Momentum Transfer

Momentum Transfer

The study of the motion of fluids and the forces that produce these motions

The study of the motion of fluids and the forces that produce these motions forces ← pressure, shear stress ← molecular transfer of momentum

fluid;

- a substance that deforms continuously under the action of a shear stress - a continuous distribution of matter (continuum)

property at a point;

lim

V V

m



V



 

 



minimum volume

surrounding the point for which statistical averages are meaningful

surface tension; a property of the surface of a liquid that allows it to resist an external force

 

 



 





2 1

o i

1 1

R p R

p 

2

2

r P r

     2

P r

  

For a spherical gas bubble

R

₁

 R

₂

 R

p  2R



 For a bubble of radius 1mm

For a bubble of radius 1μm

N/m

2

001 100 . 0

) 05 . 0 (

2 



p

atm

p 1 10 N/m 1

10 1

) 05 . 0 (

2

_{5 2}

6

  

 



_

capillarity; the ability of liquid to flow against gravity where liquid spontaneously rises in a narrow space such as a thin tube

2   r cos     g r h

2

2 cos

h gr

 

 

for mercury and glass for mercury in air

  130  0.44 N m /

 

Fluid statics

the only forces acting on the fluid at rest are those due to gravity and pressure Newton’s 2^nd law of motion;

 F  m a

pressure forces acting on a static fluid element

   



^{y y y}



^{x x}



^{xz z z}



x y z P P y z

P P x z P P x y



_

 

      

        

x

y z

g e

e e 0

 g   P

D

2

Dt P

 v       

g v

U-tube manometer

 g   P

dP g

dy   

integrating between C and D in manometer fluid

integrating between B and A in tank fluid

as elevations B and C are equal,

atm C m 2

P  P    gd

A B T 1

P  P    gd

A atm m 2 T 1

P  P   gd   gd

gage pressure

uniform rectilinear acceleration

  P   g a  

2 2

dP g a

dy    g a     

integrating between y=0 and y=d

2 2

atm B

P  P   g  a ( d) 

2 2

B atm

P  P   g  a (d)

forces on submerged surfaces

we must specify;

1. the magnitude of the force 2. the direction of the force

3. the line of action of the force

sin

dF    g  dA

G

sin

P    gy    g 

sin sin

F   g   

A

dA   g  A   1 A 

^A

 dA

centroid of area

pressure at centroid

the point at which this force acts (center of pressure) is not the centroid of the area

total force on the plate must be concentrated in order to produce the same moment as the distributed pressure

. .

c p G

F   

A

 P dA

2

. .

sin

c p A

F     g  dA F   g sin  A

2 . .

1

_aa

c p A

dA I

A A

 

 

  

moment of area about

the surface

parallel axis theorem

2

aa bb

I  I   A

_{c p. .}

I

^bb

  A

  

the center of pressure is located below the centroid

Ex 4; circular viewing port is located below the surface of the tank -> find the magnitude and location of the force acting on the window

Ex 5; rainwater collector; determine the force and center of pressure on the wall

buoyancy



_{1 2}



_{y B y}

d F  P  P dA e   gh dA e

(

_B

)

_y

d F     ghdA e (  

_B

) gV

_y

 

F e

fundamental physical laws

except relativistic and nuclear phenomena 1. the law of conservation of mass -> continuity equation

2. Newton’s second law of motion ; momentum balance equation 3. the first law of thermodynamics ; energy balance equation