1. 4. Pressure Forces, Buoyancy
CH EN 374: Fluid Mechanics

2. Isotropy Isotropic = same in all directions
Pressure at any point of a fluid is isotropic

3. Pressure Forces on Surfaces
Isotropy means pressure forces from fluids cancel
Except for a line normal to the surface
So pressure forces are always normal to surfaces

4. Pressure Forces on Surfaces

5. Types of Pressure Force Problems
1. Boundaries—one side of a container or wall
2. Submerged objects

6. Submerged Surface – Force on Floor
P is constant!
𝐹=𝑃𝐴 𝑨 𝒉
𝑃= 𝑃 0 +𝜌𝑔ℎ
𝐹= 𝑃 0 +𝜌𝑔ℎ 𝐴

7. Submerged Surface – Force on Wall
P is not constant!
𝐹=𝑃𝐴 𝑨 𝒉
𝑃= 𝑃 0 −𝜌𝑔𝑧
𝑑𝐹=𝑃𝑑𝐴

8. Net Force on Wall and Atmosphere
𝑃= 𝑃 0 −𝜌𝑔𝑧
𝑃= 𝑃 0
Net pressure on wall: 𝑃 𝑛𝑒𝑡 =𝜌𝑔𝑧
Pressure forces from atmosphere cancel out

9. Net Force on Wall – Rectangular𝒉
𝒅𝑨=𝑾𝒅𝒛 𝒅𝒛 𝑾

10. Force on Wall – Rectangular𝒉 𝒅𝒛 𝑾
𝒅𝑨=𝑾𝒅𝒛
𝑑𝐹=𝑃𝑑𝐴
𝑃 𝑛𝑒𝑡 =−𝜌𝑔𝑧
𝑑𝐴=𝑊𝑑𝑧
𝑑𝐹=−𝜌𝑔𝑧𝑊𝑑𝑧
𝐹=− 1 2 𝜌𝑔 𝑊ℎ 2
0 𝐹 𝑑𝐹 =−𝜌𝑔𝑊 0 −ℎ 𝑧𝑑𝑧
𝐹= 1 2 𝜌𝑔ℎ𝐴

11. Force on a Rectangular Wall
𝐹= 1 2 𝜌𝑔ℎ𝐴
Pressure Halfway Up Wall =
Average Pressure

12. Problem
In a mudslide, mud with density 𝜌=2000 𝑘𝑔/ 𝑚 3 piles up against a retaining wall. What is the net force on each 1 m the wall when the mud is 1 m deep?

13. Problem 𝑩
A triangular canal is crossed by a retaining gate shown to the right. The gate is also a triangle with height H and width B (for base) at its widest point. Water can be assumed to come up to the top of the gate. Find the net horizontal pressure force. 𝑯

15. Completely Submerged Objects
𝐹 𝑝 ?

16. Completely Submerged Objects
𝐹 𝑝 =𝜌𝑉𝑔
𝑊=𝑚𝑔
=𝜌𝑉𝑔

17. Completely Submerged Objects
What happens if
𝜌 𝑜 > 𝜌 𝑓
𝜌 𝑜 = 𝜌 𝑓
𝜌 𝑜 < 𝜌 𝑓
𝐹 𝑝 = 𝜌 𝑓𝑙𝑢𝑖𝑑 𝑉𝑔
𝑊= 𝜌 𝑜𝑏𝑗𝑒𝑐𝑡 𝑉𝑔

18. Partially Submerged Objects
𝐹 𝑝 = 𝜌 𝑓 𝑉 𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 𝑔
𝑊= 𝜌 𝑜 𝑉 𝑡𝑜𝑡𝑎𝑙 𝑔
𝜌 𝑓 𝑉 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 𝑔= 𝜌 𝑜 𝑉𝑔

19. Problem
It is estimated that 90 percent of an iceberg’s volume is below the surface, while only 10 percent is visible above the surface. For seawater with a density of 1025 kg/m3, estimate the density of the iceberg.

20. Problem
The hull of a boat has a volume of 180 m3, and the total mass of the boat when empty is 8560 kg. Determine how much load this boat can carry without sinking in a lake with water density about 1000 kg/m3.