Fluid Statics and Pascal's Law

• Fluid Mechanics Lecture 2a covers fluid statics and surface forces.
• Pressure and stress concepts are discussed.
• Pascal’s law and its applications are explored.

Class Objectives

• Describe pressure and pressure distribution.
• Differentiate between pressure and stress.
• Describe gauge, absolute, and vacuum pressure.
• Perform basic pressure measurement calculations.
• Define Pascal’s law and its application.

Introduction to Fluid Statics

• A fluid continuously deforms under shear stress, no matter how small.
• Fluids at rest are thus in a state of zero shear stress known as hydrostatic stress.
• Pressure, a physical quantity, originates from the hydrostatic state of stress.
• Pressure acts equally from all directions and is comprehensive in sense.

Pressure Defined

• Pressure is the normal force exerted by a fluid per unit area.
• In solids, the equivalent is called normal stress.
• The unit of pressure is N/m², also known as Pascal (Pa).
• 1 Pa equals 1 N/m².
• 10^5 Pa = 1 bar = 0.1 MPa = 100 kPa.
• 101.325 Pa = 1 atm.

Pressure vs. Stress

• Pressure is the external force applied per unit area.
• Stress is the internal resistive force per unit area.
• Pressure is measurable using gauges and manometers.
• Stress is calculated by measuring strain or elongation, not directly measurable.
• Pressure acts perpendicularly to a surface.
• Stress can act perpendicularly or parallel to a surface
• Pressure magnitude is the same in all directions at a point.
• Stress magnitude varies by direction.
• Pressure is a scalar (magnitude only).
• Stress is a vector/tensor (magnitude and direction).

Pressure Scales

• Pressure is expressed relative to absolute zero pressure (complete vacuum) or local atmospheric pressure.

Gauge Pressure

• Gauge pressure is relative to local atmospheric pressure.
• Atmospheric pressure refers to the pressure exerted by the Earth's atmosphere at a specific location.
• Gauge pressure indicates how much system pressure exceeds atmospheric pressure.
• A gauge pressure of 0 means the pressure equals atmospheric pressure.
• A gauge pressure of 32 psi means 32 psi above atmospheric pressure.
• P(gauge) = P(absolute) - P(atmospheric).

Absolute Pressure

• Absolute pressure includes atmospheric pressure.
• Absolute/total pressure references absolute zero, a perfect vacuum.
• A perfect vacuum lacks any gas molecules, hence, zero pressure.
• Measuring 50 psia means pressure is 50 psia above theoretical zero.
• Absolute pressure gauges are common in gas systems.
• P(absolute) = P(gauge) + P(atmospheric).

Vacuum Pressure

• Vacuum pressure gauges (e.g., Bourdon tube, McLeod) measure pressures below atmospheric.
• If pressure is less than atmospheric, gauge pressure reads negative, indicating vacuum pressure.
• Vacuum pressure is encountered in vacuum chambers, distillation, space simulation, and various industrial processes.
• P(vacuum) = P(atmospheric) – P(absolute).

Fluid Pressure Factors

• Fluid pressure arises from the weight of the fluid.
• Fluid depth affects pressure; pressure increases with depth.
• Fluid density influences pressure; denser fluids exert greater pressure.

Fluid Pressure Variation with Elevation

• The differential equation governs pressure change rate with elevation in a fluid.
• Pressure changes with vertical distance but remains constant laterally.
• The pressure varies inversely with elevation.
• Pressure decreases when moving upward and increases when going downward in a fluid.

Total Pressure in Liquids

• Focuses on finding total pressure at a depth in liquids, like water in a swimming pool.
• Considers Ps (pressure at the free surface).
• Considers atmospheric pressure
• Derives an expression assuming constant density (ρ) and gravitational acceleration (g).

Linear Pressure Increase

• Predicts linear pressure increase with depth from the free surface.
• Significant pressure results at great depths, impacting deep-sea divers.

