Liquid velocity after a filter Consider a water piping system of constant flowrate and diameter. The water passes through a filter and experiences a pressure drop (https://www.orangeresearch.com/why-differential-pressure-gauges.php)

Online I see mentioned that there is an inverse relationship between pressure and velocity. Perhaps this is an oversimplification on certain sites of what i believe is the Bernoulli equation, but does this mean that the velocity of the water increase after the filter (which is quite hard for me to get my head around). I would like to know whether this is true or not
 A: The continuity equation must apply in this situation, because that equation represents conservation of mass.  For the general case,
$\rho_1 A_1 v_1 = \rho_2 A_2 v_2$
where $\rho$ represents the density of the fluid, $A$ represents the cross-sectional area of the pipe that the fluid is flowing through, and $v$ represents the velocity of the fluid.  Based on this equation, there are several cases to consider.
Looking at density
For an incompressible fluid where the temperature doesn't change across the filter, there is no change in density when the fluid experiences a pressure drop.  If there is a temperature change across the filter, there will be a density change across the filter.
For a compressible fluid (e.g., air), the density of the fluid will change when there is pressure drop and when there is a temperature change.
Looking at cross-sectional area
If the pipe diameter upstream of the filter equals the pipe diameter downstream of the filter, the cross-sectional areas of both pipes will be equal, which will not change the fluid velocity in the piping system.  However, if the pipe diameters are different, the cross-sectional area will be different and the filter entrance velocity will be different than the filter exit velocity.
