Chapter 2. Fluid Statics

� Topics

� Fluid at rest: No relative motion/deformation: no shearing stress

� Hydrostatic pressure at a point: Pascal’s Law

� Pressure field/pressure variation for fluids at rest

� Measurement of pressure

� Hydrostatic force on surfaces� Vertical/Horizontal planes

� Inclined surfaces

� Curved surfaces

� Buoyancy, flotation, and stability

Pressure Field in Static Fluids

� Static fluids

� no shearing stress

� no fluid deformation

� “F = m*a” can be applied on a control volume as if on a rigid body

� We are interested in the balance between the hydrostatic force and body forces on a control volume

Pressure at a point

� Pascal’s Law: px = py = pz at any point in static fluid

i.e. pressure is isotropic

WU1

Slide 3

WU1 On whiteboard, show the derivationWindows User, 1/6/2012

Hydrostatic Force on a

Small Control Volume

� dfs = ∇p*dv

� −�� − ��� = �

∫=A

Apdfvv

∫=A

xx pdAf

WU2

Slide 4

WU2 How the pressure in a fluid in which there are no shearing stresses vary from point to point?

Show derivation, introduce the pressure derivativeWindows User, 1/6/2012

Pressure Field for Incompressible Fluids

� −�� − ��� =0

� ∆p = ρgh

� How about compressible fluids?

r1r2

Slide 5

r1 rzr11001, 1/8/2012

r2 show derivation

hydrostatic distribution; The pressure difference between two points can be specified by the distance h, which is called pressure headrzr11001, 1/8/2012

Example: atmosphere

r3

Slide 6

r3 show the derivation for isothermal case with ideal gas lawrzr11001, 1/8/2012

Figure 2.9 (p. 51) Piezometer tube.

Pressure Measurement:

Piezometer-Tube Manometer

• Absolute Pressure:relative to vacuum

• Gage Pressure:relative to ambient

� = −�1ℎ1

r4r5

Slide 7

r4 Manometers use vertical or inclined liquid columns to measure pressurerzr11001, 1/8/2012

r5 Disvantages:1. the pressure in the container must be greater than atmospheric pressure2. pressure difference relative smallrzr11001, 1/8/2012

Figure 2.10 (p. 51)Simple U-tube manometer.

Pressure Measurement:

U-Tube Manometer

� = �2ℎ2 − �1ℎ1

Figure 2.12 (p. 54)Inclined-tube manometer

Pressure Measurement:

Inclined-Tube Manometer

� − �� = �2�2���θ

r6

Slide 9

r6 To measure small pressure differencerzr11001, 1/8/2012

Hydrostatic Force on a Surface

� Differential form

� Integral form:

Apdfdvv

=

∫=A

Apdfvv

∫=A

xx pdAf

Ex: Hydrostatic Force on Planes

� Horizontal planes � Vertical planes

r7

Slide 11

r7 resultant force

Derive Equation 2.18

Explain pressure prism. Only good for rectangularrzr11001, 1/8/2012

Ex:

Hydrostatic Force on Curved Surfaces

r8

Slide 12

r8 Show derivation by considering the eqiulibrium of the fluid volume enclosed by the curved surface of interest and thr horizontal and vertial projections of this surface.rzr11001, 1/8/2012

Buoyancy

� Buoyancy: integrated hydraulic force on the surface of an object

� Archimedes principle:

� Buoyancy force = weight of the displaced volume

r9

Slide 13

r9 show the derivation procedure on Page 69rzr11001, 1/8/2012

Stability

Pressure Variation under Acceleration

Linear Acceleration

Rigid Body Rotation

−�� − ��� = �