Fluids

Question 1

Density

Answer 1

• Density: Mass to Volume ratio; ρ = m/V.
• Units are g/cm³ or kg/m³.
• For water, ρ = 1 g/cm³ = 1000 kg/m³.

• Weight of substance: F = ρVg.

• Specific Gravity: Density of given fluid divided by density of water; SG = ρ/(1 g/cm³).

Question 2

Pressure

Answer 2

• Pressure: Force per unit area; Pₙₑₜ = Fₙₑₜ/A.
• Units are Pascal (Pa) = N/m².
• 1 atm = 760 torr = 760 mmHg = 101.3 kPa.

• Hydrostatic Pressure: Total pressure exerted on an object that is submerged in a liquid or gas; P = P₀ + ρgh.
• Guage Pressure: Difference between absolute pressure and atmospheric pressure (pressure that is actually felt); P = ρgh.
• P is absolute pressure, P₀ is incident pressure at surface, ρ is density of fluid, h is depth of object.

Question 3

Hydrostatics

Answer 3

• Pascal’s Law: A change in pressure will be transmitted undiminished to every portion of the fluid and to the walls of the containing vessel (see below image).
• P = F₁/A₁ = F₂/A₂
• V = A₁d₁ = A₂d₂
• W = F₁d₁ = F₂d₂

• Archimedes’ Principle: A body wholly or partially submerged in a fluid will be buoyed upward by a force equal to the weight of the fluid that is displaced.
• F(bouy) = ρ(fl) × V(disp fl) × g = ρ(fl) × V(subm) × g
• If object is floating, buoyant force of fluid equals the object’s weight.

Question 4

Fluid Dynamics

Answer 4

• Poiseuille’s Law: Rate of laminar flow.
• Q = (π r⁴ ΔP)/(8 η L)
• Q is volume flowing over time, r is radius of tube, η is viscosity, and L is tube length.

• Turbulent Flow (rough and disorderly flow) occurs when the Critical Speed of the fluid is exceeded. Laminar flow continues to occur only in a thin layer of fluid adjacent to the wall, called the Boundary Layer.

• Continuity Equation: Flow rate (Q) is constant for any closed system, but linear speed of fluid will increase with decreasing cross-sectional area.
• Q = v₁A₁ = v₂A₂

• Bernoulli’s Principle: The sum of Static Pressure and Dynamic Pressure will be constant between any two points in a closed system.
• First term is absolute pressure of the fluid, and second term is GPE/V, which together describe Static Pressure.
• Third term is KE/V and describes Dynamic Pressure.
• Equation describes energy density (J/m³).
• P₁ + ρgh₁ + ½ρv₁² = P₂ + ρgh₂ + ½ρv₂²

Question 5

Venturi Effect

Answer 5

• As the cross-sectional area decreases from point 1 to point 2, the linear speed must increase according to the continuity equation. As the dynamic pressure increases, the absolute pressure must decrease at point 2. With a lower absolute pressure, the column of fluid sticking up from the Venturi tube will be lower at point 2.
• Pressure exerted on the walls of the tube is directly related to cross-sectional area and inversely proportional to linear speed.

Question 6

Pitot Tube

Answer 6

• The opening against fluid flow measures Stagnation (Total) Pressure, where v = 0 and h = 0; therefore, stagnation pressure is equivalent to the absolute pressure at the stagnation point, and to the total pressure found anywhere else in the system (due to constant flow rate).
• Other opening perpendicular to fluid flow measures Static Pressure.
• Total Pressure = Static - Dynamic.

• Stagnation P - Static P = Dynamic P
• ΔP = ½ρv²
• v = √(2ΔP/ρ)