# Using the continuity equation against gravity

Studying Fluid Mechanics right now and in my textbook there is an example of getting water up to a bathroom in a house. We're given the diameter of the inlet pipe and bathroom pipe, but only the velocity at the inlet pipe. Why does the continuity equation apply if gravity is accelerating the water down? The pipe is getting smaller so velocity increases, but then velocity also decreases because gravity is acting against the water flow.

• Continuity equation is the conservation of mass -- why would mass be created or destroyed just because there is gravity? – tpg2114 Apr 6 '15 at 0:15
• @tpg2114 My thought process was that because the velocity decreases, dV/dt is also decreasing, which is what the continuity equation is. Where am I wrong? – malabeh Apr 6 '15 at 0:18
• I think my answer addresses it -- $\partial V /\partial t$ is not changing (assuming you mean $V$ is volume) because water is typically assumed to be incompressible. Even if they didn't say that outright, that is probably what they are assuming. – tpg2114 Apr 6 '15 at 0:21

This started as a comment but I will flesh it out some. The continuity equation is:

$$\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_i}{\partial x_i} = 0$$

and is an expression of the conservation of mass. Effectively it is saying "What comes in, must go out, or density must increase/decrease accordingly."

The example your book gives is using water. It is probably (without telling you as much) assuming that the water is incompressible. This is a good assumption, but nothing is truly incompressible.

Anyway, if it is incompressible then the density cannot increase nor can it decrease. So you are left with "What comes in, must go out." That is why you can still use the 1D simplification of $A_1 u_1 = A_2 u_2$ which is likely what your book is doing.

• Ok, so if there is a pipe of constant radius, standing vertical, the mass of water going in is the mass of water going out. That would mean pressure is being converted into gravitational potential energy? This is due to A and v being equal at the top and the bottom of the pipe. Am I right? – malabeh Apr 6 '15 at 0:24
• Pressure is not an energy so it can't be converted to energy. But the work the pressure is doing is becoming potential energy, yes. Pressure is a force (per unit area) and shows up in the momentum equation as $-\partial P/\partial x_i$ so for fluid to move, you need a gradient in pressure. This gradient in pressure can be a pump at the bottom of a pipe increasing the pressure and atmospheric at the top of the pipe so the water flows up. It could also be due to the weight of water at the top of a pipe and atmospheric at the bottom, which is how water towers work. – tpg2114 Apr 6 '15 at 0:27
• None of which, by the way, has anything to do with the continuity equation. The only terms relating to pressure and gravity show up in the momentum and energy equations (and the energy equation may not have it, if you lump potential energy in with internal energy). – tpg2114 Apr 6 '15 at 0:28
• @malabeh And also, because I remember when I was just starting in fluids and got all confused, velocity happens because of pressure gradients, not the other way around! Low pressure isn't because the flow decided to move faster -- the flow decided to move faster because the pressure is lower. If there are no pressure gradients, there is also no velocity. – tpg2114 Apr 6 '15 at 0:31