Consider pressure forces acting on an elemental prism volume of an incompressible fluid at rest. Prove that the pressure in any horizontal plane through the prism is constant, but through any vertical plane is not.
Work:
The sum of forces Fx in the x-direction = 0
The sum of forces Fy in the y-direction = 0
The Sum of forces Fz in the z-direction = 0
Fx = Fy = 0
Fz = N - g = 0, where N = normal force and g = gravitational force = 9.8 m/s^2
N = g
W = (integral) p*dA, where W = weight = specific weight*dx*dy*dz, p = pressure, and A = area
p = W/A
Solution
dp/dx = dp/dy = 0
dp/dz = -ro*g, where ro = density of fluid