2.1. Mathematical Model
Only one of the many hybridization wells, connected with a microchannel, is considered in the three-dimensional model (Figure 1). The fluid containing target DNA flows from left to right where the incoming flow profile is characterized by fully developed laminar flow, i.e., it is parabolic with zero velocity at the channel walls. Fluid flow in the channel follows the Navier–Stokes equation, (1) ρ∂u∂t−η∇2u+ρ(u⋅∇)u+∇p=F (2) ∇⋅u=0 where u denotes the velocity field vector, ρ is density, η is the dynamic viscosity, and p is pressure. At steady-state, the first time-dependent term disappears. The external force, F, such as gravity can be neglected in such miniaturized systems. 
The incoming flow has a small concentration of the target DNA which is described by a convection–diffusion equation of the form, (3) ∂c∂t+u⋅∇c=D∇2c where c denotes the concentration of solution-phase targets, u is the identical velocity field vector given in Equation (1). The diffusion coefficient, D, was estimated using Table 1 of Chan et al. [23], where it was reported that the diffusion coefficient of DNA depends on its length and decreases with increasing base pairs [23].
Once the target DNA reaches the bottom of the hybridization well, the association and dissociation kinetics are described by using an ordinary differential equation, (4) dBdt=konc(Rt−B)−koffB where kon represents the association rate constant, koff is the dissociation rate constant, Rt is the total surface concentration of probes, and B is the surface concentration of bound targets at time, t. The association rate constant, kon, depends on the DNA sequence involved, the temperature, and the ionic strength of the medium but not to the point that it would drastically affect the order of magnitude. Based on the experimental results reported in literature [11,24,25], it was assumed that the association rate constant does not change appreciably over the temperature range used in our simulation (25–55 °C). Therefore, a value of 106 M−1s−1 for kon was used in these simulations. The dissociation rate constant, koff, was calculated using the thermodynamic model presented in literature [12]. In this model, standard Gibbs free energy and hybridization energy for different DNA sequences was used to calculate the dissociation rate constant at a specific temperature. The model also assumes that the DNA targets do not diffuse on the surface and that there is no leakage of the molecules at the edges of the surface.