1

Lecture 4

Pressure distribution in fluids.Pressure and pressure gradient.

Hydrostatic pressure

2Engenharia do AmbienteMecânica dos Fluidos Ambiental

Basic Principles of Fluid Mechanics

• Mass conservation

• =>

• Lei de Newton:

• =>

0dt

dm jj

vxt

F m

dv

dt

ij

i

jij

ij

i

x

zg

x

v

xx

p

x

vv

t

v

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Methods for Fluid Mechanics

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Chapter 2

• Concept of pressure and pressure gradient,• Hydrostatics and centripetal force.• Lei de Newton: Acceleration as the result of

the applied forces.

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Forces in Fluids

• Surface or volumetric (or mass)• Surface forces can be normal (pressure) or

tangential (friction)• Tangential forces are always parallel to

velocity. Their equation can be complex if velocity is not parallel to any reference axis.

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Pressure is a scalar

• Could be different in one point?nyx ppp ,,

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In a fluid at rest:

Esta conclusão é independente de θ.

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Pressure Gradient

• It will be shown that the pressure gradient can generate acceleration, but not the pressure by itself.

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Pressure force resultant

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Force resultant (including friction and weight)

dxdydzgWeight

dyy

dyyyx y

u

y

yxy y

u

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Summing up

ix

ppressure

2

2

y

u

dxdydz

yu

yu

dxdz

Frictionydyy

dxdydzgweight

ij

i

i

i gx

u

x

p

dt

du

2

2

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General equation

ij

i

i

i gx

u

x

p

dt

du

2

2

ij

i

ij

ij

ii gx

u

x

p

x

uu

t

u

dt

du

2

2

At rest the fluid has null velocity and so are its derivatives. The pressure is hydrostatic.

Se a aceleração vertical for nula a pressão continua a ser hidrostática.

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Hydrostatic pressure

• If the vertical axes (zz) points downwards gravity acceleration is positive, otherwise is negative.

zgz

p

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Hydrostatics

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If was constant:

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In case of gas

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Hydrostatic Forces

• Objectif: • Compute the hydrostatic force over flat and curve

surfaces and the application point (pressure centre).

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Força Hidrostática

sinh

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Computing using former knowledge

dAApF a sin

dAApF a sin

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OU

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Pressure Centre (Measured from the gravity centre)

yGC ..

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Pressure Centre (x coordinate)

On surfaces with vertical simetry CP lays on the axis.

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Resumo