Chapter 2 Pressure Distribution in a fluid ( Fluid Statics Basic )

1. Chapter 2Pressure Distributionin a fluid (Fluid StaticsBasic) White: Chapter 2 潘锦珊： 第一章

2. Definition: Unit: (SI) (Pound per Squire Inch) Properties of Pressure • Vertical to the surface and point into it. • At any point, pressure is independent of orientation. Pressure

3. (up) When Verification: At any point in a static fluid, pressure is independent of orientation. Forces on left and up surface:

4. Fluid Dynamics Fluid in motion Fluid Statics Aerodynamics Fluid at rest Fluid Mechanics

5. Applications: • Dams（水坝） • Buoyancy related instrument（利用浮力的装置） • Fluid power system（液压驱动系统） • Connected vessel （连通器） • …… § 2.1 Fluid @ rest • Pressure is the only surface force. • Pressure distribution relates to body force only.

6. Consider a cube in a static fluid Pressure at the center is p; Body forces are dz dy dx § 2.2 Equilibrium of a Fluid Element

7. Pressure: Body force: dz dy dz dx dy dx

8. In x direction: Force on left surface Body force in x direction Force on right surface Euler Equilibrium Equations (Euler 1775)

9. Equipressure surface（等压面） Surfaces in fluid with same pressure, vertical to body force everywhere, in gravity field it is a horizontal plane. Pressure increase in the direction of body force.

10. 1 Basic rule: z p0 h z0 z General solution x 2 Boundary condition: § 2.3Pressure Distribution under Gravity

11. Pressure at free surface Pressure due to weight on top equipressure surface（等压面） Any point with the same depth h under free surface has the same pressure. Free surface is an equipressure surface

12. Pressure measurement p0 Absolute pressure （绝对压力） Relative pressure（相对压力） Gauge pressure（表压力） Vacuum degree（真空度） h p h+p0(mmH2O) Pressure source Connected water tube h (mmH2O) Water manometer （水柱压力计） Application ? pA 绝压 pG 表压

13. Homework: P2.7

14. p0 α h hc x y A C yc y x Centroid（形心，重心） § 2.4 Hydrostatic Force on Plane Surface Find total force P

15. p0 α h hd hc x y A C yc y D yd x Center of pressure Is the center of pressure at centroid ? hc:depth of centroid The force on a submerged plane equals the pressure at the plate center times the plate area, independent of the shape of the plate or the angle .

16. Moment to x axis p0 α h hd hc x y A C yc y D yd x

17. Example • The gate is 5m wide,is hinged at point B,and rest against a smooth wall at point A. • Find • The force on the gate exerted by seawater pressure, • The horizontal force Px exerted by the wall at point A

18. (b) Lc L Solution: (a) Centroid: 3m above B

19. Hoover Dam Channel § 2.5 Hydrostatic Forces on Curved Surfaces • Select a dA and find the three forces on it • Integration

20. x O 1. Horizontal forces: Ax A z O 2. Vertical forces: V z Conclusion: x

21. y 45o H  R O x F Example: Find the forces acting on the hemi-spherical covers. Solution:

22. Gravity Body force Inertia Force § 2.6.1 Uniform Linear Acceleration （恒加速度直线运动） x a X= -a g § 2.6 Fluid in rigid body motion

23. Euler Equilibrium Equations Boundary condition: At free surface（自由液面） Equipressure surface（等压面）

24. It will not spill out ! Example: A cup of coffee is 7cm deep at rest. 1. Whether it will spill out while ax=7m/s2? 2. Gage pressure of point A? Solution:

25. z Body force: ω2r g f O y ω2y f R ω2x r y θ x x O ω § 2.6.2 Rigid body rotation （整体旋转） Equipressure surface: dp=0

26. z Free surface: ω2r z0 g f O 旋转抛物面体的体积是同底面积和高的圆柱体积的一半。 Parabola dish （抛物面） How to find z0?

27. Homework: P2.64 P2.97 (selective) P2.147 P2.152