Chapter_3._Fluid_Statics

8/7/2019 Chapter_3._Fluid_Statics

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Introduction to Fluid Mechanics

Fluid Statics



Main Topics

The Basic Equations of FluidStatics

Pressure Variation in a Static Fluid

Chapter 3: Fluid StaticsENGR 361/4X : Fluid Mechanics-I 2

Hydrostatic Force on SubmergedPlane and Curved Surfaces

Buoyancy & Stability





The Basic Equations of Fluid Statics

Surface Force





Surface Force





Surface Force





Total Force





Newton’s Second Law





Pressure-Height Relation




Pressure

Pressure is defined as a normal force exerted by a fluid per unit area .

Units of pressure are N/m2, which is calleda pascal (Pa).


encountered in practice, kilopascal (1 kPa= 103 Pa) and megapascal (1 MPa = 106

Pa) are commonly used.Other units include bar , atm, kgf/cm 2 ,lbf/in 2 =psi .



Pressure at a Point

“Pressure at any point in a fluid at rest or

in motion, is independent of direction aslong as there is no shearing stresspresent”- Pascal’s law.


Pressure has a magnitude, but not aspecific direction, and thus it is a scalarquantity.



Absolute, gage, and vacuum pressures

Actual pressure at a give point is called

the absolute pressure.Most pressure-measuring devices arecalibrated to read zero in the atmosphere,


and therefore indicate gage pressure,Pgage=Pabs - Patm.

Pressure below atmospheric pressure arecalled vacuum pressure, Pvac=Patm - Pabs.



Absolute, gage, and vacuum pressures





Incompressible Fluid: Manometers





Compressible Fluid: Ideal Gas


Need additional information, e.g., T (z )for atmosphere



Hydrostatic Force on Submerged Surfaces

Plane Submerged Surface









y´


Figure 2.17 (p. 58)Notation for hydrostatic force on an inclined plane surface of arbitrary shape.

x´




Magnitude of

Resultant force,FR is given by

To compute location of Resultant

force, FR

y´ x´

Integral Equation

Algebraic equation

R

A

x F x p dA′ = ∫ R

A

y F y p dA′ = ∫ R

A

F p dA= ∫


Figure 2.17 (p. 58)Notation for hydrostatic force on an inclined plane surface of arbitrary shape.

the submerged sideonly

Algebraic equationfor resultant net

force when ambientpressure P0 on theother side of thesurface as at thefree surface

ˆ ˆ xx

c R

y y F ′ = +

ˆˆ xy

c R

x x F ′ = +

ˆ ˆ xxc

c

I y y Ay′ = +

ˆˆ xy

c

c

I

x x Ay′ = +

( ) R c absF P A=

( ) R c gageF P A=



Resultant/Net force of a static fluid is due to thehydrostatic pressure on the submerged plane surface.


The total hydrostatic pressure force on any submerged plane

surface is equal to the product of the surface area and the

pressure acting at the centroid (C.G.) of the plane surface.


The resultant force acts compressively, normal to thesurface, through a point called centre of pressure (CP).This centre of pressure is not necessarily to be the

centroid.This resultant force is independent of the shape ofthe surface or its angle of inclination.



• Pressure forces acting on a plane surface are distributed overevery part of the surface.

• They are parallel and act in a direction normal to the surface.• They can be replaced by a single resultant force F R = γ hcA.

acting normal to the surface.



• The point on the plane surface at which this resultant force actsis known as the center of pressure (C.P.) .

• The center of pressure of any submerged plane surface is

always below the centroid of the surface (y´> y c ).

2

2

ˆˆ ˆ ˆ

R

xx xx c xx A Ac

R c xx c

y dF y dA I I Ay I

y yF Ay M Ay Ay

+′ = = = = = +

∫ ∫



If the surface is horizontal, the centre of pressurecoincides with centroid.

If the surface become more deeply submerged, centreof pressure approaches centroid.

When the surface area is symmetric about either axis,



ˆˆ 0 xy

A I xy dA= =



The centroid, area, and moment of inertia with respect tothe centroid of some common geometrical plane surfaces




The centroid, area, and moment of inertia with respect tothe centroid of some common geometrical plane surfaces




Hydrostatic Force on Submerged Curved

Surfaces





Surfaces


The hydrostatic force on a curved surface can be best analyzed by

resolving the total pressure force on the surface into its horizontal and

vertical components.

Then combine these forces to obtain the resultant force and its direction.




SurfacesResultant force

The overall resultant force is found by combining thevertical and horizontal components vectorialy:

22

V H F F F +=


The angle the resultant force makes to the horizontal is:

The position of F is the point of intersection of thehorizontal line of action of F H and the vertical line ofaction of F V .

=

−

H

V

F

F 1tanθ



Pressure distribution on a semi-cylindrical gate




Buoyancy and Stability

When a body is submerged or floating in a static fluid, the resultantpressure force exerted on it by the fluid is called the buoyancy force.

Archimedes' Principle

The weight of a submerged body is reduced by an amount equal

to the weight of the liquid displaced by the body .


The buoyant force has a magnitude equal to the weight of the fluiddisplaced by the body and is directed vertically upward.

This force will act vertically upward through the centroid of the volume offluid displaced, known as the centre of buoyancy.

Bodyby Displaced Fluid b V F γ =





Buoyancy of a submerged body

2 1( )b Fluid Displaced by Body

F V h h dAγ γ = = −




Buoyancy force F b is equalonly to the weight of

displaced fluid ρ f g V displaced

Three scenarios possible:

1. ρ bod < ρ fluid : Floating body/


positively buoyant body

2. ρ body = ρ fluid : Neutrally buoyantbody

3. ρ body > ρ fluid : Sinking body/ negatively buoyant body(gravitational pull is greaterthan buoyant force)



Stability of Immersed Bodies


Rotational stability of immersed bodies depends uponrelative location of center of gravity G and center of buoyancy B .

G below B : stableG above B : unstable

G coincides with B : neutrally stable.



Stability of Floating Bodies

The equilibrium of a floating body may be:• Stable Equilibrium : if when displaced returns to

equilibrium position.

• Unstable Equilibrium: if when displaced it returns to anew equilibrium position.


• Neutral Equilibrium if when displaced it moves furtherfrom it

The stability depends upon whether, when given a smalldisplacement, it tends to return to the equilibrium position,move further from it or remain in the displaced position.




When the body is displaced through an angle θ , the centerof buoyancy move from B to B ̀ and a turning moment isproduced.

M (metacentre) is the point at which the line of action of theupthrust F b intersects the vertical line through G.


GM known as the metacentric height.

Then:

Moment generated W GM Sinθ = × ×




If M lies above G, body is bottomheavy, it is always stable, Positivemetacentric height, The couple soproduced sets in a restoring couple

equal to W*GM*Sin θ opposing thedisturbing/overturning moment andthereby bringing the body to its originalposition. The body is said to be in


.

If M is below G, Negative metacentricheight, The moment of the couplefurther disturbs the displacement andthe body is in unstable equilibrium.BM< BG

G & M Coincide, zero metacentricheight. The body floats stably in itsdisplaced position. This condition ofneutral equilibrium exists when BM=BG

Measure of stability is the metacentricheight GM . If GM/BG >1, floating body isstable.



Sea level Conditions of US Standard Atmosphere




Useful Equations


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Chapter_3._Fluid_Statics