Elementary Mechanics of Fluids CE 351Majed Abu-Zreig Pressure Shear ForcesNormal Forces (pressure) where F is a force normal to area A Absolute Pressure, Gage Pressure, and Vacuum Figure 3.3 (p. 34) Example of pressure relations Pressure in a vacuum isp = 0. Absolute pressure is referenced to perfect vacuum. Gage pressure is referenced to another pressure, typically atmospheric pressure (most gages measure relative pressures).Pressure Pressure is scalar quantity Fluid at rest can not support shear stress Normal stress on any plane is pressure (+ for compression) It acts normal to the surface and all direction x n n x xp p l y p l y p Fdirection X= A A A A = =o o sin sin 0Z - direction Divide through by to getNow shrink the element to a point:This can be done for any orientation o, so Pressure transmission (pascals law) In a closed system a pressure change produced at one point in the system will be transmitted throughout the entire system Hydraulic jack system EXAMPLE 3.1 LOAD LIFTED BY A HYDRAULIC JACK A hydraulic jack has the dimensions shown. If one exerts a force F of 100 N on the handle of the jack, what load, F2, can the jack support?Neglect lifter weight. Solution 1. Moment equilibrium 2. Force equilibrium (small piston) Thus 3. Force equilibrium (lifter) Note that p1 = p2 because they are at the same elevation (this fact will be established in the next section). Apply force equilibrium:

Pressure Variation with Elevation Static fluid pressure varies only with elevation in the fluid. o sin ) (0l A A p p A pF F FFweight right leftlA A A A + A = ==dldzdldplzlp o =AA = =AAor sin =dzdpPressure Variation with Elevation If is a constant =dzdpz pz z p pdz dpzzppA = A = =} }) (1 2 1 22121head) ic (piezometr constant = + = zphz = 0 1 2 z = z1, p = p1 z = z2, p = p2 z p A = A 2211zpzp+ = + Elevation head Pressure head Piezometric head Piezometric Head z = 0 1 2 1z1p /2p2zconstant = + zpOpen Tank Piezometric Head Pressurized Tank z = 0 3 1z1p2zconstant = + zp3z2p3p2 1 Example (3.6) Tank contins fluid with A > B Which graph depicts the correct pressure distribution?Hydrostatic Paradox Example Leaking Storage Tank Water has seeped in to the depth shown Find: Pressure at gas-water interface, pressure at bottom of tank Assume: Fluids are at rest water Gasoline SG=0.68 17 ft 3 ft 0 1 2 bbaazpzp+ = + psi ft lbf ppz z SG p pw01 . 5 144 / 721) 3 20 )( 4 . 62 )( 68 . 0 ( 0) (2111 0 1 0= = = = psi ft lbf ppz z p pz z p pww31 . 6 144 / 908) 0 3 )( 4 . 62 ( 721) () (2222 1 1 22 1 2 1= = + = + = = Pressure transmission (pascals law) In a closed system a pressure change produced at one point in the system will be transmitted throughout the entire system What is maximum force F2 that can be supported? Problem (3.13) 222 1112 1 1 22 1 2 11/ 500 , 142) 2 0 )( 9810 )( 85 . 0 () 04 . 0 (4200) () () (200m Nz z SGAFz z p pz z p pN Fwoiloil= + = + = + = = =tN FA p F119 , 1) 1 . 0 )(4( 500 , 142222 2 2== =tCompressible Fluids (i.e Gases) Density is nearly proportional to pressure constant and but = = = gdzdp For perfect gasses gRTpdzdpRTpRT p = = = Need to know T = T(z) Constant (Isothermal): Linear:.0const T T = =) (0 0z z T T = oCompressible Fluids Assume linear and integrate } =}zzppz z TdzRgpdp0 0) (0 0oRgTz z Tp z poo((

=00 00) () ( Assume constant and integrate } =}zzppdzRTgpdp0 0000) (0) (RTz z ge p z p =Temperature variation with altitude for the U.S. standard atmosphere