Hydrostatic Equation

• Specific weight γ = ρg (density multiplied by gravitational acceleration).
• P = Ps + γH (pressure at depth equals surface pressure plus specific weight times depth).
• Hydrostatic equation predicts pressure variation with depth.
• It applies for constant density fluids.

Constant Density Fluid Pressure

• Pressure difference between two points in a constant density fluid is proportional to the vertical distance (ΔH) between the points and the fluid density (ρ).
• p(below) = p(above) + γ * ΔH.

Stacked Fluid Layers

• Pressure at the bottom (P₁) of stacked fluid layers can be calculated starting from the surface pressure Patm.
• When all fluids have the same density the equation simplifies to: P₁ = Patm + ρg(h₁ + h₂ + h₃)

Pressure Variation in Gases

• Gas density is not constant, but is a function of pressure and slightly of temperature.
• Equations for liquids are valid in gases only for "small changes in elevation" and with "the assumption of constant density".

Moderate or Large Changes in Elevation

• Density (ρ) is given by ρ = MwP/RT or ρ = MwP/ZRT.
• Mw is the molecular weight of the gas, P is the pressure, R is the ideal gas constant, T is the temperature, and Z is the compressibility factor.

Gas Pressure Example

• Using ideal gas behavior, for small values of Mwgz/RT, the last term is an insignificant second-order effect (compressibility effects are unimportant): p ≈ p0 - ρ0gz.

Pressure Measurement Instruments

• Barometer
• Bourdon-tube gauge
• Piezometer
• Manometer
• Transducer

The Barometer

• Barometers use a single mercury column to measure atmospheric pressure.
• The atmosphere exerts pressure on the liquid.
• This exerted pressure keeps the liquid in the glass to be at a cetain height
• Measures height to determine atmospheric pressure.
• Atmospheric pressure supports a 760 mm mercury column.
• 1 atm = 760 mm Hg = 101.325 kPa = 14.7 psi = 1.013 bar.
• Hydrostatic pressure is defined as P = hρg, where h is height, ρ is density, and g is gravity.
• At sea level, atmospheric pressure is 101325 Pa.
• The hydrostatic pressure at sea level equals the hydrostatic pressure due to the weight of the air column.
• Patm = hρg

Manometers

• Manometers use a liquid column.
• One end connects to a pipe/container of fluid (A).
• Measures exerted pressure of gas in U tube or a U tube containing one or more fluids.
• The lower U-tube part has liquid immiscible with fluid A and greater fluid density than A, which is manometric fluid.
• Manometers measure pressure difference between measured fluid vs atmospheric pressure or measure pressure between fluids
• The instrument relies on balancing the pressure to be measured against that exerted by a liquid column (mercury or water)
• Manometers also determine the pressure if the vacuum gauge is closed, and liquid density is low

Using a Manometer

• Relates the height of the liquid in the manometer to pressure.
• Pressure in a continuous static fluid is the same, is equal to the manommeter fluid pressure, at any horizontal level, so equating pressures at B & C in terms of the heights of fluids above those points:
  • pressure at B = pressure at C
  • LHS: PB = PA + 𝜌gh
  • RHS: PC = Patm + 𝜌gh

Pascal's Law

• A fluid must be at static equilibrium and not flowing.
• If the fluid is water, it is said to be hydrostatic equilibrium.
• For a fluid in static equilibrium, the net force on any part of the fluid must be zero; otherwise, the fluid will start to flow.
• Pascal observed that a change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to all fluid points is called (Pascal’s Principle).
• The exerted pressure is thus transmitted equally at all points on the fluid.

Hydraulics

• Application: At the same depth, 𝑃₁ = 𝑃₂.
• The ratio of output force to input force will be P₁ = P₂ → F₁/A₁ = F₂/A₂ → F₂/F₁ = A₂/A₁.
• The area ratio A₂/A₁ ,is referred to the ideal mechanical advantage of the hydraulic lift.

Lecture on fluid statics covers pressure, stress, and Pascal's law. Explains pressure distribution, measurement calculations, and differences between pressure types. Introduces hydrostatic stress and pressure as normal force per unit area.