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Viscously dominated flows

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Fully developed (velocity profile does not change with position) ... Co-ordinate system. Assumptions. Integration. Boundary conditions. Manipulation. Parameter values ... – PowerPoint PPT presentation

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Title: Viscously dominated flows


1
Viscously 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).

2
Creeping 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?

3
How to solve a problem
Solution
Differential equations of motion in each direction
Differential equation for class of problems
General transport equations
Velocity profile
4
Assumptions for viscous flows
  • Incompressible
  • Viscosity constant.
  • Gravity negligible
  • Steady flow
  • Two-dimensional
  • Fully developed

5
Flow development
Fully developed. Profile does not further change
shape.
Entry length, about fifty times pipe diameter
6
Consequences 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.
7
Creeping flow between flat plates
H
Fully developed. Profile does not further change
shape.
Entry length, about fifty times pipe diameter
8
Creeping flow in a film on a wall
y0, u0
yH, ?0
?
?x
y
g
H
9
Creeping flow in a circular pipe
dr
r
dA2?rdr
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