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.
[Physics] Using the continuity equation against gravity
fluid dynamics
Best Answer
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.