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Example of application of

general transport equations:viscous flows

Resources

Massey, chapter 6,

Alexandrou, chapter 7.

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Viscously dominated flowsLow Reynolds numbers. Sometimes called

creeping flows.Assumptions are:

Incompressible.

Viscosity constant.

Gravitational forces negligible or driving flow.

Steady flow.

Fully developed(velocity profile does not change

with position).

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Creeping flow in a circular pipe:control volume approach

u(r ) only

x

r

r=0,u/r=0

r=R, u=0

R

p

r

dx

p+dp/dxx

What is the velocity distribution?

What is the pressure drop?How does it vary with flow rate?

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How to solve a problem

Generaltransport

equations

Differentialequation for

class ofproblems

Differentialequationsof motionin eachdirection

Velocity

profile

Solution

AssumptionsCo-ordinatesystem

Assumptions

Integration

Boundary

conditions

Manipulation

Parameter

values

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Flow development

Entry length, about

fifty times pipediameter

u(y) only

u

v u=0

x

y

Fully developed.Profile does not

further changeshape.

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Consequences of fully developed

flow1. The gradients ofu, v in the axial direction (u/x, v/y)

must be zero (otherwise the velocity profile will change inan incompressible fluid).

2. By using the equation for the conservation of mass, the

gradient ofv in the transverse direction is equal to zero(i.e. v is a constant).

3. The value ofv at the wall is zero; thereforev=0

everywhere.The resulting momentum equation is thecreeping flow

equation for whenRe0.

u2= p

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Creeping flow between flat plates

Entry length, about

fifty times pipediameter

H

u(y) only

x

y

y=0, u/y=0

y=H/2, u=0

Fully developed.Profile does not

further changeshape.

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Creeping flow in a film on a wall

y=0, u=0 y=H, =0

xy

g

H

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Hydrodynamic lubrication

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Slipper bearingMassey 6.6.1

The analysis is the same as for Couette flow and aviscous film, except that the boundary conditions arey=0, u=V; y=h, u=0, to give

( )

+=h

yVhyy

dx

dpu 1

2

1

or integrating to give the volume flow rate per

unit width, Q, and rearranging

=32

2

12

h

Q

h

V

dx

dp

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Integrating again alongx using the boundary conditionx=0, p=pa and findingQ using the other boundaryconditionx=l, p=pa, then eventually

( ) ( ),

2)(6

22 laxa

xlVxpp a

+=

whereh=(a-x).

The force or thrust per unit width on the slipper isthen

( )

==

2/1/1

1//ln6

20 lalalaVdxppT l a

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Journal bearing

Force parallel to OC

( )22223

1

c

RL

Force perpendicular to OC

2

322

3

14

c

RL

And volume flow rate of oilper unit length

ce= cQ =

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Creeping flow in a circular pipe

drr

dA=2rdr

u(r ) only

x

r

r=0, u/r=0

r=R, u=0R

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Videos of fluid flowsMultimedia Fluid Mechanics, G.M.Homsy et al.

Cambridge University Press (2004) 15.99