Fluid Statics
Analyze fluid at rest, including pressure variation in a static fluid and forces on submerged surfaces.
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Why it matters
Hydrostatics determines loads on gates and tanks, pressure readings and buoyancy. Begin by choosing a pressure reference and defining vertical depth.
Key ideas
A fluid at rest has no viscous shear stress. Pressure acts normal to a surface and is isotropic at a point. In a connected static liquid of uniform density, points at the same elevation have equal pressure. Gauge pressure is measured relative to local atmospheric pressure; absolute pressure includes it.
Formulas
- With z positive upward,
dp/dz = −ρg. For constant density at depth h below an open surface,p_gauge = ρgh. - On a submerged plane with atmospheric pressure cancelled on the two sides, resultant force is
F = ρg A h_c, where h_c is centroid vertical depth. - Centre-of-pressure vertical depth:
h_p = h_c + I_G sin²θ/(A h_c), with θ measured from the horizontal and I_G about the centroidal in-plane axis parallel to the free surface. - Buoyancy:
F_B = ρg V_displaced, acting through the centroid of displaced fluid volume. Floating equilibrium requires buoyancy equal to weight; rotational stability needs a separate check.
Worked example
A vertical rectangular gate is 2 m wide and 3 m high. Its top is at the water surface, and its opposite side is at atmospheric pressure. Take ρ = 1000 kg/m³ and g = 9.81 m/s².
A = 2 × 3 = 6 m², h_c = 1.5 m.
F = 1000 × 9.81 × 6 × 1.5 = 88290 N = 88.29 kN.
I_G = bh³/12 = 2 × 3³/12 = 4.5 m⁴.
h_p = 1.5 + 4.5/(6 × 1.5) = 2.0 m.
Answer: A horizontal resultant of 88.29 kN acts 2.0 m below the surface. Its moment about a hinge at the gate top is 176.58 kN·m.
Common mistakes
- Using distance along an inclined gate instead of vertical depth in pressure.
- Placing the resultant at the centroid when pressure varies with depth.
- Adding atmospheric pressure to one side while ignoring it on the other.
- Treating density in kg/m³ as unit weight in N/m³.
Quick check
- Gauge pressure 4 m below a water surface? 39.24 kPa.
- Does vessel shape change pressure at a given depth in the same static liquid? No.
- Is centre of pressure below the centroid of a vertical submerged plane under this triangular/trapezoidal gauge-pressure distribution? Yes.
Reference
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