Fluid Flow

Pressure, momentum flux and viscosity.

Viscosity-fluid property that influences the rate of fluid flow under stress.

The viscosity is the ratio between the shear stress and the velocity gradient between the plates, or

dy

xdv )(

y

How can we attack this problem?

(1) Define viscosity

Newton’s Law of Viscosity

yvx

d

d

Pa

m

s

m

Pa s

Units of viscosity

Layers of fluid particles with top layer moving faster than bottom layer

energy

Position

Activation energy

o expactivation_energy

RT

log log o activation_energyR

1

T

log log o activation_energy

R

1

T

Momentum transfer

The top plate drags the fluid particles in the top fluid layer, which then drag the particles in the adjacent lower layer, which then drag the particles in the next lowest layer, and so on—thus giving rise to momentum transfer

Units on shear stress and pressure

Force/area=mass*length/time2 * 1/area =mass * length/time * 1/time * 1/area

= mass*velocity * 1/(time * area)

= momentum/(time*area)

= momentum flux

Forces balance at steady state (equilibrium)

Rate of momentum in =rate of momentum out at steady state.

A momentum flux (or stress) multiplied by a cross sectional area is a FORCE!

‘control volume’

y

xz

2. Determine velocity profile. If flow is fully developed, the fluid velocity only depends on y.

Papplied

Patm

x x x

y

y y

V(x)

y

yx yv x( )d

d

0

yxy yxy y

xz P x Px x yz

0-

High Pressure Low Pressure

Hydrostatic pressure varies with x while shear stress varies with y! We only have to consider the shear stress acting normal to the xz plane and the hydrostatic pressure component acting normal to the yz plane.

yxy yxy y

xz P x Px x yz

0

yxy yxy y

y

P x Px x

x

0

y yx

d

d

xPd

d

Velocity can only depend on y

Function of x unless P=constant or P linear with

x.

y yx

d

d

PL

yy yx

d

d

d yPL

d yxPLy C

-

-

C is the integration constant from indefinite integration

C=0

yx yv x( )d

d

PLy y

yv x( )d

d

d yPLy

d

v x( )P

2L y2 C1

C1P

2L 2

v x( )P

2L 2

y2

If the velocity is a local maximum at y=0 (center in between plates), then