fluid statics

FLUID MECHANICS

FLUID STATICS

Learning Outcomes

determine the pressure at various locations in a fluid at rest.

explain the concept of manometers and apply appropriate

equations to determine pressures.

Upon the completion of this lecture, you will beable to:

EE038-3.5-3 FLUID MECHANICS FLUID STATICS 2

equations to determine pressures.

calculate the hydrostatic pressure force on a plane or curved

submerged surface.

calculate the buoyant force and discuss the stability of floating or

submerged objects

Analyze the rigid-body motion of fluids in containers during linear

acceleration or rotation

2.1 What is Fluid Statics?

deals with problems associated with fluids atrest.

used to determine the forces acting on floatingor submerged bodies and the forcesdeveloped by devices like hydraulic pressesand car jacks.


Hydrostatics when the fluid is a liquid.

Aerostatic when the fluid is a gas.

The only stress in fluid statics is the normal stress.

The variation of pressure is only due to the weight ofthe fluid.

2.2 Pressure Pressure is defined as

normal force exertedby a fluid per unitarea.

1 Pa = 1 N/m2

Figure 2.1


1 bar = 105 Pa = 0.1 MPa = 100 kPa

1 atm = 101325 Pa = 101.35 kPa = 1.01325 bars

1 kgf/cm2 = 9.807 N/cm2 = 9.807 104 N/m2 = 9.807 104 Pa

= 0.9807 bar

= 0.9679 atm

2.3 Scalar Quantity

Pressure is a scalar quantity, and it is the same in all direction at a point (Pascal Law)


2.4 Pascal Law


Figure 2.2 Infinitesimal fluid element

2.4 Pascal Law


2.5 No-Shear Stress element


Figure 2.3 Surface and body forces acting on small fluid element.


This is the resultant surface force acting on a small fluid element whichdepends only on the pressure gradient if there are no shearing stressespresent.

(pressure gradient)


(pressure gradient)

(the gradient vector operator)

(the resultant force per unit volume)


The weight of the fluid in z-axis direction is given by

The negative sign indicates it acts downward.

From Newtons second law of motion


Resultant force acting on the element

Element mass

Element acceleration


The total force is given by

General Equation for the motionof fluid without shearing stress


of fluid without shearing stress

2.5.1 Pressure variation in a Fluid at rest

when a = 0 then

or in component form

(General Equation)


or in component form

Indicating that the pressure does not depend on x or y,thus z-point on x-y plane is considered.

2.5.1 Incompressible Fluid


Figure 2.4 Hydrostatics Pressure Distribution.

Example 2.1

Because of a leak in a buried gasoline storage tank, water hasseeped in to the depth shown in Figure E2.1. The specific gravity ofthe gasoline is SG=0.68. Determine the gauge and absolutepressures ata) Gasoline-water interfaceb) Bottom of the tank


5.0 m

0.5 m

Figure E2.1

Example 2.2

Calculate the elevation difference, between the water levels in thetwo open tanks shown in Figure E2.2


Figure E2.2

Exercise 2.1

The 500-kg load on the hydraulic lift shown in Figure P2.1raised by pouring oil of density 780 kg/m3 into a thin tube.Determine how high h should be in order to begin to raise theweight. Answer h = 56.7 cm


Figure P2.1

Exercise 2.1

Calculate the gage pressures in chambers A and B shown inFigure P2.2. The fluid in both chambers are the same andseparated by a piston of 50 N. Answer h = 56.7 cm


Figure P2.2

2.6 Manometer

U-Tube Manometer

Piezometer Tube


Inclined-Tube Manometer

2.6 Manometer Simple but

Suitable only if PA > Patm, else air would besucked into the system

The pressure to be measured must berelatively small so the required height of thecolumn is reasonable.

The fluid in the container inwhich the


Piezometer Tube

The fluid in the container inwhich thepressure is to be measured must be a liquidrather than a gas

2.6 Manometer

Fluid can be different from the fluid in the container in which the pressure is to be determined.

Use to measure pressure between two points.


U-Tube Manometer

2.6 Manometer

Find the Pressure Gage


2.6 ManometerDifferential U-tube manometer


2.6 ManometerInclined-Tube Manometer

Used for small pressure difference

For relatively small angle



2.6 ManometerFind the differential pressure of the system below



2.6 Manometer

An elevation change of Dz in a fluid at rest corresponds to DP/g.

