III. Fluid Statics

Contents 1. Pressure in Static Fluid 2. Governing Equation 3. Units and Scales of Pressure 4. Pressure Measurement 5. Fluid Force on Plane Surface 6. Fluid Force on Curved Surface 7. Relative Equilibrium

1. Pressure in Static Fluid

There is only inward-normal stress (pressure) on any contact area at a point in static fluid

The magnitude of the pressure at any point in static fluid is independent of the direction

Forces

Forces Balance

Force Balance

1. 1.There is only inward-normal stress (pressure) on any contact area at a point in static fluid (by definition) 2. 2.The magnitude of the pressure at any point in static fluid is independent of the direction (by force balance) Stress state in static fluid can be described by a SCALAR

State of Stress In generalIn static fluids

2. Basic Equation

Forces on a fluid element

Force balance

Force Balance

Governing Equation

The gradient of pressure is balanced by the body force

In the case of conservative force Y = Force potential

When gravity force is the only external force ( z axis is vertically upward )

Let h = -z ( h is the depth below a datum ) Pressure in a static fluid is linearly proportional to the depth below a datum

z = constant Horizontal plane in static fluid is necessarily equi-pressure plane p = constant

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Piezometric head in a static fluid is constant Position Head Pressure Head Piezometric head

1 2

3. Units and Scales of Pressure

Reference of pressure is taken at ZERO Absolute Pressure P A Relative Pressure P R Reference of pressure is taken at p 0 P R = Gage pressure P atm - P A = Vacuum pressure

Pressure measured by fluid force per unit area Pressure measured by equivalent height of liquid 1 atm = at = Pa = 760 mm Hg = m H 2 O Pressure measured by atmospheric pressure

4. Pressure Measurement

Barometer

Piezometer h

Manometer h1h1h1h1 h2h2h2h2

Manometer for negative pressure h1h1h1h1 h2h2h2h2

Differential Manometer hBhBhBhB hAhAhAhA DhDhDhDh DzDzDzDz

5. Fluid Force on Plane Surface

x y

Centroid

Centroid

Centre of pressure

PropertiesofAreas

Example Rock-fill Porous Water Impermeable Curtain Impermeable Layer Concrete Cover 2 1 h l

Example On the slope On the bottom

6. Fluid Force on Curved Surface

S AzAz AxAx AyAy p

Fluid Force

Example 30° d

Horizontal Force No net force h

Vertical Force

Force Balance

Buoyancy

Stability of Floating Bodies

7. Relative Equilibrium

Static state of fluid in a moving coordinate system is called relative equilibrium

Basic Equation D’Alembert’s Principle

Uniform Acceleration Equi-pressure surface

Uniform Rotation Equi-pressure surface