02- Fluid Statics - 01

7/23/2019 02- Fluid Statics - 01

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2-Estática do fluido

Mecânica dos Fluidos Aula 5

7/23/2019 02- Fluid Statics - 01


Outline• Hydrostatic Force on a Plane Surface

• Pressure Prism

• Hydrostatic Force on a Cured Surface• !uoyancy" Flotation" and Sta#ility

• $i%id !ody Motion of a Fluid

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Hydrostatic Force on a Plane Surface: Tank Bottom

Sim&lest Case' (an) #ottom *it+ a uniform &ressure distri#ution

atm patm phγ p -,-

h p γ =

o*" t+e resultant Force'

R F , &A

Acts t+rou%+ t+e Centroid

A , area of t+e (an) !ottom

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Hydrostatic Force on a Plane Surface: General Case

.eneral S+a&e' Planar

/ie*" in t+e 0-y &lane

θ

is the angle the plane makes

with the free surface.

y is directed along the plane

surface.

The origin is at the FreeSurface.

! is the area of the surface.

d! is a differential element

of the surface.

dF is the force acting on

the differential element.

C is the centroid.

CP is the center of Pressure

F" is the resultant force

acting through CP

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Hydrostatic Force on a Plane Surface: General Case

(+en t+e force actin% on t+e differential element'

(+en t+e resultant force actin% on t+e entire surface'

1it+ γ and θ ta)en as constant'

1e note" t+e inte%ral &art is t+e first moment of area a#out t+e 0-a0is

1+ere yc is t+e y coordinate to t+e centroid of t+e o#ect3

#e note h $ ysinθ

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Hydrostatic Force on a Plane Surface: %ocation

&ow' we must find the location of the center of Pressure where the "esultant Force !cts:(The )oments of the "esultant Force must *+ual the )oment of the ,istri-uted Pressure Force

#e note'

)oments a-out the /0a/is:

Then'

Second moment of 1ntertia' 1/

Parallel !/is Thereom:

1/c is the second moment of inertia through the centroid

Su-stituting the parallel !/is thereom' and rearranging:

#e' note that for a su-merged plane' the resultant force always acts -elow the centroid of the

plane.

!nd' note h $ ysinθ

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Hydrostatic Force on a Plane Surface: %ocation

)oments a-out the y0a/is: ∫ = A R R xdF x F

!nd' note h $ ysinθ

#e note'

Then'

Second moment of 1ntertia' 1/y

Parallel !/is Thereom:

1/c is the second moment of inertia through the centroidcc xyc xy y Ax I I +=

Su-stituting the parallel !/is thereom' and rearranging:

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Hydrostatic Force on a Plane Surface: Geometric Properties

Centroid Coordinates

Areas

Moments of 4nertia

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Hydrostatic Force: 2ertical #all

Find t+e Pressure on a /ertical 1all usin% Hydrostatic Force Met+od

Pressure 3aries linearly with depth -y the hydrostatic e+uation:

The magnitude of pressure at the -ottom is p $ h

The width of the wall is (- into the -oard

The depth of the fluid is (h into the -oard

By inspection' the a3erage pressure

occurs at h45' pa3 $ h45

The resultant force act through the center of pressure' CP:

( )

h

hh

y

h

bhh

bh y

R

R

3

2

26

2

212

3

=+=

+=

O

y$ , 26+

y0coordinate: 3

12

1bh I xc =

2

h yc =

bh A =

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Hydrostatic Force: 2ertical #all

/0coordinate:

( )

2

2

2

0

b x

b

bhh

x

R

R

=

+=0= xyc I

2

b yc =

bh A =

Center of Pressure'

3

2,

2

hb

(+e &ressure &rism is a second *ay of analy7in% t+e forces on a ertical *all3

o*" *e +ae #ot+ t+e resultant force and its location3

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Pressure Prism: 2ertical #allPressure Prism' A %ra&+ical inter&retation of t+e forces due to a fluid actin% on

a &lane area3 (+e 8olume9 of fluid actin% on t+e *all is t+e &ressure &rism and

e:uals t+e resultant force actin% on t+e *all3

( )( )bhh F R γ 2

1=

/olume

( ) Ah F R γ 2

1=

$esultant Force'

;ocation of t+e $esultant Force" CP'

The location is at the centroid of the 3olume of the

pressure prism.

Center of Pressure'

3

2,

2

hb

O

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Pressure Prism: Su-merged 2ertical #all

( )12

hhb A −=

( ) Ah F 11

γ =

Trape6oidal

( )( ) Ahh F 122

2

1−= γ

The "esultant Force: -reak into two (3olumes %ocation of "esultant Force: (use sum of moments

Sol3e for y!

y7 and y5 is the centroid location for the two

3olumes where F7 and F5 are the resultant forces of

the 3olumes.

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Pressure Prism: 1nclined Su-merged #all

&ow we ha3e an incline trape6oidal 3olume. The methodology is the

same as the last pro-lem' and we affi/ the coordinate system to the

plane.

The use of pressure prisms in only con3enient if we ha3e regular

geometry' otherwise integration is needed

1n that case we use the more re3ert to the general theory.

