《流体力学》课程教学资源：（英文版）Fluid statics

Fluid statics Study of fluid at rest i e. not in motion or not flowing On any plane, shear force is zero(no velocity gradient or shear deformation hence only pressure forces exist Sx ss Pasca’sLaw For fluid at rest, pressure at a point is the same in all direction

Fluid Statics • Study of fluid at rest, i.e. not in motion or not flowing • On any plane, shear force is zero (no velocity gradient or shear deformation) hence only pressure forces exist Pascal’s Law • For fluid at rest, pressure at a point is the same in all direction P = Px = Py = Pz

Proving pascal’sLaw P,8 δxδz 6 As element is in equilibrium ∑F,=0 P、δxδz=P6 x SS Sin0 Ss sin e= sz Hence Similarly,forΣF=0

Proving Pascal’s Law As element is in equilibrium : Fy = 0 Py x z = Ps x s Sin  s Sin  = z Hence : Py = Ps Similarly, for Fz = 0 Pz = Ps

Basic hydrostatic equation for Pressure field Rectangular free body P+↓↑ of top area a W P+ dP Body of fluid in equilibrium ∑F,=0 PA+W=(P+ dP).A .A+pgadh=(P+ dP)a dP/dh=pg=y dP/dy =-pg-Y

Basic Hydrostatic Equation for Pressure Field Body of fluid in equilibrium :  Fy = 0 P.A + W = (P + dP) . A P.A +  g A dh = (P + dP) A dP/dh =  g =  dP/dy = -  g = -   h dh P P + dP W Rectangular free body of top area A y

Incompressible fluid Pressure difference between two points in a body of fluid dP/dh=pg dP=pg dh Integrating from P, to P2 dp=p gdh h If fluid is incompressible and homogeneous p=constant 2-PI=pg(h2-h1=pg(Ah)

Incompressible Fluid Pressure Difference between two points in a body of fluid dP/dh =  g dP =  g dh Integrating from P1 to P2 : If fluid is incompressible and homogeneous  = constant P2 – P1 =  g (h2 – h1 ) =  g ( h)  P1 P2 h1 h2 h   = 2 1 2 1 h h P P dP  gdh

Incompressible fluid If Pi is on free water surface, i.e. PI=Pat P=P 2-Pi=pg h P P-P atm p g P2.abs =pgh+P atm For gauge pressure measurement, Patm=0 gauge pg h + P ,ga auge Gauge pressure at a point h below free surface =pg h

Incompressible Fluid If P1 is on free water surface, i.e. P1 = Patm : P2 - P1 =  g h P2 - Patm =  g h P2,abs =  g h + Patm For gauge pressure measurement, Patm = 0 P2,gauge =  g h P2,abs = P2,gauge + Patm Gauge pressure at a point h below free surface =  g h  h P2 P1=Patm

Definition of pressure head Pressure=P Datum Pressure head at point=Plpg(m) Piezometric head=p/ g +y (m) Piezometric pressure =P+pgy h In fluid statics, gauge pressure head at depth h below free surface Lpg=pgh/pg-h

Definition of Pressure Head Pressure head at point = P/ g (m) Piezometric head = P/ g + y (m) Piezometric pressure = P +  g y In fluid statics, gauge pressure head at depth h below free surface = P/ g =  g h /  g = h Datum Pressure = P y  h

Example I KN 0.2m 0.05m 2.51 dial di lam P1=F/A=1000/(x0.0252)=5093kPa 2=P1-pg(2.5)=48477kPa f=PA =48477x兀0.12=1523kN

Example P1 = F/A = 1000 / ( 0.0252 ) = 509.3 kPa P2 = P1 -  g (2.5) = 484.77 kPa F = P2 A2 = 484.77 x  0.12 = 15.23 kN 2.5m 1 kN F ? 0.2 m diam 0.05 m diam P1 P2

Pressure at liquid interface Liquid 1 Liquid 2 p2 Liquid 3 p 1=Pgh 22=pgh+ p2gh2 P3-P1gh1 p2gh2 pgh

Pressure at liquid interface P1 = 1gh1 P2 = 1gh1 + 2gh2 P3 = 1gh1 + 2gh2 + 3gh3  Liquid 1 1 Liquid 2 2 Liquid 3 3 h1 h2 h3 P1 P2 P3

Pressure measurement Absolute Pressure: Measured relative to perfect vacuum Perfect vacuum: 0 absolute pressure Gauge Pressure: Measured relative to local atmospheric pressure If no specified, pressure reading is usually assumed to be gauge Gauge pressure is positive if higher than atmospheric pressure, and negative if lower than atmospheric pressure Negative gauge pressure is also known as suction or Ⅴ acuum pressure

Pressure Measurement • Absolute Pressure : Measured relative to perfect vacuum • Perfect vacuum : 0 absolute pressure • Gauge Pressure : Measured relative to local atmospheric pressure • If no specified, pressure reading is usually assumed to be gauge • Gauge pressure is positive if higher than atmospheric pressure, and negative if lower than atmospheric pressure • Negative gauge pressure is also known as suction or vacuum pressure

Atmospheric Pressure ↓↓↓4↓ Mercury Atmospheric pressure is usually measured using a mercury barometer Patm =phg g ha 101, 300 N/m2 abs 0N/m2 gauge ≈1.013 bar abs ≈760 mm hg abs ≈10.3 m water abs

Atmospheric Pressure Atmospheric pressure is usually measured using a mercury barometer Patm = ρHg g h  101,300 N/m2 abs = 0 N/m2 gauge  1.013 bar abs  760 mm Hg abs  10.3 m water abs