**2.2 Velocity field modeling**

The fluid, supposed to be Newtonian and incompressible, flows in a laminar and isothermal regime. The Navier-Stokes equations are thus used to determine the pressure and velocity fields of the fluid in the microchannel (Eqs. (1) and (2)):

**Figure 1.** *Schematic of the used microfluidic biosensor setup with integrated flow confinement.* *Enhancement of SARS-CoV-2 Detection Time for Integrated Flow Confinement Microfluidic… DOI: http://dx.doi.org/10.5772/intechopen.104802*

$$\nabla.\mathbf{u} = \mathbf{0} \tag{1}$$

$$
\rho(\mathfrak{u}.\nabla)\mathfrak{u} = -\nabla p + \mu \nabla^2 \mathfrak{u} \tag{2}
$$

where u is the flow velocity field, *ρ* and *μ* are, respectively, the fluid's density and dynamic viscosity, and *p* is the pressure.

#### **2.3 Analyte concentration modeling**

The transport of aimed analytes by diffusion and convection is modeled by the following Fick's second law (Eq. (3)):

$$\frac{\partial[A]}{\partial t} + \mathfrak{u}.\nabla[A] = D\Delta[A] \tag{3}$$

where [A] and D designate the concentration and the diffusion coefficient of the target analyte, respectively.

#### **2.4 Analyte-ligand concentration modeling**

The first-order Langmuir–Hinshelwood adsorption model [18, 19] was employed to calculate the concentration of analyte-ligand complexes formed on the reaction surface (Eq. (4)):

$$\frac{\partial [AB]}{\partial t} = k\_{on} \left[ A\_{surf} \right] \left\{ B\_{max} - \left[ AB \right] \right\} - k\_{off} \left[ AB \right] \tag{4}$$

where ½ � *AB* is the bound analyte-ligand concentration and *Bmax* is the immobilized ligand concentration on the reaction surface. *Asurf* is the concentration of analytes at the reaction surface, *kon* is the adsorption rate constant and *koff* is the desorption rate constant.

#### **2.5 Boundary and initial settings**

**Table 1** and **Figure 2** recapitulate all the boundary settings used in this study. For the laminar flow, the inlet fluid flows inside the main microchannel with a parabolic profile where the average velocity, *uave,* was set at 50 μm/s, that which enters through the confining channel, its velocity was set to *uconf* and at the outlet, the flow was


**Table 1.**

*Boundary settings of velocity field and concentration of analyte for the reaction surface, walls, inlets, and outlet.*

**Figure 2.**

*Boundary conditions. (a): boundary conditions for the velocity field. (b): boundary conditions for analyte concentration.*

assumed fully developed. For mass transport, a constant concentration of analytes *c*0, was imposed at the inlet of the microchannel and the outlet the condition *n* !*:*ð Þ *<sup>D</sup>*∇½ � *<sup>A</sup>* was applied. Considering the non-interaction of the analyte with the rest of the walls, the homogeneous Neumann condition was used [17]. On the sensitive surface, the diffusive flux condition generated by the binding adsorption reaction between analytes and ligands was employed.

Initially the velocity of the fluid within the microchannel, the bulk analyte concentration, ½ � *<sup>A</sup>* ð Þ *<sup>t</sup>*¼<sup>0</sup> , and the complex concentration formed on the reaction surface, ½ � *AB* ð Þ *<sup>t</sup>*¼<sup>0</sup> were assumed to be zero. *<sup>A</sup>*ll the parameters used in the modeling [20] are presented in **Table 2**.


**Table 2.** *Simulation parameters [20].* *Enhancement of SARS-CoV-2 Detection Time for Integrated Flow Confinement Microfluidic… DOI: http://dx.doi.org/10.5772/intechopen.104802*

## **2.6 Numerical method**

The proposed model equations were solved using Galerkin finite element analysis [21]. We used 1923 triangular geometric elements for the complete 2D domain, including the refined elements of the reaction surface. To prove that the convergence is reached and that the calculated results are independent of the mesh, the profile of the velocity field at x = 140 μm of the microchannel was plotted, in **Figure 3**, for several meshes (1328, 1923, 1958, and 2612 elements). The variations obtained by using different numbers of elements are significantly the same. All the stages of the model resolution are recapitulated as presented in **Figure 4**. The pressure and the velocity fields were calculated by solving stationary Eqs. (1) and (2) at once. The concentration of analytes and that of the analyte-ligand complexes, appearing on the binding area, were simulated from the coupled time-dependent Eqs. (3) and (4).

**Figure 3.** *Velocity field at x = 140 μm of the microchannel for different mesh grids.*

**Figure 4.** *Model simulation flowchart.*
