. Steady Flow and 2. Incompressible liquid - and I'm sure this isn't relevant to any acceleration due to gravity.

Can someone please explain this?

Thanks for the help!

 

[lendav_rott]

Is this a concrete amount of fluid that is poured down the pipe or do you keep pouring and pouring, increasing the total amount of it as time passes by? If you visualise this in your head, say you have a small diameter pipe of like...0.5 cm? Let some amount of fluid into it, the speed doesn't increase, it moves the same all throughout the pipe but ofcourse, there are different kinds of fluid, I always thought it has also to do with the viscosity of it.

 

[Chestermiller]

Is this a concrete amount of fluid that is poured down the pipe or do you keep pouring and pouring, increasing the total amount of it as time passes by? If you visualise this in your head, say you have a small diameter pipe of like...0.5 cm? Let some amount of fluid into it, the speed doesn't increase, it moves the same all throughout the pipe but ofcourse, there are different kinds of fluid, I always thought it has also to do with the viscosity of it.

You're right about the viscosity. The fluid is not all traveling downward at the same rate. The downward velocity at the wall is zero, and the downward velocity at the center of the tube is maximum. This velocity gradient with respect to radial position times the viscosity translates into a shear stress in the fluid. The shear stress distribution supports the weight of the fluid, so that there is no downward acceleration (aside from the entrance effects). The net downward force on each parcel of fluid is zero, so it travels downward at constant velocity.