Transcript A reaction-advection-diffusion equation from chaotic chemical mixing Junping Shi

A reaction-advection-diffusion
equation from
chaotic chemical mixing
Junping Shi 史峻平
Department of Mathematics
College of William and Mary
Williamsburg, VA 23187,USA
Math 490 Presentation, April 11, 2006,Tuesday
Reference Paper 1:
Neufeld, et al, Chaos, Vol 12, 426-438, 2002
Reference paper 2: Menon, et al, Phys.
Rev. E. Vol 71, 066201, 2005
Reference Paper 3:
Shi and Zeng, preprint, 2006
Model
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The spatiotemporal dynamics of interacting biological or chemical substances is governed
by the system of reaction-advection-diffusion equations:
where i=1,2,….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.
Some numerical simulations
by Neufeld et. al. Chaos paper
Arguments to derive a new
equation
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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).
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 A+B -> 2A.
We also assume the zero boundary conditions for C at infinity.
Numerical Experiments
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Software: Maple
Algorithm: build-in PDE solver
Spatial domain: -20<x<20 (computer can’t 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
Numerical Result 1:
D=0.5, u(0,x)=exp(-(x-2)^2)+exp(-(x+2)^2)
Numerical Result 2:
D=1.5, u(0,x)=exp(-(x-2)^2)+exp(-(x+2)^2)
Numerical Result 3:
D=10, u(0,x)=exp(-(x-2)^2)+exp(-(x+2)^2)
Observation of numerical results: different D
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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 Let’s compare different initial conditions
Numerical Result 4:
D=40, u(0,x)=exp(-(x-2)^2)+exp(-(x+2)^2)
Numerical Result 5:
D=40,
u(0,x)= 2*exp(-2*(x-2)^2)+exp(-(x+2)^2)+3*exp(-(x-5)^2)
Numerical Result 6:
D=40, u(0,x)= cos(pi*x/40)
Observation of numerical results:
different initial conditions
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Solution tends to the same equilibrium solution for
different initial conditions, which implies the
equilibrium solution is asymptotically stable.
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More experiments can be done to obtain more
information on the equilibrium solutions for different D
Numerical results in Menon, et al, Phys.
Rev. E. Vol 71, 066201, 2005
Mathematical Approach
More solutions of u”+xu’+Du=0
Stability of the zero equilibrium
Bifurcation analysis
Numerical bifurcation diagram
in Menon’s paper
Rigorous Mathematical Result
(Shi and Zeng)
Traveling wave solutions?
Comparison with Fisher’s equation
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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.
Numerical solution of Fisher equation
D=40, u(0,x)=exp(-(x-2)^2)+exp(-(x+2)^2)
Stages of a research problem
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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