Download 1D channel flows I

Geodynamics
Basics of fluid mechanics
Lecture 9.2 - 1D channel flows I
Lecturer: David Whipp
[email protected]
Geodynamics
www.helsinki.fi/yliopisto
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Goals of this lecture
•
Introduce channel flows
•
Present the forces acting on a 1D channel of fixed width
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Channel flows in the Earth
Downloaded from http://sp.lyellcollection.org/ at Dalhousie University on April 25, 2012
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D. GRUJIC
variations (e.g. Bertotti et aL 2000; Lehner 2000);
(c) lithostatic pressure in crust with uniform density.
This third level approximation is the simplest case
and is assumed in the following discussions about
channel flow in the crust.
The flux U of material in the channel can be
expressed by integrating the velocity u of the
material over the channel thickness:
•
•
Channel flows in the Earth occur when a fluid flows
within a channel, between two solid “walls”
hch
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Such channels
can be found in a number of geological
U =
(3)
u(z) dz
0
settings:
Applied to Equation 2 this leads to the flux U
given by:
•
•
•
•
•
U--
h3h dp
12/xch dx
(4)
Counterflow in the asthenosphere
(Turcotte & Schubert 2002). For a linear viscous
fluid Turcotte & Schubert (2002, equation 6 - 1 7 )
give the following equation for the mean velocity
of flow (equal to flux/thickness) in a parallelsided channel:
Grujic, 2006
Fig. 2. Idealized picture of channel flow. Here the
Lower crustal flow
Intra-crustal
channels (figure on left)
h~h dp
_
uo
12/Xch dx +5-
(5)
Subduction channels
Equation 5 accounts for both Couette flow and
Poiseuille flow (for the latter if Uo = 0). In a crust
with uniform densities the principal factors that
influence channel flow are therefore the relative velocity of the bounding plates, the thickness of the
channel, the viscosity of the channel material and
the pressure gradient along the channel. Which of
the end-members will be dominant in the hybrid
velocity profile is a function of threshold value for
a particular parameter (e.g. England & Holland
1979; Mancktelow 1995). For a constant plate convergence rate, pressure gradient and channel thickness, lowering of channel viscosity below a
Salt tectonic channels
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1D channel flows
Fig. 6.1, Turcotte and Schubert, 2014
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The most simple fluid flow we can consider is flow of a fluid in
one direction within a channel of fixed width
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1D channel flows
Fig. 6.1, Turcotte and Schubert, 2014
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Fluid is flowing with velocity 𝑢 in the 𝑥 direction, and the flow
velocity 𝑢 is a function of distance across the channel 𝑦
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Flow results from
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a pressure gradient (𝑝0 - 𝑝1)/𝑙, and/or
motion of the side wall of the channel 𝑢0
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1D channel flows
Fig. 6.1, Turcotte and Schubert, 2014
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Shear, or a gradient in the velocity, in the channel results in a
shear stress 𝜏 that is exerted on horizontal planes in the fluid
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For a Newtonian fluid with a constant dynamic viscosity 𝜂 we
can state
du
⌧ =⌘
dy
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1D channel flows
Fig. 6.1, Turcotte and Schubert, 2014
•
Shear, or a gradient in the velocity, in the channel results in a
shear stress 𝜏 that is exerted on horizontal planes in the fluid
•
For a Newtonian fluid with a constant dynamic viscosity 𝜂 we
can state
du
⌧ =⌘
dy
strain rate
stress
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1D channel flows
Fig. 6.1, Turcotte and Schubert, 2014
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We can now determine the flow in the channel using the
equation of motion, based on the force balance on a layer of
fluid of thickness 𝛿𝑦 and length 𝑙
•
The net pressure force on the element in the 𝑥 direction is
(p1 p0 ) y
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1D channel flows
Fig. 6.1, Turcotte and Schubert, 2014
•
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Because the shear stress 𝜏 and velocity 𝑢 are both only a
function of distance 𝑦, the shear force on the upper boundary
of the element is
⌧ (y)l
The equivalent shear force on the lower boundary is
✓
◆
d⌧
⌧ (y + y)l = ⌧ (y) +
y l
dy
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Let’s see what you’ve learned…
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If you’re watching this lecture in Moodle, you will now be
automatically directed to the quiz!
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References
Grujic, D. (2006). Channel flow and continental collision tectonics: an overview. Geological Society, London,
Special Publications, 268(1), 25-37.
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