Rigorous justification of Taylor Dispersion via Center Manifold theory
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Abstract
Imagine fluid moving through a long pipe or channel, and we inject dye or solute into
this pipe. What happens to the dye concentration after a long time? Initially, the
dye just moves along downstream with the fluid. However, it is also slowly diffusing
down the pipe and towards the edges as well. It turns out that after a long time,
the combined effect of transport via the fluid and this slow diffusion results in what
is effectively a much more rapid diffusion process, lengthwise down the stream. If
0 <nu << 1 is the slow diffusion coeffcient, then the effective longitudinal diffusion
coeffcient is inversely proportional to 1/nu, i.e. much larger. This phenomenon is called
Taylor Dispersion, first studied by GI Taylor in the 1950s, and studied subsequently
by many authors since, such as Aris, Chatwin, Smith, Roberts, and others. However,
none of the approaches used in the past seem to have been mathematically rigorous.
I'll propose a dynamical systems explanation of this phenomenon: specifically, I'll
explain how one can use a Center Manifold reduction to obtain Taylor Dispersion as
the dominant term in the long-time limit, and also explain how this Center Manifold
can be used to provide any finite number of correction terms to Taylor Dispersion as
well.