Hydrostatic Laws , Units and scales of pressure measurements

Ex1: A closed tank contains 1.5 m of SAE 30 oil, 1 m of water, 20 cm of mercury, and an air space on top, all at 20?C. If pbottom ??60 kPa, what is the pressure in the air space? ?oil=8720N/m3, ?water=9790N/m3, ?mercury=133100N/m3.
Solution: Apply the hydrostatic formula down through the three layers of fluid:

P bottom = Pair + ?oil * hoil + ?water *hwater+ ? mercury *h mercury
60000 Pa= Pair + (8720 N/m )(1.5 m)+ (9790)(1.0 m) +(133100)(0.2 m)
Solve for the pressure in the air space: Pair ?10500 Pa Ans.
This means: 1) There is no pressure change in the horizontal direction. 2) There is a vertical change in pressure proportional to the density, gravity and depth change in the fluid (i.e. the weight of the column of the fluid above the point). Note: in the limit as the fluid wedge shrinks to a point, ?z goes to zero, we have:
Px=Pz= Pn=P Thus, pressure in a static fluid is a point property. Pressure force on a fluid element
Assume the pressure vary arbitrarily in a fluid, p=p(x,y,z,t). Consider a fluid element of size ?x, ?y, ?z as shown in Fig. 2. The net force in the x?direction is given by:
dFx=Pdydz- (P+?P/?x dx)dydz=-?P/?x dxdydz