An important application of micro fluidics is in the area of electro kinetic separations, in which an electric field is applied along the length of a micro channel to generate differential velocities for various components (molecules, nano particles, cells, etc.) within the channel. As we will see when discussing micro fluidic separations, the electro kinetic particle velocity is given by u = µE where u is the velocity (units of m/s), µ is the electro kinetic mobility (a constant associated with the particle, with units of V/s) and E is the electric field (units of V/m).

a. Consider a micro channel with a constant voltage difference applied between the inlet and outlet ports. How does the time required for a particle to traverse the length of the channel (t) vary with the generalized size scale of the channel (L) when all dimensions are uniformly reduced by the same factor?

b. What is the scaling relationship if the system is operated with constant electric field (instead of constant voltage)?

c. How do the scaling relationships in (a) and (b) change if only the channel length is reduced, while keeping width and height constant?

d. In practice, Joule heating during electro kinetic separations limits the electric field that can be applied to some maximum value. If heat removal from the channel scales linearly with channel surface area, what is the scaling relationship between generalized size scale (L) and separation time when the separation is operated at maximum (not constant) electric field?