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A particle laden flow is a multiphase flow where one phase is the fluid and the other is dispersed particles. Governing equations for both phases are implemented in Fluent. To run a meaningful simulation, a review of the theory is necessary.
Fluid Phase:
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In
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the
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simulations
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considered
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for
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this
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tutorial,
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the
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fluid
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flow
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is
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a
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2D
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perturbed
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periodic
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double
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shear
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layer
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as
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described
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in
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the
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first
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section.
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The
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geometry
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is
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Lx
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=
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59.15m,
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Ly
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=
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59.15m,
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and
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the
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mesh
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size
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is
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chosen
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as
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{latex}{\large$$\Delta x = L_x / n_x$$}{latex} |
...
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order
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to
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resolve
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the
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smallest
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vorticies.
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As
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a
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rule
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of
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thumb. One
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typically
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needs
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about
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20
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grid
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points
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across
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the
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shear
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layers,
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where
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the
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vorticies
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are
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going
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to develop.
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The
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boundary
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conditions
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are
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periodic
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in
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the
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x
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and
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y
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directions.
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The
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fluid
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phase
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satisfies the
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Navier-Stokes
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Equations:
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-Momentum Equations
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Equations {latex} {\large \begin{eqnarray*} \rho_f (\frac{d \mathbf{u}_f}{dt}+\mathbf{u}_f \cdot \nabla \mathbf{u}_f)=- \nabla p + \mu \nabla ^2 \mathbf{u}_f + \mathbf{f} \end{eqnarray*} } {latex} |
-Continuity Equation
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Equation {latex} {\large \begin{align*} \frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \mathbf{u}_f)=0 \end{align*} } {latex} where |
where
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{latex}{\large$$\mathbf{u}$$}{latex} |
...
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the
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fluid
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velocity,
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{latex}{\large$$p$$}{latex} |
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pressure,
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{latex}{\large$$\rho_f$$}{latex} |
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fluid
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density
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and
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{latex}{\large$$\mathbf{f}$$}{latex} |
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a
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momentum
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exchange
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term
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due
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to
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the
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presence
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of
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particles.
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When
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the
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particle
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volume
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fraction
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{latex}{\large$$\phi$$}{latex} |
...
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the
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particle
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mass
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loading
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{latex}{\large$$M=\phi \rho_p/\rho_f$$}{latex} |
...
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very
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small,
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it
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is
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legitimate
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to
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neglect
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the
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effects
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of
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the
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particles
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on
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the
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fluid:
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{latex}{\large$$\mathbf{f}$$}{latex} |
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be
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set
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to
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zero.
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This
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type
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of
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coupling
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is
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called
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one-way.
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In
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these
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simulations
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the
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fluid
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phase
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is
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air,
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while
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the
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dispersed
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phase
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is
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constituted
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of
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about
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400
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glass
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beads
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of
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diameter
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a
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few
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dozens
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of
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micron.
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This
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satisfies
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both
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conditions
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{latex}{\large$$\phi \ll 1$$}{latex} |
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{latex}{\large$$M \ll 1$$}{latex} |
One way-coupling is legitimate here. See ANSYS documentation (16.2) for further details about the momentum exchange term.
Particle Phase:
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The
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suspended
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particles
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are
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considered
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as
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rigid
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spheres
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of
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same
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diameter
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d,
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and
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density
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{latex}{\large$$\rho_p$$}{latex} |
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Newton’s second
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law
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written
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for
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the
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particle
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i
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stipulates:
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{latex}{\large $$m_p \frac{d \mathbf{u}_p^i}{dt}=\mathbf{f}_{ex}^i$$}{latex} where |
where
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{latex}{\large$$\mathbf{u}_p^i$$}{latex} |
...
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the
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velocity
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of
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particle
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i,
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{latex}{\large $$\mathbf{f}_{ex}^i$$}{latex} |
...
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forces
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exerted
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on
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it,
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and
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{latex}{\large $$m_p$$}{latex} |
...
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mass.
