Date post: | 03-Apr-2018 |
Category: |
Documents |
Upload: | zakria-toor |
View: | 220 times |
Download: | 0 times |
of 14
7/28/2019 Vis c Slides
1/14
Example of application of
general transport equations:viscous flows
Resources
Massey, chapter 6,
Alexandrou, chapter 7.
7/28/2019 Vis c Slides
2/14
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).
7/28/2019 Vis c Slides
3/14
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?
7/28/2019 Vis c Slides
4/14
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
7/28/2019 Vis c Slides
5/14
Flow development
Entry length, about
fifty times pipediameter
u(y) only
u
v u=0
x
y
Fully developed.Profile does not
further changeshape.
7/28/2019 Vis c Slides
6/14
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
7/28/2019 Vis c Slides
7/14
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.
7/28/2019 Vis c Slides
8/14
Creeping flow in a film on a wall
y=0, u=0 y=H, =0
xy
g
H
7/28/2019 Vis c Slides
9/14
Hydrodynamic lubrication
7/28/2019 Vis c Slides
10/14
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
7/28/2019 Vis c Slides
11/14
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
7/28/2019 Vis c Slides
12/14
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 =
7/28/2019 Vis c Slides
13/14
Creeping flow in a circular pipe
drr
dA=2rdr
u(r ) only
x
r
r=0, u/r=0
r=R, u=0R
7/28/2019 Vis c Slides
14/14
Videos of fluid flowsMultimedia Fluid Mechanics, G.M.Homsy et al.
Cambridge University Press (2004) 15.99