II' Kinematics of Fluid Motion

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CEE 262A HYDRODYNAMICS
Lecture 11 Autumn 2007-2008
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Exact solutions to the Navier Stokes Equations

• Real flow problems are too complicated for us to
  be able to solve the NS equations subject to
  appropriate BCs We must simplify matters
  considerably!
• Our method
• Reduce real flow to a much simpler, ideal flow
• Solve model problem exactly
• Extract important dynamical features of solutions
• Apply lessons learned to real flow

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Generally, the model flow illustrates 1 of 3
possible two-way force balances (2 terms
involved)
Three way force balances (3 terms) are much
harder to deal with
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Poiseuille Couette Flows
Consider the non-rotating, constant density flow
between two infinite parallel plates, one of
which moves the other of which is fixed
Moving plate
x3 H
U0
Constant pressure gradient
Fixed plate
x30
Because the plates are infinite,
except pressure
everything
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So, continuity tells us that
Thus, most generally, u3fn(x1,x2). But since u3
0 on the top (or bottom plates), it must be 0
everywhere. Additionally since there is no
pressure gradient or plate motion in the x2
direction, u2 0 . Thus, the only non-zero flow
component is u1(x3)
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Now we turn to the Navier Stokes equation to see
what can be eliminated
No rotation (by assumption)
Steady (by assumption)
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So what is left is
Subject to the boundary conditions that
Thus if we integrate twice with respect to x3
And use the boundary conditions
(a) u10 at x30 B0
(b) u1U0 at x3H
We find that
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If we divide both sides by U0 we can rewrite our
velocity distribution in a convenient
non-dimensional form
where
The parameter P represents the relative
importance of the imposed pressure gradient and
the moving surface
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-3
-10
-1
0
1
10
5
3
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We can calculate the flow rate
Thus, Q 0 when P 3, whereas Q gt 0 when P lt 3,
and Q lt 0 when P gt 3
Wind
P 3 wind driven flow in a lake
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Two Special Cases
1. Plane Couette Flow
whence
Stress on plane parallel to plates
Tangential Stress i1
is constant over depth
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2. Plane Poiseuille Flow U00
whence
and
Tangential Stress
Shear stress varies linearly with depth
Wall stress opposes p.g.