Title: Viscously dominated flows
1Viscously dominated flows
- Low 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).
2Creeping flow in a circular pipe control volume
approach
dx
u(r ) only
rR, u0
R
r
r0, ?u/?r0
pdp/dx ?x
p
?
- What is the velocity distribution?
- What is the pressure drop?
- How does it vary with flow rate?
3How to solve a problem
Solution
Differential equations of motion in each direction
Differential equation for class of problems
General transport equations
Velocity profile
4Assumptions for viscous flows
- Incompressible
- Viscosity constant.
- Gravity negligible
- Steady flow
- Two-dimensional
- Fully developed
5Flow development
Fully developed. Profile does not further change
shape.
Entry length, about fifty times pipe diameter
6Consequences of fully developed flow
1. The gradients of u, v in the axial direction
(?u/?x, ?v/?y) must be zero (otherwise the
velocity profile will change in an incompressible
fluid).
2. By using the equation for the conservation of
mass, the gradient of v in the transverse
direction is equal to zero (i.e. v is a
constant).
3. The value of v at the wall is zero therefore
v0 everywhere.
7Creeping flow between flat plates
H
Fully developed. Profile does not further change
shape.
Entry length, about fifty times pipe diameter
8Creeping flow in a film on a wall
y0, u0
yH, ?0
?
?x
y
g
H
9Creeping flow in a circular pipe
dr
r
dA2?rdr