A device based on this is called a manometer.

1 2

2 atm

P P

P P gh


A manometer consists of a U-tube containing one or more fluids such as mercury, water, alcohol, or oil.

Heavy fluids such as mercury are used if large pressure differences are anticipated.

2.6 Manometer For multi-fluid systems

Pressure change across a fluid column of height h is DP = gh.

Pressure increases downward, and decreases upward.

Two points at the same elevation in a continuous fluid are at the same


2 1 1 2 2 3 3 1P g h g h g h P

continuous fluid are at the same pressure.

Pressure can be determined by adding and subtracting gh terms.

This measures the gauge pressure!

Absolute pressure ?

2.7 Hydrostatic Force on a Plane Surface

Force exerted by static fluid on a surface when submerged in it

Since there are no shearing stress present, the only force acting on thesurface is Normal Force.

For incompressible fluid, pressure increases linearly with depth.

The Forces involved are important for the design of storage tanks, ships,dams, and other hydraulic structures.


Pressure distribution and resultant hydrostatic force on the bottom of an open tank. FR = pA

Pressure distribution on the ends of an open tank.

Example 2.3

A 4-m-high, 5-m-wide rectangular plate blocks the endof a 4-m-deep freshwater channel, as shown. Theplate is hinged about a horizontal axis along its upperedge through a point A and is restrained from opening bya fixed ridge at point B. Determine the force exertedon the plate by the ridge


Exercise 2.2

The two sides of a V-shaped water trough are hinged to eachother at the bottom where they meet, as shown, making anangle of 45 with the ground from both sides. Each side is0.75 m wide, and the two parts are held together by acable and turnbuckle placed every 6 m along the length ofthe trough. Calculate the tension in each cable. when the troughis filled to the rim.


Example 2.4

The 4-m-diameter circular gate shown in Figure is located in theinclined wall of a large reservoir containing water The gate ismounted on a shaft along its horizontal diameter, and the waterdepth is 10 m above the shaft


Exercise 2.3


2.8 Hydrostatic Force on a Curve Surface

FR on a curved surface ismore involved since itrequires integration of thepressure forces that changedirection along the surface.

Fv


direction along the surface.

Easiest approach is

determine horizontal andvertical components FH andFV separately.

FH

2.8 Hydrostatic Force on a Curve Surface

0

0

0

Y

x

2H

H2

F

FF

FF

F


W

0W

0Y

1V

1V

FF

FF

F

VgW

AhF c1

where

Example 2.5

The water side of the wall of a 100-m-long damis a quarter circle with a radius of 10 m.Determine the hydrostatic force on the damand its line of action when the dam is filled to therim.


Example 2.5

Solution

The horizontal Force FH on Vertical Surface

MN05.492

2A

2

H

aveH

LR F

RLR

PF


Free Body DiagramThe vertical Force FH on the horizontal surface since it coincides with the free surface

2H

LR

F

F

V

V

MN05.774

g

VgmgW

2

FFF 2V

2R H

MN3.91

The line of action of the hydrostaticforce passes through the center of thecurvature of the dam, making 57.5downwards from the horizontal.

2.9 Buoyancy

Buoyancy is due to the fluiddisplaced by a body.

FB= mg = fgV.

Archimedes principle:


The buoyant force actingon a body immersed in afluid is equal to the weightof the fluid displaced by thebody, and it acts upwardthrough the centroid of thedisplaced volume.

Buoyancy force FB is equal only to the displaced volume fgVdisplaced.

Three scenarios possible

2.9 Buoyancy


1. bodyfluid: Sinking body

2.9 Buoyancy


2.9 Buoyancy


weight of the dashed fluid

Total volume of the parallelepiped

2.9 Buoyancy


Example 2.6

The 0.2 kg lead fish sinker shown inFigure (a) is attached to a fishingline as shown in Figure (b) Thespecific gravity of the sinker isSGsinker = 11.3. Determine thedifference between the tension inthe line above and below the sinker


the line above and below the sinker

Figure (a) Figure (b)

Example 2.6

A 1-m-diameter cylindrical mass, M, is connectedto a 2-m-wide rectangular gate as shown. Thegate is to open when the water level, h, dropsbelow 2.5 m. Determine the required value for M.Neglect friction at the gate hinge and the pulley.


FLUID MECHANICSLearning Outcomes

fluid statics

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