A reaction-advection-diffusion equation from chaotic chemical mixing

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A reaction-advection-diffusion equation from
chaotic chemical mixing

• Junping Shi ???
• Department of Mathematics
• College of William and Mary
• Williamsburg, VA 23187,USA

Math 490 Presentation, April 11, 2006,Tuesday
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Reference Paper 1Neufeld, et al, Chaos, Vol 12,
426-438, 2002
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Reference paper 2 Menon, et al, Phys. Rev. E.
Vol 71, 066201, 2005
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Reference Paper 3Shi and Zeng, preprint, 2006
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Model

• The spatiotemporal dynamics of interacting
  biological or chemical substances is governed by
  the system of reaction-advection-diffusion
  equations

where i1,2,.n, C_i(x,t) is the concentration of
the i-th chemical or biological component, f_i
represents the interaction between them. All
these species live a an advective flow v(x,t),
which is independent of concentration of
chemicals. Da, the Damkohler number,
characterizes the ratio between the advective and
the chemical or biological time scales. Large Da
corresponds to slow stirring or equivalently fast
chemical reactions and vice versa. The Peclet
number, Pe, is a measure of the relative strength
of advective and diffusive transport.
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Some numerical simulationsby Neufeld et. al.
Chaos paper
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Arguments to derive a new equation

• At any point in 2-D domain, there is a stable
  direction where the spatial pattern is squeezed,
  and there is an unstable direction where the
  pattern converges to
• The stirring process smoothes out the
  concentration of the advected tracer along the
  stretching direction, whilst enhancing the
  concentration gradients in the convergent
  direction.
• In the convergent direction we have the following
  one dimensional equation for the average profile
  of the filament representing the evolution of a
  transverse slice of the filament in a Lagrangian
  reference frame (following the motion of a fluid
  element).

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One-dimensional model
Here C(x,t) is the concentration of chemical on
the line, and we assume F(C)C(1-C), which
corresponds to an autocatalytic chemical reaction
AB -gt 2A. We also assume the zero boundary
conditions for C at infinity.
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Numerical Experiments

• Software Maple
• Algorithm build-in PDE solver
• Spatial domain -20ltxlt20 (computer cant do
  infinity)
• Grid size 1/40
• Boundary condition u(-20,t)u(20,t)0
• Strategy test the simulations under different D
  and different initial conditions

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Numerical Result 1 D0.5, u(0,x)exp(-(x-2)2)
exp(-(x2)2)
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Numerical Result 2 D1.5, u(0,x)exp(-(x-2)2)
exp(-(x2)2)
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Numerical Result 3 D10, u(0,x)exp(-(x-2)2)e
xp(-(x2)2)
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Observation of numerical results different D

• For small D, the chemical concentration tends to
  zero
• For larger D, the chemical concentration tends to
  a positive equilibrium
• The positive equilibrium nearly equal to 1 in the
  central part of real line, and nearly equals to 0
  for large x. The width of its positive part
  increases as D increases.
• Now Lets compare different initial conditions

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Numerical Result 4 D40, u(0,x)exp(-(x-2)2)e
xp(-(x2)2)
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Numerical Result 5 D40, u(0,x)
2exp(-2(x-2)2)exp(-(x2)2)3exp(-(x-5)2)
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Numerical Result 6 D40, u(0,x) cos(pix/40)
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Observation of numerical results different
initial conditions

• Solution tends to the same equilibrium solution
  for different initial conditions, which implies
  the equilibrium solution is asymptotically
  stable.
• More experiments can be done to obtain more
  information on the equilibrium solutions for
  different D

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Numerical results in Menon, et al, Phys. Rev. E.
Vol 71, 066201, 2005
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Mathematical Approach
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More solutions of uxuDu0
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Stability of the zero equilibrium
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Bifurcation analysis
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Numerical bifurcation diagramin Menons paper
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Rigorous Mathematical Result(Shi and Zeng)
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Traveling wave solutions?
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Comparison with Fishers equation

• In Fisher equation (without the convection term),
  no matter what D is, the solution will spread
  with a profile of a traveling wave in both
  directions, and the limit of the solution is
  u(x)1 for all x
• In this model (with the convection term),
  initially the solution spread with a profile of a
  traveling wave in both directions, but the
  prorogation is stalled after some time, and an
  equilibrium solution with a phase transition
  interface is the asymptotic limit.

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Numerical solution of Fisher equationD40,
u(0,x)exp(-(x-2)2)exp(-(x2)2)
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Stages of a research problem

• Derive the model from a physical phenomenon or a
  more complicated model
• Numerical experiments
• Observe mathematical results from the experiments
• State and prove mathematical theorems