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!tmospheric Pressure on a 2ertical #all

Gage Pressure !nalysis !-solute Pressure !nalysis But'

So" in t+is case t+e resultant force is t+e same as t+e %a% &ressure analysis3

4t is not t+e case" if t+e container is closed *it+ a a&or &ressure a#oe it3

4f t+e &lane is su#mer%ed" t+ere are multi&le &ossi#ilities3

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Hydrostatic Force on a Cur3ed Surface• General theory of plane surfaces does not apply to cur3ed surfaces

• )any surfaces in dams' pumps' pipes or tanks are cur3ed• &o simple formulas -y integration similar to those for plane surfaces• ! new method must -e used

4solated /olume

Bounded -y !B an !C

and BC

(+en *e mar) a F3!3<3 for t+e olume'

F7 and F5 is the hydrostatic force on

each planar face

FH and F2 is the component of the

resultant force on the cur3ed surface.

# is the weight of the fluid 3olume.

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Hydrostatic Force on a Cur3ed Surface

o*" #alancin% t+e forces for t+e E:uili#rium condition'Hori7ontal Force'

/ertical Force'

$esultant Force'

(+e location of t+e $esultant Force is t+rou%+ O #y sum of Moments'

H H

V V c

x F x F

x F Wx x F

==+

22

11=-a0is'

>-a0is'

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Buoyancy: !rchimedes8

Principle

!rchimedes 95;0575 BC< Story

•!uoyant force is a force t+at results from a floatin% or su#mer%ed #ody in a fluid3

•(+e force results from different &ressures on t+e to& and #ottom of t+e o#ect•(+e &ressure forces actin% from #elo* are %reater t+an t+ose on to&

o*" treat an ar#itrary su#mer%ed o#ect as a &lanar surface'

!r-itrary Shape

/

Forces on t+e Fluid

Arc+imedes? Princi&le states t+at t+e #uoyant

force +as a ma%nitude e:ual to t+e *ei%+t of

t+e fluid dis&laced #y t+e #ody and is directed

ertically u&*ard3

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Buoyancy and Flotation: !rchimedes8 Principle

!alancin% t+e Forces of t+e F3!3<3 in t+e ertical <irection'

( )[ ]V AhhW

−−= 12

γ

# is the weight of the shaded area

F7 and F5 are the forces on the plane surfaces

FB is the -ouyant force the -ody e/erts on the fluid

(+en" su#stitutin%'

Sim&lifyin%"

(+e force of t+e fluid on t+e #ody is o&&osite" or ertically

u&*ard and is )no*n as t+e !uoyant Force3

(+e force is e:ual to t+e *ei%+t of t+e fluid it dis&laces3

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Sum t+e Moments a#out t+e 7-a0is'

Find *+ere t+e !uoyant Force Acts #y Summin% Moments'

1e find t+at t+e #uoyant forces acts t+rou%+t+e centroid of t+e dis&laced olume3

(+e location is )no*n as t+e center of #uoyancy3

2T is the total 3olume of the parallelpiped

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1e can a&&ly t+e same &rinci&les to floatin% o#ects'

1f the fluid acting on the upper surfaces has 3ery small specific weight 9air<'

the centroid is simply that of the displaced 3olume' and the -uoyant force is

as -efore.

1f the specific weight 3aries in the fluid the -uoyant force does not pass

through the centroid of the displaced 3olume' -ut through the center ofgra3ity of the displaced 3olume.

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Sta-ility: Su-merged -=ect

Sta#le E:uili#rium' if *+en dis&laced returns to e:uili#rium &osition3

@nsta#le E:uili#rium' if *+en dis&laced it returns to a ne* e:uili#rium &osition3

Sta#le E:uili#rium' @nsta#le E:uili#rium'

C C." 8Hi%+er9 C B C." 8;o*er9

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Buoyancy and Sta-ility: Floating -=ect

Sli%+tly more com&licated as t+e location of t+e center #uoyancy can c+an%e'

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Pressure 2ariation' "igid Body )otion: %inear )otion

.oernin% E:uation *it+ no S+ear $i%id !ody MotionD'

(+e e:uation in all t+ree directions are t+e follo*in%'

Consider" t+e case of an o&en container of li:uid *it+ a constant acceleration'

*stimating the pressure -etween two closely spaced points apart some dy' d6:

Su-stituting the partials

!long a line of constant pressure' dp $ >: 1nclined free

surface for ay? >

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Pressure 2ariation' "igid Body )otion: %inear )otion

o* consider t+e case *+ere ay , " and a7 '

0=∂∂ x

p$ecall" already'

( ) z a g z

p

y

p

+−=∂∂

=∂∂

ρ

0(+en"

So" on-Hydrostatic

Pressure will 3ary linearly with depth' -ut 3ariation is the com-ination of gra3ity ande/ternally de3eloped acceleration.

! tank of water mo3ing upward in an ele3ator will ha3e slightly greater pressure at the

-ottom.

1f a li+uid is in free0fall a6 $ 0g' and all pressure gradients are 6ero@surface tension is all

that keeps the -lo- together.

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Pressure 2ariation' "igid Body )otion: "otation

.oernin% E:uation *it+ no S+ear $i%id !ody MotionD'

1rite terms in cylindrical coordinates for conenience'

Pressure Gradient:

!ccceleration 2ector:

Motion in a $otatin% (an)'

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(+e e:uation in all t+ree directions are t+e follo*in%'

*stimating the pressure -etween two closely spaced points apart some dr' d6:

Su-stituting the partials

!long a line of constant pressure' dp $ >:

*+uation of constant pressure surfaces:

The surfaces of constant pressure are para-olic

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o*" inte%rate to o#tain t+e Pressure /ariation'

Pressure 3aries hydrostaticly in the 3ertical' and increases radialy

02- Fluid Statics - 01

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