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

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

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

Pressure Forces on Surfaces

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

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

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

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

Net Force on Wall – Rectangular 𝒉 𝒅𝑨=𝑾𝒅𝒛 𝒅𝒛 𝑾

Force on Wall – Rectangular 𝒉 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

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

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?

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

Completely Submerged Objects 𝐹 𝑝 ?

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

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

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

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.

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.