ME 305 Fluid Mechanics I

Part 2

Fluid Statics

These presentations are prepared by

Dr. Cüneyt Sert

Mechanical Engineering Department

Middle East Technical University

Ankara, Turkey

csert@metu.edu.tr

Please ask for permission before using them. You are NOT allowed to modify them.

2-1

Fluid Statics

• Fluids at rest can not support any shear stress (they are free of shear).

• For static fluids we can only talk about normal stress which is equal to pressure.

• Determining the pressure distribution within the fluid is the main task here.

• Applications include

• Pressure distribution in atmosphere and oceans.

• Pressure measurement using manometers.

• Forces acting on submerged solid bodies.

• Bouyancy and stabilitiy of floating bodies.

• Fluids in rigid body motion are also free of shear

forces and will be studied in this chapter.

Clyde Dam http://www.cromwell.org.nz/aerial_photos/pages/Clyde Dam_jpg.htm

2-2

𝑝𝑎𝑡𝑚

Hydrostatic Forces Acting on Submerged Surfaces

• Pressure always acts perpendicular to a surface.

Exercise : Show the variation of pressure force acting on the walls of the following containers.

𝑝𝑎𝑡𝑚

• The task is to find

• the total pressure force acting on a submerged surface and

• point of application of the resultant pressure force. 2-16

Hydrostatic Forces Acting on Planar Surfaces

Exercise : Calculate the resultant pressure force Fr acting on the upper surface of the arbitrarliy shaped plate and its point of application (xCP, yCP).

y

x dA

CP

xCP

yCP xG

yG

G

y

x

Normal view

y

z h dF

𝛽

Fr

hCP

CP

G

hG

p0

Side view

2-17

Hydrostatic Forces Acting on Planar Surfaces (cont’d)

• Resultant force Fr is equal to the pressure at G times the area of the surface.

• Fr acts at a point called the center of pressure (CP).

y

z

𝛽

Fr

CP : center of pressure

G : centroid

p0

2-18

𝐹𝑟 = 𝑝𝐺𝐴

𝑥𝐶𝑃 =1

𝐹𝑟 𝑥 𝑝 𝑑𝐴

𝐴

𝑦𝐶𝑃 =1

𝐹𝑟 𝑦 𝑝 𝑑𝐴

𝐴

Exercise : Solve the previous problem by considering a triangular gate as shown. Base length and height of the gate are 6 m and 3 m, respectively.

Hydrostatic Forces Acting on Planar Surfaces (cont’d)

Exercise : Semicircular gate is hinged along B and held by horizontal force FA applied at A. The liquid on the left of the gate is water. Calculate the force FA required for equilibrium. 𝐹𝐴

A

B

8 m

3 m Gate Side view

2-19

𝐹𝐴 A

B

Hydrostatic Forces Acting on Planar Surfaces (cont’d)

Exercise : A 2 m wide plane gate of uniform thickness holds back a depth of water as shown. It is hinged from along its top side. Find the minimum weight needed to keep the gate closed.

2-20

30o

3 m

Water

Exercise : As water rises on the left side of the rectangular gate, the gate will open automatically. At what depth D above the hinge will this occur ? How will the result change if the mass of the gate is considered ?

Hinge

Gate

1.5 m D

Pressure Prism Method

• This is an alternative (and easy) technique to calculate hydrostatic forces acting on submerged planar surfaces.

• Consider an imaginary prism with the surface of interest being its base and the amount of pressure acting on the surface being its height.

2-22

𝑑𝐹 = 𝑝𝑑𝐴 = 𝑑∀𝑃𝑃

𝐹𝑟 = 𝑑𝐹

𝐴

𝐹𝑟 = ∀𝑃𝑃 = volume of the

pressure prism

(xCP, yCP) corresponds to the coordinates of the centroid of the pressure prism.

y

hA

dF Fr

CP

p0

hB

h

𝑝0 + 𝜌𝑔ℎ𝐴

𝑝0 + 𝜌𝑔ℎ𝐵 dA 𝑑∀𝑃𝑃= 𝑝𝑑𝐴

Hydrostatic Forces Acting on Curved Surfaces

Exercise : Consider the following curved surface. Derive expressions to calculate the components of the resultant force in x, y and z directions.

2-23

z

𝑑𝐴

x

y

𝑑𝐴𝑥 𝑑𝐴𝑦

𝑑𝐴𝑧

Hydrostatic Forces Acting on Curved Surfaces (cont’d)

For the following simplified curved surface which is aligned with one of the Cartesian axis, we need to calculate only one horizontal and one vertical force component.

2-24

Curved surface

𝐹𝑉

𝐹𝐻

𝐹𝐻 Projected planar surface

𝐹𝑉

𝐹𝑉 = 𝜌𝑔∀𝑎𝑏𝑜𝑣𝑒

∀𝑎𝑏𝑜𝑣𝑒

2-25

Exercise : A Trainer gate used to control water flow from a dam to a river is shown. The gate width is 35 m. Determine the magnitude and line of action of the force from the water acting on the gate.

Hydrostatic Force on Curved Surfaces (cont’d)

http://www.ciltug.com http://www.discover-net.net/ ~dchs/history/gate_ani.gif

10 m 20 m