## 3 Results  
### Time-Dependent Study  
In order to gain understanding of the physical phenomena developed in the microfluidic channel, (namely the flow field, movement and forces exerted on the crystal), numerical simulations were performed on COMSOL Multiphysics software in two-dimensional space. Two reference coordinate systems are of special interest in the description of the governing equations. Those are the material frame (reference, initial) and the spatial frame (present,  
Figure 4: Parabolic velocity profile. Figure 5: Forces acting on a zeolite crystal in a parabolic velocity profile. actual). They are used to describe initial geometry, and deformed geometry respectively. Computational fluid dynamic simulation coupled with solid mechanics [15] allows study of the unconstrained movement of the zeolite crystal inside the microfluidic channel. The flow in the channel, if assumed incompressible (constant \(\rho\)), for low Reynolds number (\(\mathrm{Re}\ll 1\), the inertial term is neglected) is described in the spatial frame by the simplified Navier-Stokes equations,  
\[\rho_{f}\frac{\partial u_{f}}{\partial t}=\
abla\cdot\left[-p_{f}I+\mu_{f} \Big{(}\
abla u_{f}+(\
abla u_{f})^{T}\Big{)}\right]+F, \tag{2}\]  
\[\mathrm{and}\quad\rho_{f}\
abla\cdot u_{f}=0, \tag{3}\]  
where \(\rho_{f}\) is the density of the fluid [\(\mathrm{kg/m^{3}}\)], \(u_{f}\) is the fluid velocity field [\(\mathrm{m/s}\), \(\mathrm{m/s}\)] in the spatial (deformed) frame, \(p_{f}\) is the pressure [\(\mathrm{Pa}\)], \(\
abla\cdot()\) is the divergence operator, \(\
abla()\) is the gradient operator, \(\mu_{f}\) is the fluid dynamic viscosity [\(\mathrm{Pa\cdot s}\)], and \(F\) is the volumetric force [\(\mathrm{N/m^{3}}\)]. The governing equation for solids  
\[\rho_{s}\frac{\partial^{2}u_{s}}{\partial t^{2}}=\
abla\cdot\sigma_{s}+F_{s}, \tag{4}\]  
where \(\rho_{s}\) denotes the solid density [\(\mathrm{kg/m^{3}}\)], \(u_{s}\) is the solid displacement field [\(\mathrm{m}\), \(\mathrm{m}\)], \(\sigma_{s}\) is the Cauchy stress tensor defined in the spatial frame, and \(F_{s}\) is the volumetric force [\(\mathrm{N/m^{3}}\)]. The force generated by the fluid on the solid and the solid displacement affecting the fluid velocity is  
\[F_{s}=-n\cdot\Big{[}-p_{f}I+\mu_{f}\Big{(}\
abla u_{f}+(\
abla u_{f})^{T} \Big{)}\Big{]}, \tag{5}\]  
\[\mathrm{where}\quad u_{f}=\frac{\partial u_{s}}{\partial t}, \tag{6}\]  
where \(n\) is the outward normal to the boundary and \(F_{s}\) is the volumetric force on the solid boundary in the spatial (deformed) frame. The time-dependent simulation of Equations (2)-(6) allowed study of the free motion of the crystal in the channel. The three-dimensional zeolite crystal (\(15~{}\upmu\mathrm{m}\times 15~{}\upmu\mathrm{m}\times 75~{}\upmu\mathrm{m}\)) orthogonal to the flow direction, was represented in two-dimensional space as a central transversal slice, by a square which is free to translate and rotate in the plane, with side length of \(15~{}\upmu\mathrm{m}\) and a separation \(d=6~{}\upmu\mathrm{m}\) between a side parallel to the channel floor. This separation \(d\) allowed the square to translate and rotate without touching the floor, with a minimum distance of about \(1.8~{}\upmu\mathrm{m}\) between a downward- pointing corner and the floor. The distance between the centroid of the square and the floor (\(y\) axis direction) of \(13.5~{}\upmu\mathrm{m}\) remained nearly constant as the square translated (along the \(x\) axis) and rotated, with variations \(|\delta|<1~{}\upmu\mathrm{m}\). Narrowing the aforementioned separation leaded to errors in simulations or prohibitively large computation time. Figure 6 depicts a crystal slice near the bottom of the microchannel translating and rotating from \(0\) to \(90^{\circ}\) from time \(t_{0}\) to \(t_{1}\), with velocity \(v\).