Pressure in liquids

There were some queries a while back about liquids and the Venturi effect; why pressure is lower when a fluid is compressed through a smaller aperture, etc.

This is easier to get a grip on if you think about density. Density and pressure are related - if the atmosphere was denser, then pressure at the earth's surface would be higher, because there would be more air above it pushing downward.
Likewise pressure increases the deeper you go in fluids, but density in a fluid is constant. If this wasn't true water would get "thicker" the deeper you went. Since it doesn't - water has the same density at 3 miles down as at the surface - pressure is entirely gravitational. The relation of depth to pressure is linear because water is a homogenous density 'surface' or manifold.

What "manifold" means here is that fluids (in fact all forms of matter) are folded up - they are able to 'fold' density into themselves and remain incompressible. Molecules in a fluid translate away from applied pressure and dissipate any - they 'fold' along lines of intermolecular interaction, so this is the underlying 'pressure and density' structure.

Since density is constant it's divergence-free and solenoidal (q.v). Velocity then 'measures' pressure/density. Pressure changes because density doesn't.
For the relation to stay equal. velocity increasing means pressure decreasing. This can be a little nonintuitive because it's easier to understand compression (and so increases in density) in a gas.