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In
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order
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to
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know
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accurately
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the
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hydrodynamic
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forces
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exerted
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on
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a
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particle
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one
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needs
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to
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resolve
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the
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flow
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to
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a
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scale
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significantly
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smaller
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than
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the
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particle
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diameter.
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This
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is
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computationally
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prohibitive.
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Instead,
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the
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hydrodynamic
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forces
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can
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be
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approximated
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roughly
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to
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be
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proportional
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to
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the
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drift
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velocity
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ref3:
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]: {latex}{\large $$\frac{d \mathbf{u}_p^i}{dt}=\frac{\mathbf{u}_f-\mathbf{u}_p^i}{\tau_p}$$}{latex} where |
where
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{latex}{\large $$\tau_p=\rho_p D^2/(18\mu)$$}{latex} |
...
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known
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as
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the
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particle
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response
...
time,
...
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{latex}{\large $$\rho_p$$}{latex} |
...
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particle
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density
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and
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D
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the
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particle
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diameter.
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This
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equation
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needs
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to
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be
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solved
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for
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all
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particles
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present
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in
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the
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domain.
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This
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is
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done
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in
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Fluent
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via
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the
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module:
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Discrete
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Phase
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Model(DPM).
Choosing the Cases:
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The
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particle
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response
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time
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measures
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the
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speed
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at
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which
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the
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particle
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velocity
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adapts
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to
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the
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local
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flow
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speed.
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Non-inertial
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particles,
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or
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tracers,
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have
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a
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zero
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particle
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response
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time:
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they
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follow
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the
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fluid
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streamlines.
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Inertial
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particles
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with
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{latex}{\large$$\tau_p \neq 0$$}{latex} |
...
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adapt
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quickly
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or
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slowly
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to
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the
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fluid
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speed
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variations
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depending
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on
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the
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relative
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variation
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of
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the
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flow
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and
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the
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particle
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response
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time.
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This
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rate
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of
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adaptation
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is
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measured
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by
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a
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non-dimensional
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number
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called
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Stokes
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number
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representing
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the
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ratio
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of
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the
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particle
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response
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time
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to
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the
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flow
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characteristic
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time
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scale.
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{latex}{\large$$St = \frac{\tau_p}{\tau_f}$$}{latex} |
In
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these
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simulations,
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the
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characteristic
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flow
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time
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is
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the
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inverse
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of
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the
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growth
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rate
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of
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the
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vortices
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in
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the
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shear
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layers.
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This
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is
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also
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predicted
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by
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the
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Orr-Sommerfeld
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equation.
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For
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the
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particular
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geometry
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and
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configuration
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we
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used
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in
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this
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tutorial,
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the
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growth
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rate
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is
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{latex}{\large$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}$$}{latex} |
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When
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St
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=
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0
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the
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particles
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are
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tracers.
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They
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follow
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the
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streamlines
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and,
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in
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particular,
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they
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will
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not
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be
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able
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to
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leave
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a
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vortex
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once
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caught
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inside.
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When
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{latex}{\large$$St \gg 1$$}{latex} |
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particles
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have
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a
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ballistic
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motion
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and
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are
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not
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affected
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by
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the
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local
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flow
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conditions.
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They
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are
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able
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to
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shoot
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through
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the
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vorticies
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without
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a
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strong
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trajectory
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deviation.
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Intermediate
...
cases
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|---|
{latex}{\large$$St \approx 1$$}{latex} |
...
...
a
...
maximum
...
coupling
...
between
...
the
...
two
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phases:
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particles
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are
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attracted
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to
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the
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vorticies,
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but
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once
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they
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reach
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the
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highly
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swirling
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vortex
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cores
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they
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are
...
ejected
...
due
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to
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their
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non
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zero
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inertia.
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In
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this
...
tutorial,
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we
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will
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consider
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a
...
nearly
...
tracer
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case
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St
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=
...
0.2,
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an
...
intermediate
...
case
...
St
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=
...
1
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and
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a
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nearly
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ballistic
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case
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St
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=
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5.
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