Boundary conditions

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Boundary conditions

• What separates a submarine from a turtle flowing
  through the same fluid medium?

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Boundary conditions

• Answer?
• The physics on the boundary of the fluid medium

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Boundary conditions

• To examine the properties on the boundary,
  consider an infinitesimal cylinder or pillbox
  of height dl

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Boundary conditionsVelocity field

• Application of conservation of mass (Boussinesq)
  at the boundary of this infinitesimal cylinder
  shows us that the normal velocity is continuous
  at the interface
• Discontinuity would imply a shock at the interface

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Boundary conditionsVelocity field

• Due to the transport of any differences of the
  velocity field along a boundary leading to a
  continuous equilibrium state, we can assume the
  tangential velocity is also continuous.

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Boundary conditionsVelocity field

• What if one of the second medium is a stationary
  solid?
• Then
• This is called the no-slip condition.
• The no-slip condition then tells us that fluid
  medium must be stationary along the solid surface.

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Example 1
Recall in chapter 5 we learned that laminar
viscosity is extremely small under most normal
circumstances in the atmosphere and ocean when
compared to pressure forces but that it was very
important along the boundaries. In particular,
water would not be wet if not for
viscosity Let us use our understanding of
boundary conditions and viscosity to explain why
water is wet To understand this problem we
need to review the Reynolds number.
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Example 1
Recall the Reynolds number compares the magnitude
of inertial contributions to frictional If
Re gtgt 1 Then the viscous effects are not
important. If Re ltlt 1 Then the viscous
effects are important the fluid is sticky.
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Example 1
To restate our question then. Given the Reynolds
number, and our understanding of boundary
conditions so far, when (or where) is water
wet?
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Example 1- Answer
The key is that we need to look at small lengths
scales from the surface of contact. Then we can
consider the flow to be a low Reynolds number or
locally viscous and therefore sticky. This is
true of any fluid with non-zero viscosity
provided you that look at it at a small enough
length scale. (and that you do not violate the
continuum hypothesis) This local region of
viscous flow is called the Viscous Boundary
layer Provided that water is locally sticky,
then water is also wet due to the no-slip
boundary condition in that there is no relative
motion at the surface of contact. The profile of
water from the surface generally looks like
(assuming in this case relative motion between
the solid and water of course)
surface
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Boundary conditionsForces along the boundary

• Forces along a fluid boundary is the same as any
  force applied to a surface. Therefore we must
  examine the continuity of the stress tensor.
• In the tangential direction, the stress tensor is
  continuous along the boundary

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Boundary conditionsForces along the boundary

• Due to the different attractive potential forces
  of the various molecules in the two separate
  mediums, an additional surface force is required
  along the boundary in the normal direction called
  surface tension.
• It is shown, without proof, that the surface
  tension force is proportional to the curvature of
  the boundary
• Where R1 and R2 are the principal radii of
  curvature and
• g is the surface tension coefficient.

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Boundary conditionsForces along the boundary

• The boundary condition of the stress tensor in
  the normal direction is then

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Boundary conditionsOther properties

• Most properties obey the standard requirement
  that, in equilibrium, the properties are
  continuous along the boundary.
• For example, for temperature, we can show that
• Where k1 and k2 are thermal diffusivity
  coefficients in the two different mediums.
• The one exception, as we have seen, is the normal
  component of the stress tensor across the
  boundary.

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How do we apply these continuity conditions?

• As we know from partial differential equations,
  you need a set of boundary conditions to uniquely
  determine a solution to a physical system.
• We utilize our understanding of the 4 previously
  mentioned continuity conditions at the boundary
  to deduce (or measure) the physical requirements
  at the boundary and thus establish a unique
  solution to the system which we can then evolve
  through time (forecast).
• The next example will be helpful in understanding
  this concept

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Example 2
Describe what boundary conditions are important
for a 3-D Ocean Basin Specifically consisting
of the ocean surface to the ocean bottom and
along the continental boundaries.
From- http//www.meer.org/atldet1.jpg
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Example 2 - Answer
Describe what boundary conditions are important
for a 3-D Ocean Basin Specifically consisting
of the ocean surface to the ocean bottom and
along the continental boundaries. Answer The
answer really depends on what one wants to see
but the usual suspects are No slip condition
and no normal flow along the bottom and sides.
(Exceptions would be a seismic event such as an
earthquake- What flow would no longer be 0 along
the boundary in this case?) On the ocean surface
the dominant boundary condition is normally
the wind stress However, surface tension
effects (The normal stress tensor boundary
condition) are important if one considers effects
such as capillary waves
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Example 2
Random image of ocean capillary waves
From- http//earthsci.org/processes/weather/waves/
Waves_files/capwaves.jpg