FluidStatics

8/9/2019 FluidStatics

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3. Fluid Statics

3. Fluid Statics


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3. Fluid Statics

3.1 Definition of Fluids and Viscosity

3.2 Pressure at a Point : Pascal s Law

3.3 Pressure Distribution in a StaticFluid under Gravity


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3. Fluid Statics

3.1 Definition of Fluids and Viscosity

A fluid is a substance that deforms

continuously under the action of an appliedshear forces, or stress,of any magnitude . A F /!X

Fig. 3.1

F


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3. Fluid Statics

Q : What is the difference between solidand fluid?

A : For a constant shear force or shears thedeformation of solid, the shear angle, isconstant, but for the fluid the time rate ofthe shear angle is constant .

solid : fluid :


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Consequence of the fluid deformation

1. Fluid can be at rest, only when no shearingstress acts.

2. Fluid can resist shear only when moving.


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What is viscosity?

Viscosity is a fluid property which indicates howhigh is the deformation rate(shear rate), of afluid for a given shear stress,

For Newtonian fluids : a linear relationbetween and , ,

For non-Newtonian fluids : non-linear,


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Fig. 3.2

Fig. 3.3


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3.2 Pressure at a Point: Pascal

s Law

According to the deformation of fluids there are no

shear stresses acting on fluid elements at rest.Therefore, on the surfaces of a fluid element at rest

of any shape only normal stresses can be present.

The surface force from the normal stress on a fluidelement at rest under gravity must be in equilibriumwith the volume force of the fluid element due to thegravity.


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From this equilibrium the Pascal s law results :

The pressure at any point in a fluid at rest hasa single value, independent of direction.

Pressure is a scalar quantity


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Fig. 3.4

From geometry,

The equation of motion in the y and z direction are ,


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The equation of motion can be rewritten as

We take the limit as and approachzero

y x H H , z H

The pressure at a point in a fluid at rest, or inmotion, is independent of direction as long as there areno shearing stresses present. Pascal s Law


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3.3 Pressure Distributionin a Static Fluid under

Gravity

or

Fig. 3.5

If we let the pressure at the center ofthe element be designated as p, then the

average pressure on the various forces canbe expressed as Fig. 3.5 .We are using a Taylor series expansion ofthe pressure at the element center.


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The resultant surface force acting on the element can be expressed in vectorform as

or (3.1)The group on terms in parentheses in Eq. 3.1 represent in vector form thepressure gradient

where

The symbol is the gradient or del vector operator. Thus ,

Since the z axis is vertical, the weight of element is

Newton s second law, applied to the fluid element, can be expressed

or

Therefore, (3.2)


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3. Fluid Statics

Incompressible Fluid

Changes in are caused either by a change in or . Formost engineering applications the variation in is negligible, so ourmain concern is with the possible variation in the fluid density. Forliquids the variation in density usually negligible so that theassumption of constant specific weight when dealing with liquids isgood one.

Where p1 and p2 are pressure at the vertical elevations z1 and z2,as is illustrated in Fig. 3.5. Eq. 3.3 can be rewritten as

or

or

(3.3)

(3.4)

(3.5)

K V g g


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Fig. 3.6


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3.4 Applications

3.4.1 Communicating Tube

Fig. 3.7


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3.4.2 Pascal

s Paradox

Fig. 3.8


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3.4.3 Hydraulic Jack

Fig. 3.9

(3.6)


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3.4.4 ManometerA) U-Tube Manometer

(3.7)

(3.8)

(3.9)

Fig. 3.10


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B) Prandtl-Manometer

(3.10)

(3.11)

Fig. 3.11


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C) Betz-Manometer

Fig. 3.12


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D) Manometer for Small Pressure Difference

(3.13)

(3.14)

(3.15)

(3.12)

Fig. 3.13


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3.4.5 Hydraulic Siphon

(3.16)

Fig. 3.14


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3.4.6 Chimney

Fig. 3.15


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3.5 Hydrostatic Force on a Plane Surface

Fig. 3.16

Fig. 3.17


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The magnitude of the resultant force can be found by summingthese differential forces over entire surface.

(3.17)The integral is the first moment of the area with respect to the x axis

or (3.18)

The moment of the resultant force must equal the moment of thedistributed pressure force, or

Therefore,

By parallel axis theorem

Thus,

The integral is the second moment of the area with respect to axisformed by the intersection of the plane surface and the free surface.

(3.19)


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3.6 Buoyancy : Archimedes

Principle

(3.20)

(3.21)

(3.22)

Fig. 3.18


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3.7 Hydrostatic Forceon a Curved

Surface(3.23)

(3.24)

Fig. 3.19


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3.8 Aerostatics

The temperature variation in the troposphere (3.26)

(3.25)

Eq.3.27 Eq.3.26 and integrating,(3.27)

(3.28)

(Pc at the loweredge of thestratosphere Zc)

Fig. 3.20

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