Perturbation solution to the convection–diffusion equation with moving fronts

Abstract

AbstractElectrophoresis of a solute through a column in which its transport is governed by the convection – diffusion equation is described. Approximate solutions to the convection – diffusion equation in the limit of small diffusion are developed using perturbation methods. The diffusion coefficient and velocity are assumed to be functions of space and time such that both undergo a sudden change from one constant value to another within a thin transition zone that itself translates with a constant velocity. Two cases are considered: (1) the thickness ϵf of the transition zone is negligible compared to the diffusional length scale, so the zone may be treated as a singular boundary across which the diffusion constant and velocity suffer discontinuous changes; (2) the transition zone is considerably wider than the diffusional length scale, so the diffusion coefficient and velocity, although sharply varying, are smooth functions of position and time. A systematic perturbation expansion of the concentration distribution is presented for case 1 in terms of the small parameter ϵ = 1/Pe. A lowest order approximation is given for case 2. A suitably configured system analyzed here can lead to progressive accumulation, or focusing, of the transported solute. The degree of focusing in case 1 scales with ϵ−1, whereas in case 2 it scales with (ϵfϵ)−1/2, and thus increases much more weakly with increasing Pe. A separation based on this concept requires development of materials and devices that allow dynamic tuning of the mass‐transport properties of a medium. This would make it possible to achieve progressive focusing and separation of solutes, such as proteins and DNA fragments, in electrophoretic media with an unprecedented degree of control.