 We're now going to take a look at a case where we have no fluid motion or a fluid at rest or at a constant velocity and this will enable us to come up with the hydrostatic pressure distribution equation. So this applies for fluid at rest or constant velocity. So by that that would mean for example a velocity field that looks something like this and with that the acceleration vector is zero and we can write del squared v is equal to zero as well. Consequently the equation that we had for a fluid with this the acceleration term goes away that goes away and we're left with the special case where the gradient of pressure is just the density times the gravity vector. Now typically gravity is oriented down unless you're doing space fluid mechanics. So what we can say is that g is minus gk and with that writing out the gradient terms what we have is dp by dx is zero, dp by dy would be equal to zero because there's no component on the right hand side in either the i or the j direction and finally dp by dz is the only term that would exist and that would equal minus rho times g the gravity vector or the scalar value of the gravity and with this that means that pressure is only a function of z so we can rewrite that as an ordinary differential and we get dp by dz is equal to minus rho g. So this is going to be the equation that we're going to work with for hydrostatic pressure distribution and what we'll do let me write that out again so we had dp by dz equals minus rho g and what we're going to do we're going to integrate this so let's rewrite that we would have dp on the left is minus rho g dz and we're going to integrate that so that would be p2 minus p1 is equal to the integral between two limits of rho g dz. Now notice I've written this out within the integral we have density as well as the gravity vector or the scalar value of gravity now typically for the problems that we're going to look at gravity won't change however density can change and so we'll look at two special cases of that in the next two segments but the integration so we're going to assume that the gravity scalar value g is going to be constant and not changing was the however density can and we're going to look at two special cases in the next few segments the first one will be a liquid and for a liquid for the most part the density does not change and the second thing we'll look at is gas and in particular we'll look at air now that is one where the density can change with elevation and we'll look at the values within the atmosphere so those are two special cases that we'll look at and we will apply that to this equation here and that will then enable us to get the hydrostatic pressure within fluids either liquids or gases