**3.2 Governing equation**

Laminar flow interface is used to model and simulate fluid mechanics for laminar and incompressible fluids by using Navier-Stokes equation. Since the fluid flow is laminar flow in the microreactor, this interface is suitable to be implemented in

*Computational Fluid Dynamics of Mixing Performance in Microchannel DOI: http://dx.doi.org/10.5772/intechopen.89928*

this simulation work. The Navier-Stokes equation for incompressible flow is given as [20]:

$$\begin{aligned} \rho \frac{\partial v}{\partial t} + \rho (v \bullet \nabla) v &= \nabla \bullet \left[ -pI + \mu \left( \nabla v + (\nabla v)^{T} \right) \right] + F\\ \rho \nabla \bullet v &= 0 \end{aligned} \tag{7}$$

where *v* is velocity vector (SI unit: m/s); *p* is the pressure (SI unit: Pa); *ρ* is the density (SI unit: kg/m<sup>3</sup> ); *F* is the volume force vector (SI unit: N/m<sup>3</sup> ); *μ* is the dynamic viscosity (SI unit: Pa.s); and *T* is the absolute temperature (SI: K).

The density and the viscosity data are those of water (*<sup>ρ</sup>* = 1 � <sup>10</sup><sup>3</sup> kg/m<sup>3</sup> and <sup>μ</sup> = 1 � <sup>10</sup>�<sup>3</sup> Pa s).

The driving force for the fluid to flow through the mixing slot to the outlet is the applied inlet velocity boundary conditions on the inputs while the pressure boundary condition is assumed to be equal to zero. Meanwhile, the chamber wall is assumed to have a nonslip boundary condition. Mixing is obtained by diffusion of various species in the fluid. The species are diluted in the water, thus having material properties like water. The transfer equation is then taken as the convection-diffusion equation with a reaction term as shown below [20]:

$$\frac{\partial \mathcal{L}}{\partial t} + \nu \bullet \nabla \mathcal{c} = \nabla \bullet (D \nabla \mathcal{c}) + R \tag{8}$$

**Figure 3.** *The meshing for corrugated microchannel.*

**Figure 4.** *The meshing for straight microchannel.*

the physics setting of the model, while the user-controlled meshing builds the mesh based on the user input of size, element type, etc. [19, 20]. **Figures 3** and **4** are the meshed geometry domains with different types of the mesh element can be seen. The mesh element of corrugated microchannel has a high number at the shape of the corrugated section, while the straight microchannel has a high number at the

*(a) The photograph of SSIMM and (b) the mixing element; (c) the flow principle of SSIMM [18].*

Laminar flow interface is used to model and simulate fluid mechanics for laminar and incompressible fluids by using Navier-Stokes equation. Since the fluid flow is laminar flow in the microreactor, this interface is suitable to be implemented in

entrance of the discharge slit.

*Schematic diagram of the simulation system.*

*Computational Fluid Dynamics Simulations*

**3.2 Governing equation**

**Figure 2.**

**108**

**Figure 1.**

where *c* is the concentration of the species (SI unit: mol/m<sup>3</sup> ); *D* is the diffusion coefficient (SI unit: m<sup>2</sup> /s); *R* is a reaction rate expression for the species (SI unit: mol/(m<sup>3</sup> s)); and *v* is velocity vector (SI unit: m/s).

In this model, *R* = 0, because there is no reaction occurred. The species is introduced at different concentration from the range of 0–1 mol/m<sup>3</sup> where one species is at a concentration of 1 mol/m<sup>3</sup> on one of the input boundaries, while the other is at zero concentration. At the output boundary, the substance flows through the boundary by convection [21].
