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Transcript of 02- Fluid Statics - 01
7/23/2019 02- Fluid Statics - 01
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2-Estática do fluido
Mecânica dos Fluidos Aula 5
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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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Buoyancy and Flotation: !rchimedes8 Principle
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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Buoyancy and Flotation: !rchimedes8 Principle
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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Pressure 2ariation' "igid Body )otion: "otation
(+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