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4.2.2: Pressure, Pressure Force, and Fluid Motion Without Flow

[Q2 and Q3] 1. Area as A Vector

Component of Area Vector – Projected Area

Net Area Vector for A Two-Dimensional Surface

2. Resultant Due to Pressure

Resultant Force and Moment (on A General Curved Surface)

Questions of Interest

Q1: Given the pressure field/distribution , find the net/resultant pressure force and

moment on a finite surface

-------------------

4.2.2

Q2: Given the pressure field/distribution , find the net pressure force (per unit

volume) on an infinitesimal volume

Q3: Given a motion (fluid motion without flow), find the pressure field/distribution

),( txp

),( txp

),( txp

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Q2: Given the pressure field/distribution , find the net pressure force

(per unit volume) on an infinitesimal volume

Q3: Given a motion (fluid motion without flow), find the pressure field/distribution

Governing Equation of Motion

Differential Change in p

Calculation: Substitute and integrate (line integral)

Very Brief Summary of Important Points and Equations

),( txp

),( txp

Volume

Forcep

dV

F p = Net pressure force on an

infinitesimal volume per unit volume

Integral)(Line)()()(

)(

2

1

2

1

12

x

x

x

x

xdagxdpxpxp

agxddp

)2(:inchangealDifferenti pxddpp

)1(:

Volume

ForceagpamF

g

a

p

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x

x x+dx

• Magnitude of force at x on -x-plane =

Use Taylor series expansion, we have

• Magnitude of force at x+dx on +x-plane =

• Net x-force =

Q2: Given the pressure field/distribution , find the net pressure force (per unit volume) on an infinitesimal volume

)( xx pAF

dxx

pApAdFF x

xxx

)(

)(

dxx

ppdpp

In this FBD, only the x component is shown.

)( xx pAF

p

dxx

pApAdFF x

xxx

)(

)(

),( txp

dVdxAdVx

pF

dxx

pA

dxx

pApApAF

xx

x

xxxx

,

)(

)()()(

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p = Net pressure force on an infinitesimal

volume per unit volume

• Similarly, we find the resultant forces on y-planes and z-planes and in the y- and z-directions,

respectively,

dVz

pF

dVy

pF

dVx

pF

z

y

x

= Net pressure force on an infinitesimal volume per unit volume

Volume

Forcep

dV

F

dVpdVz

p

y

p

x

pF

)(,,

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Here, we are interested in the motion of fluid where the only forces are

• Surface force Pressure force alone (no friction)

• Body force mg alone

Q3: Given a motion, find the pressure field/distribution

),( txp

g

a

p

Volume

Forcea

Volume

F

ForceamF

][

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Examples of Fluid Motion where The Only Forces are Pressure Force and mg

1. Fluid Motion without Flow

a. Static Fluid

b. Fluid in Rigid-Body Motion

2. Inviscid Flow

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Shear Deformation (Flow)

No Deformation (Flow) No shear

From Surface forces = Pressure +

Viscous/Friction,

then the only surface force in fluid motion without flow is the

pressure force.

Definition of Fluid: A fluid is a substance that deforms continuously

under the application of a shear (tangential)

stress no matter how small the shear stress

may be. (Fox, et al., 2004)

(t)

Fluid Motion without Flow- Fluid in Rigid Body Motion - Static Fluid

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1. From: Newton’s Second law

if evaluated per unit volume, we have

or,

where = net surface force, = net body force

2. In the case of fluid motion without flow, since

• the only surface force is pressure,

• the only body force present is gravitational force,

3. We therefore have the equation of motion for fluid motion without flow

Surface Force:

Resultant force due to pressure per unit volume

Body Force:

Resultant gravitational force per unit volume

g

ap

Direction of maximum spatial rate of increase in pressure

Direction of maximum spatial rate of decrease in pressure

Direction of constant pressure

Incr

easin

g p

Decre

asin

g p

BS FF

gm

pVFS /

Volume

Forceagp

VmaVF /;/ amF

aVFVF BS

//

gVFB

/

for a fluid motion without flowamF

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Net force due to pressure force and gravitational force results in an

acceleration of a fluid element.

Pressure, p, has the maximum spatial rate of change in the direction of,

and has constant value in the direction perpendicular to,

)( agp

Volume

Forceagpam

meunit voluper meunit voluper force nalgravitatioNet

meunit voluper force pressureNet

)( ag

Incr

easin

g p

Decre

asin

g pSurface Force:

Resultant force due to pressure per unit volume

Body Force:

Resultant gravitational force per unit volume

Direction of constant pressure

p

g

Direction of maximum spatial rate of decrease in pressure

a

Direction of maximum spatial rate of increase in pressure

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Q3: Given a motion (fluid motion without flow),

- find the pressure field/distribution , or

- find the pressure difference between any two

points in the flow

Two Main Equations

),( txp

Integral)(Line)()()(

](2)into(1)e[Substitut)]([

.......................

)2(:inchangealDifferenti

)1(:

2

1

2

1

12

x

x

x

x

xdagxdpxpxp

agxddp

pxddpp

Volume

ForceagpamF

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Example: Finding The Pressure Difference between Two Points / Pressure Field of Fluid in Rigid Body Motion with Constant Linear Acceleration

ox

p

g

a

x

xz

y

ox

xProblem: Find the pressure difference between any two points

(a reference point and any point ) within a

fluid in rigid body motion that moves with constant

linear acceleration.

NOTE: Since is any point in the flow, we in effect solve

for the pressure field

Assumption 1: Fluid in rigid body motion.

Assumption 2: The only body force is the gravitational force.

Assumption 3:

ANS

NOTE: 1) In general, the “flow” is not steady with respect to the stationary frame of reference.

2) Here, we analyze the effect of the change in space at any one particular fixed time t.

Surface of constant pressure can be found from letting 0)()()();();( xfxxagtxptxp oo

x

),( txp

)()();();(

Integral)(Line)();();(:)3(

)3()(:)2()1(

)2(:inchangealDifferenti

)1(:

oo

x

x

o

x

x

xxagtxptxp

xdagtxptxp

agxddp

pxddpp

agpamF

oo

constant,, ag

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Example: Finding Pressure Difference between Two Points / Pressure Field of Fluid in Rigid Body Motion

Rotating with Constant Angular Velocity

ox

xProblem: Find the pressure difference between any two points

(a reference point and any point ) within a

fluid in rigid body motion rotating in a cylinder with

constant angular velocity.

p

zegg ˆ

rera ˆ2

x

oxx

z

ree

ze

Assumptions: 1) Fluid in rigid body motion. 2) The only body force is the gravitational force. 3: , g, = constant

oo

ooo

r

r

z

z

x

x

zrrz

zr

x

x

txp

txp

tx

tx

gzrgzr

rrzzgtxptxp

rdrgdz

edzerdedrereg

edzerdedrxdxdagtxdp

xdagdp

txpxddp

agtxpagtxp

oo

o

ooo

2222

222

2

2

);(

);(

);(

);(

2

1

2

12

1)();();(

ˆˆˆˆ)(ˆ)(

ˆ)(ˆ)(ˆ)(,)(),(:)3(

)3()(:)2()1(

)2(),(

)1()(),(),(

Surface of constant pressure can be found from letting

0),(2

1

2

1);();( 2222

zrfgzrgzrtxptxp ooo