text: One may derive the Taylor equation using method of averages, first introduced by Aris. The result can also be derived from large-time asymptotics, which is more intuitively clear. In the dimensional coordinate system , consider the fully-developed Poiseuille flow  flowing inside a pipe of radius , where  is the average velocity of the fluid. A species of concentration  with some arbitrary distribution is to be released at somewhere inside the pipe at time . As long as this initial distribution is compact, for instance the species/solute is not released everywhere with finite concentration level, the species will be convected along the pipe with the mean velocity . In a frame moving with the mean velocity and scaled with following non-dimensional scales
  where  is the time required for the species to diffuse in the radial direction,  is the diffusion coefficient of the species and  is the Peclet number, the governing equations are given by
  Thus in this moving frame, at times  (in dimensional variables, ), the species will diffuse radially. It is clear then that when  (in dimensional variables, ), diffusion in the radial direction will make the concentration uniform across the pipe, although however the species is still diffusing in the  direction. Taylor dispersion quantifies this axial diffusion process for large .
  Suppose  (i.e., times large in comparison with the radial diffusion time ), where  is a small number. Then at these times, the concentration would spread to an axial extent . To quantify large-time behavior, the following rescalings
  can be introduced. The equation then becomes
  If pipe walls do not absorb or react with the species, then the boundary condition  must be satisfied at . Due to symmetry,  at .
  Since , the solution can be expanded in an asymptotic series,  Substituting this series into the governing equation and collecting terms of different orders will lead to series of equations. At leading order, the equation obtained is
  Integrating this equation with boundary conditions defined before, one finds . At this order,  is still an unknown function. This fact that  is independent of  is an expected result since as already said, at times , the radial diffusion will dominate first and make the concentration uniform across the pipe. 
  Terms of order  leads to the equation
  Integrating this equation with respect to  using the boundary conditions leads to
  where  is the value of  at , an unknown function at this order.
  Terms of order  leads to the equation
  This equation can also be integrated with respect to , but what is required is the solvability condition of the above equation. The solvability condition is obtained by multiplying the above equation by  and integrating the whole equation from  to . This is also the same as averaging the above equation over the radial direction. Using the boundary conditions and results obtained in the previous two orders, the solvability condition leads to 
  This is the required diffusion equation. Going back to the laboratory frame and dimensional variables, the equation becomes
  By the way in which this equation is derived, it can be seen that this is valid for  in which  changes significantly over a length scale  (or more precisely on a scale . At the same time scale , at any small length scale about some location that moves with the mean flow, say , i.e., on the length scale , the concentration is no longer independent of , but is given by